Logarithmic abelian varieties
Kato, Kazuya
This is a joint work with T. Kajiwara and C. Nakayama. Logarithmic abelian varieties are degenerate abelian varieties which live in the world of log geometry of Fontaine-Illusie. They have group structures which do not exist in the usual algebraic geometry. By using the group structures, we give a new formulation of the work of K-W Lan on the toroidal compactification of the moduli space of abelian varieties with PEL structures.
 Thank you very much for the invitation. I hope that he will become more and more active and younger and younger. I respect him, also like my father, and I am thankful to Ruku very much. Today I am the founder of the Logarithmic Geometry and I talk about the Logarithmic Geometry and the Logarithmic Geometry. I use the document camera to work. I hope I myself cannot see so well, but I hope you can see this. This is a space with a she-hub ring and a log-she-hub ring and a she-hub monolith. This scheme was started by the great teacher, Urotan Dik of Ruku. The theory of the log scheme was started. Ruku was very much excited and probably he was happy that he was making a new world of the algebraic geometry like his teacher did. We discussed Ruku and I shared many dreams on the logarithmic geometry. I think the log-Aberian variety was already in the dream from the early days of the geometry. I am happy that today I can talk about the logarithmic-Aberian varieties after 30 years. Here is a nice textbook on the logarithmic geometry written by Oga. In the first line of this book, it is written that the logarithmic geometry was developed to deal with two fundamental and related problems in algebraic geometry. One is compactification, one is degeneration. How compactification is important and how degeneration is important. There are two famous golden things and famous observations. For the compactification, Golden Thin is by Argello Vistolo, working with non-complete ojira spaces like keeping a change in a pocket with holes. This golden thing is often quoted by Abramovic in his talks. For degeneration, the king observation by Pakesh Saito is that in the story of beauty and beast, Argelli beast became a nice man by the love of the girl. In the designate object, Argelli objects in algebraic geometry became a nice object by the power of the log and the power of love of the girl and the power of love are similar. And then here is the mysterious coincidence of the letters. Who is the first one? In the case of modular space, these two things are connected in this way. It is a nice object in the usual geometry, but it is not compact. It can be ugly in the usual geometry, but it becomes a nice object in the log geometry. The space of Argelli and Varaitis is TEL-tropical. Today I give the information of the joint work with Kajiwara and Nakayama in which the territorial compactification is understood as the space of log-Aberian Varaitis with TEL structure. Usually, the designate-Aberian Varaitis have no group structure and other Argelli, but it is nice here in the log geometry and they have group structure. And here the PEL is the polarization, the endomorphism ring, and the L is the level structure. So the endomorphism ring is fixed or something like that. The ring of this endomorphism preserves the structure of the Argelli and Varaitis. Now we can have a ring because two endomorphisms can be added because log-Aberian Varaitis can have a group structure. So without a group structure, it is very hard to consider the endomorphism of the degenerate-Aberian Varaitis. But we can have such a nice object in the logarithmic world. So it becomes very natural and nice. The formulation becomes very transparent and nice with the log-Aberian Varaitis. And at present, our work is only the reinterpretation of the work of Ran, but I hope that in the future the log-Aberian Varaitis will have good applications because the theory becomes so transparent. And then in the case of the endomorphism, the theory was made by Farting and Chai in 1990. And also I hope to say that Fujiwara had also some nice work on the case of PEL type structure for Argelli Varaitis, for the Tore Dalcon Motivation. But unfortunately this work was unpublished around 1990. And I just introduced the rough idea of log-Aberian Varaitis. So log-Aberian Varaitis over FES-log-log-scheme is a contravariant factor from the category of FES-log-scheme to the category of Arbelian Groups. And so it is naturally the structure of the group structure. And if K is a complete discrete variation field and A K is Arbelian Varaitis over K with semi-stable reduction, then A K extends to log-Aberian Varaitis over OK. Here OK has the standard is endowed with the standard log structure and A K extends to log-Aberian Varaitis over OK uniquely. And this semi-stable reduction, how to treat semi-stable reduction nicely was the starting point, motivation of the creation of the log geometry. So then in the usual Arge-Brick geometry, in this case of the theory of Arbelian Varaitis, in the usual Arge-Brick geometry, A K extends to a robust scheme with semi-stable reduction, but it doesn't have the group structure. So the group structure cannot extend to such schemes, but the Arbelian, log-Aberian Varaitis has a group structure. And for example, if A K is a teiton elliptical curve with Q invariant Q, then A is if we restrict the phonkta A to the FES log schemes over OK module Mk to the N, over this Arltinian quotient of the OK, then this restriction can be written in the very simple way. It is a quotient of GM log Q divided by the power of Q. Here Q is a QE invariant. And this presentation is very similar to the presentation of the teiton elliptical curve, an analytic presentation of the teiton elliptical curve in the analytic geometry over the field K. But that can be treated in an algebraic way, not an analytic way, an algebraic way in this log geometry. And GM log Q is, no, no, no, I made a mistake. So GM log Q is this, and sorry GM log Q is a part of GM log, such as Q to the A device T, and T device Q to the B in the log structure for some AB. And then we divide this part of the Mk group by the power of QZ. And this is very similar to the presentation of the teiton elliptical curve as a multiplicative group by the powers of Q in the analytic geometry over QK. And here the Q is the Q is the element of Mk minus 0. And it is regarded as the section of the log structure M of the spec of K, called the standard log structure. And then for log structure, the log structure is the log structure, the C of monoid of the log structure is regarded as the extension of the unit group. So we have a bigger unit group. And this is a monoid, but this monoid is embedded in the group associated with the monoid. And you have also a map, multiplicative map from MT to OT. And then here for these schemes over OK, Mojiro, Mk to the N, Q is nilpotent. And if n is 0, then Q is 0 here. But you can, in this case, so but Q is put in the log structure and then Q becomes invertible. That is, we are making an nilpotent element invertible here. And in the case n is 1, we are making 0 invertible here. So we can make 0 invertible in the log geometry. And by such such thing, then we can have this group structure of this degenerate of the Arbillian variety. So log geometry is such a theory to make 0 invertible. Yeah, so now I put one comment that a log Arbillian variety is a log algebraic space in the second sense. And the usual Arbillian space is something which is covered by a usual scheme by a termorphism. The morphism here is relatively etal. X is regarded as a functor, but this is relatively it is representable. This morphism is represented by a termorphism. And Q is also representable and you have such a suggestion. That is the algebraic space. Arbillian space is something which is covered by by by representable to put object by etal morphism. And log algebraic space. The first sense is a is a similar to a functor covered by a representable object by log etal morphisms. So it's our morphism is replaced by log with our most log Arbillian varieties is such a object. And log algebraic space in the first sense is is such X, unctah X, covered by a representable object by the classical etal morphism. And this is also equal to the algebraic space with with usual algebraic space with an infuse of structure. So there are two things, first sense and second sense and log Arbillian varieties log algebraic space in the second sense. Yeah. Now, now I, I, and it's so I don't explain more about log with Arbillian variety if I explain too much then I will lose time so. And then now I will go to the problem of Mojira with PL structure for polarization and endomorphism and level structure. So, so then in such theory, we fix a semi semi semi semi simple algebra over Q and with such a operator operator, it's two to four locations identity and it is changes the order of the multiplication and and then. So, when you say log etal, you mean the common log etal or the general the more general log etal. Yeah, yeah, I think so yeah. So, I think we have also another question from Bruno. Sorry. We have a second question. It was a mistake I apologize. So, so then the module I, so I don't explain all things because I, it is, there are too many things to be put to just I, I explain the main part and the satisfying certain conditions, including the this, this size skew symmetric non-degenerate by linear form and then another the group of similitudes for this pair of the outside. This is algebraic group over Q, algebraic group over Q, deductive algebraic group over Q and and then by some I don't explain the some condition but under some condition. Then we have a similar similar data for this algebraic group and then we have associated reflex fields and then we have we can start the module. But this is the usual things in the, in the theory of PL structure. This is something we don't know. And then, then in the following the, following the usual story, we fix the structure of B, which is the order in B subring B, which is stable under this star. And the VZ is an integral version of V, VZ stable Z lattice in V, satisfying this condition. And then we fix a set of prime numbers here from now on, I follow the notation of, of, of, of long in his work. And this square is a set of prime numbers without bad primes. So bad prime is a contents of primes for which this integral version of site. The site for this integral structure is not perfect pairings at P, at the prime. Such, such prime is a bad prime. Maybe we also we remove some, some bad primes. So then, then, now the notation is that this local, localization of Z around this square is the localization of Z by inverting the all prime numbers feature outside the square. And we have the localization of this integer ring of the, the, the reflex field by, by, by this square is just a product. And Z hat delta, delta, z hat square means the product of all the L for L prime numbers L, which do not belong, belong to, to, to, to this square. So that is, this is not, not the for L for in delta. This is the usual notation for people, people in, in the study of Shimura varieties. We put to, we put to such notation primes in upper place, then we are removing the, the, the section. And if you put to lower place the prime set of prime numbers, then we are keeping these, then, then, like this, like this. And we fix the open compact subgroup H in, in, in this group because following this integral structure we can consider define the, the such Z hat to square set points of G. And we assume that it is neat. I don't explain the meaning of neat here. By, by, by this neat condition we have nice modular space. Yeah. And so then our modularity function. We have a modularity function. So, I think to, is it to 20 minutes, 20. I forgot the time, the end of the time, time of, of, of this talk. It is until 20 or it's okay. 20 or yeah. Yeah. So, so now we can start the modularity function. We can start three modularity functions. And the first one is just a modularity function for Arbelian varieties. And these six second and third are functors for, for, for modularity functions, modularity functions for log Arbelian varieties. They are all functors from the, the Fs, category of Fs log schemes over this localization of the integer ring of the reflex field and to the category of sets. And the Fs and the Fs. So here, we first define F1, F bar and F bar, sigma is, comes later. And this, this sigma will become the, the toroidal compactification. But, and then I have to explain sigma, conical position. This sigma is conical position. But, but F1 and F bar can be defined first in a very, very simple way. That is, we can connect, we can use the same, same presentation of the definition for F1 and F bar just connecting by less. Because, because we, we have a logarithmic Arbelian varieties have, have group structure. And, and then, so we can define the endomorphism and the level structure in the, in the simple way. So, so then Arbelian, so the Fs and F bar, this F bar is such such a, a and, and u and a and p. So, for the yotais, endomorphism, it is a level structure, p is a polarization. All those things can be defined nicely by using the group structure. And, and then, so the a is a, a is a Arbelian scheme over s and this logarithmic Arbelian variety over s and yotais ring homomorphism from bz to end a. Here we are using the, the group structure of a in the quarter. Because, because endomorphism means, means homomorphism for the group structure. So, so we are using the such group structure to, to, to, to, for this. And, and then, we, a, a is a level structure. So, his isomorphism between this standard object tensor. So, this is a square part of, square, square, removing the square part of the state module. So, state module, upper with upper square is a, is a, a, a, a, a, not in, in square. And it is the inverse limit of the kernel of the multiplication by n only, where n ranges over all int, integers feature prime to, to, to square. So, here we are using the group structure and, and the multiplication by n on the log Arbelian variety. And, and so just like the case of Arbelian variety. And, and then the, the, the level structure is the isomorphism module h. So, so we, we, we, we, h is acting here. And so then, then h is acting here. Then, if two isomorphisms are connected by, by h, then we regard them as the same, same element, same level structure. And this is, and then this is a busy isomorphism on the pro etal side. This is more busy isomorphism on the pro etal, this pro log etal side. And finally, p is a polarization. It is a homomorphism from a to the, to the x to sheep of, of a and gm log. This is also defined by using the, the, the, the group structure. And so, this polarization, I don't give the precise definition of the polarization. It is such homomorphism, satisfying some, some condition. And, and then these three things, you turn the etal and p should satisfy some compatibility. Yeah, yeah. So, this already defines the, so the, the, if the definition of this is well known, that, that is a, a very unvariety with the structure. And here the, the log, log, a very unvariety for the log, a very unvariety with the, the, the structure is defined in the exactly the same style. So, the definition is very simple. So, so, then, then, actually, our, our, one of the main result is that, if f bar is, is f, this, I will explain this sigma part of the, of f bar. But if we, we don't think sigma part, then f bar is f bar sigma is coincides with coincide with the, the toroidal compactification of run. But f bar is bigger as a factor. And it is a log algebraic space in the second sense. So, this, this is the fact this is a log algebraic space in the second sense is one, one, one means, means around here. So, so, now, so, then, the remaining part, I still have a big, still a big time, but I'm all, we must finish my story because the definitions are so simple. And, and then, but still, for this sigma part, I need some, some complicated story. So, then, this part will have time, need time. Yeah, I think, yeah. So, but roughly speaking, this sigma part of f bar is element of f bar, such that all element point of s, the local monodrome of a, at s, belongs to sigma. And I have, I hope to explain this more precisely. But I am afraid that my talk finishes soon. I, I, I, I, I, because, so, it's, it's, it's, it's so simple. Story is so simple. But yeah, here, the sigma part, story of sigma part is, is not so simple. Yeah. So, so the local, what is the local monodrome? So, this part is a part of the, we are discussing the local monodrome type of local monodrome. And then the local monodrome of the Aberian variety, a local Aberian varieties is as follows. Usually, Aberian variety doesn't have local monodrome. This local monodrome appears only for local Aberian varieties. And, and then for, for, for, for, for point of s, small s, for the point of s, small s, we, we consider the separable closure s bar of the, of the point s. And so, with the inverse image of the log structure base. And so, this is why now we consider the log Aberian variety over s bar to, to, to think about the local monodrome of A over s. And then, then we consider the log Aberian variety over our spec of separable closed feed. This has some FS log structure. Then the state module, state module, this, no, no square part of the state module has a filtration, weight filtration, W0, W-1, W-2, and W-3 is 0, such that, and then this, this quotient of W0 by W-1, even as group W0 is, is, has some integral structure. And the group W-2 has also integral structure and is, and then group W is the tensor product of the, of this, this integral structure times this, this, the, the square and the group minus W-2 is similar. And then we, such, things then, then for element of the pi-1 log, log fundamental group of s bar, which acts on the state module, then we have for element of the log fundamental group, E-1 square is 0. And, and then the action of the E-1 factor in this way, it goes to group 0 by the canonical projection. And then, then it goes to group, group minus 2. And then group minus 2 is embedded in state module. And so, so the map is, goes like this, E-1 goes like this. And, and then, and furthermore, this pi-1 log s bar is, is a completion of this integral structure. And it is, if s bar is characteristic of the field s, field of s is, of characteristic 0, it is a, this is a profiling completion of this. And if we, if the ratio characteristic of s is p, then this is a profiling completion of this for the non-p part, in the case of characteristic p. And so this is a dense subspace of this subgroup of this, which has is a finitely generated Arbillian group. And if we use this, then, then this action E-1 keeps, respect this, this structure. And such thing happens in the designation, in the photo log Arbillian variety. And in the case, in the case, K is a complete discrete variation field. And K, and here, as before, so, K is a Arbillian variety over K of semi-stable reduction, and is a unique extension log Arbillian variety over K. And if s is a closed point and s bar is a separable closure of the closed point, then this pi-1 log fundamental group, pi-1 log s bar is the, the, the lower group of the, the lower group of the maximal term extension of K over the maximal undamaged extension of K. And in the, so, then such stories very well known for the, for the Arbillian variety of semi-stable reduction over, over, over, over, over, over, over complete discrete variation field. And then this state module is, is, can be taken over, is, for AK and for AR, essentially the same, they have the same state module. And, and then such stories well known for the Arbillian variety with semi-stable reduction. And, and then, for example, if A is a Q-tate curve, then this, this, we are considering, we are considering the, the, the story, here, the story of, of the special fiber of the log Arbillian variety. Yeah. And then, then, then the state module is the same as the state module of the, of the unique fiber. Yeah. And, and if A is a Q-tate curve, that is, this special fiber is, is, is written in this way. Then, then the group zero is, is the power of this QZ, Q, QZ, which is isomorph to Z, and group minus two is, is such, is also isomorph to Z. These integral structures are understood in this way. Yeah. And I, I am, I am just, maybe I am omitting the, the, the, I am sorry that I, some, some thing like this is culturally necessary here, but I am omitting this, this thing to make things description simpler. We have to fix some, some, yeah, yeah, I am omitting that, yeah, yeah, yeah. And then, then for this story of, of Sigma, we fix, fix Sigma, the, the, the, the family of condi compositions. This is such thing as follows exist. This is written in the paper work of Lang. This is, this is written in the work of Lang, but it is, it is used from the old work of Manford and other people on the, on the territorial compactification, and they consider condi compositions over complex number field. And, and they, and they consider the condi compositions. But in the work of Lang, because he did not update module, in the not have the group structure of the designate of a new variety, so the combination becomes becomes a little complicated, more complicated. Here, the combination becomes simpler. Essentially the same as, as Fat Lang did. Yeah. Condi compositions and W ranges over, I naturally generated BZ module and, and he is a surjective homomorphism of this standard one to two to W after tensor Z hat square. This is a surjective BZ module. Such that the, I don't explain this part so well the kernel, kernel, the annihilator of the, of the kernel for this side kills each, each seven such conditions should be put. And then, then, and then for each such a pair of W and D, the, the condi, we should have the condi composition of the, the space of positive semi definite symmetric by linear forms W tensor W tensor to R with rational kernels are such positive semi, semi, semi, semi, semi, dynamic partner when it comes to symmetry before moreข value belongs to甜 bolt P dot directly to elsewhere in this measurement Everything becomes in T Our standard in this measurement is gonna we are not going to have to use the figures that is this condi-composition for W and GH, H in the open compact subgroup, H is the same of the condi-composition for W and G. And for a subjective homomorphism from W to another W prime, then the condi-composition given for W should be related to the condi-composition for W in some simple way. Yeah. And then the factor F bar, F bar, F bar sigma, S, is the, is the such a yota, ah, sorry, I forgot. This is in F bar S. So vice. And then the conditions are that, for all S point S for the special fiber for the restriction of A to S bar, the separable closure of S. And there exists for each point, there is a sigma, and then here the, the, the, the, you have a map introduced by yota, yota, ah, yeah. So other, other, other, other, other, other, other, other, other, the, the level structure is the, is the isomorphism from, from this to, to, to, to, to R log t, t, t, r, e, t t square e with, with this. This is the level structure. And then, then we can have, we have a, a map to grew W, zero. And it is a grew W, zero. z times z hat. So then we have a map g for this integral object and then we are using this group as w here. So then we have a composition associated on the space of the space of semi-definite in our forms on this space and then the condition is that for each point s, we have some element of cone on the index sigma such that for all a in this sub-space of, so we have now monoid inside the integral structure of the log fundamental group. And then we have a prime from group for a here, we have a prime from group w0 to group minus 2. And then the condition is that we then we have by this a prime associated to a minus action of a minus 1, then it preserves the z structure and then we have a map from group 0w times group 0z to group 0z times group w minus 2z by this prime. And then we have a pairing to z by this polarization. Yeah, polarization gives such a map and then this belongs to this. The condition is that this map is a Pyrenean form on this space and this condition is that this map for all a, this map should belong to sigma. So that is we have a map from this monoid to this sigma by in such a way. So this is the definition of the sigma part and the main theorem is that this f bar, so we have the f bar sigma and f bar and f is a usual monoid fangota for a Pyrenean varieties. So this f is actually the other is a monoid space of a Pyrenean varieties is PEL type and this is a classical object. And this f bar sigma is log algebraic space. So f bar is a log algebraic space in the second sense. And if sigma is a log algebraic space in the first sense, it is an algebraic space with an Fs log structure. And this algebraic space is actually the toroidal compactification of run. And like run did sometime we can, we can have also not only algebraic space, but a projective, projective modular space, representable modular compactification. Such thing can be proved, but it is already proved. It is proved by run and such thing is, yeah. But the main good thing about the proof is, the main thing is that the definition becomes so simple at least for f bar, the definition is perfectly the same as the case of usual Aperian varieties. And the sigma part is a little bit complicated, but we can still use the group structure. And the search, yeah. So a group structure makes all things simpler. And the proof of this main theorem is also becomes easy. And using the algorithm criterion or log algorithm criterion. And for example, the case, so from the partings try to run without endomorphism ring to the case with endomorphism ring was a hard process. And that was a very, very hard process. But this becomes very simple, because this is by just we can prove the space that the home space from between log Aperian varieties A and A prime is representative, representable over the base. And so then if by using this is proved by some arching criteria. And then this is relatively, so if we take as the base the modular space of without endomorphism as the base and A and A prime, the universal objects over it, then such thing can give the representative such theorem. And so, because, because we use that this is a space then this case becomes algebraic space. This becomes, if we know that this is algebraic space, then we can deduce that this to the PL case is also algebraic space because it is relatively the home space is one is relatively the home space over the, over the, over the case because q, because we are, because we are consigned, endomorphism ring here. So, so just, just this is relatively home space over this, and the home space is representable so, so we can deduce the representability of this from this. So, from the type to run is becomes simpler. Yeah, so, so I had to present my this method gives only a new interpretation of the theory of longer, but I hope that in the future, the, the, this theory of longer varieties will be useful in the theory of similar varieties and some other other theories. Yeah, I, I have still several minutes but I finished the, the stories. So I finish here. Yeah, yeah. Thank you very much. Thank you very much. So, we will take a few questions. So is there any question in the. So, maybe Sophie. Can you see the, can you see the stratification of the toroidal classification in a simple way. Oh, thank you. How, how, how is this, this W. Yeah, yeah, so I, yeah, yeah, yeah, so I, that is like, yes, so I, that is, so if you consider the local Archinian case, then the log of the varieties is like, like this, so this is, this is generalized to some G, G log. Y, y, y, y, y are, for y is only G to ZN and such he is a semi-Abelian scheme and so over Archinian based, we have always over, in the case of, Yeah, yeah, yeah. So the type of the log-Aberian varieties is like, is log-Aberian varieties over such log point is Warlog, Arachnian, Arachnian. This is classified by using the dimension of this Arbillian part, Arbillian skin part and this Taurus part and this rank Y and such rank this is, I think, and T is, I think, GM to R and these are Arbillian skin. And such thing appears if we consider semi-stabilization over complete risk variation. So such R and such dimension of display becomes the, gives the stratification of the proidal compatibility, I think. Yeah, yeah. Thank you. So Luke has a question and you have two questions from the chat. So maybe your question is, we finish with your question. So we take two questions from the chat. So I have a question about, in which category do you take your X in your, for your slides when you define your F, I think F sigma, you take some text. In the pluralization, this is a category of sheaves of, or sheaves of Arbillian groups on the, on the, on the, on the, if yes, if yes, if we consider log-Aberian varieties over S and, and this sheave means a, a tartopology in the classical sense. Yeah, yeah, yeah, yeah. There's another question that, okay, I don't understand what I am reading. So do you expect to generalize from log-Aberian varieties with the best structures to a notion of log-Aberian varieties with odd cycles and give a reformulated moduli function of the real commodification of the general structure varieties in the sense of, there are similar varieties of, of Arbillian type or some general, more general, similar varieties than the similar varieties for PEA type and, and your question is about it, the possibility of the generalization to, to more general similar varieties. Yeah, actually it's not my question, the question is in the chat, but you can, you can answer the question you think too, yeah? Yeah, yeah, yeah. So, yeah, we have not yet considered such a problem and, and so, so there are some interesting things which should be considered by using log-Aberian varieties, maybe a level structures for PEA not invertible, for N level structure for N not invertible. Oh yeah, but this is already difficult for Arbillian varieties, but, yeah, yeah, yeah. In the such more general similar varieties should be considered related, the similar varieties related to Arbillian varieties, but, but not, not included in the, in the similar varieties of PEA type, so maybe it is nice to think about them, yeah, yeah, yeah, yeah, yeah, we have not yet considered such a thing. Thank you. So, Luke has a question? Yes, some time ago I think it's Nakayama Chikara who asked me the following question, suppose you're over a CDVR and you have a commutative flat group scheme, locally of an ant type, over this complete DVR, and suppose that on the, on the special fiber, it is similar billion, then is it similar billion everywhere? Ah, the question, and the answer I, so at the time it seems that you didn't know the answer, and then Erstedrik Petmar would have found some, some arguments for that, so showing roughly that if you have something really potent in the general fiber, then taking some, some closure, then you get something also unique potent in the special fiber, so we'll come back to that. And I wonder whether this had something to do with your work and the criterion, the acting criterion you mentioned at the end? Ah, so I hope Nakayama knows this well, I hope in these days Nakayama is much cleverer than me, and so Nakayama, but is it in the, in the, in the, in the, in this work or not? Is it in different direction? Is this my useful for, what is my useful or what you discussed or not? Is it different thing? Ah, I am sorry, I, I cannot catch what you explained so well, so, so maybe we, I will ask Nakayama. Because the seemingly obedient thing is stable and originalization, so that was the property, is an open property somehow, almost. So, so yeah, I could not follow this thing so, how do you are saying so well, I, I hope that you, you explained this later slowly, because I have a second question, so maybe a much older question, so now you have these log-in billion schemes, so in the log, the group structure, so is there a way to better or better understand, you understand in a simpler way, gotentic monotony pairing in SGS 7, 9? Ah, yeah, yeah, in the SGS 9 is confronted prime by prime and I think there, so of course you only have the, the Nehromodern, you don't have this, this, you have any, I think it should come out from what you, yeah, about it, about that conjecture, the my student, Suzukuma, I was kidding Takashi, yes. Takashi, Suzuki, yes. Well, he's a good expert but I, I am, I am not the good expert about it and so, ah, yeah, I'm sorry that, that Suzuki is much better than me. You have the third question, so, and the last one, so very long ago you wrote a beautiful print, unpublished and maybe you're finished, maybe incomplete, so log, log due to the NACRI, so here, log due to the NACRI for these long and billion schemes. Oh, yeah, yeah, yeah, that should be, I hope to, to complete it soon, because here are the two points of the Abelian varieties, log Abelian varieties are so important and, and I hope that log crystalline theory of crystalline convolution of log Abelian varieties are important and, and the two points are important and, and then such things should be, should be considered well, yeah, yeah, so I, I have, I have, I hope to complete that paper soon, yeah, thank you very much, thank you very much. Okay, so I think after you've had the question, so after you can ask your question, if you unmute your microphone. Ah, what was my question? Oh, yeah, so, um, yeah, the question was, what's the nature of the embedding of this F bar sigma in F bar, is it the closed immersion and open immersion? Ah, that is, that is like a log smooth, or, or some, or some, brewing up, so that it is inclusion, but in the if we consider it up, yes, yes, yes, okay, in the category of schemes then it is like a brewing up, yeah, yeah, yeah, yeah, yeah, yeah, yeah. Okay, thank you very much. So I think we have no other questions. So let's thank the speaker again.
 Thank you very much for the invitation.  I hope that he will become more and more active and younger and younger.  I respect him, also like my father, and I am thankful to Ruku very much.  Today I am the founder of the Logarithmic Geometry and I talk about the Logarithmic Geometry and the Logarithmic Geometry.  I use the document camera to work.  I hope I myself cannot see so well, but I hope you can see this.  This is a space with a she-hub ring and a log-she-hub ring and a she-hub monolith.  This scheme was started by the great teacher, Urotan Dik of Ruku.  The theory of the log scheme was started.  Ruku was very much excited and probably he was happy that he was making a new world of the algebraic geometry like his teacher did.  We discussed Ruku and I shared many dreams on the logarithmic geometry.  I think the log-Aberian variety was already in the dream from the early days of the geometry.  I am happy that today I can talk about the logarithmic-Aberian varieties after 30 years.  Here is a nice textbook on the logarithmic geometry written by Oga.  In the first line of this book, it is written that the logarithmic geometry was developed to deal with two fundamental and related problems in algebraic geometry.  One is compactification, one is degeneration.  How compactification is important and how degeneration is important.  There are two famous golden things and famous observations.  For the compactification, Golden Thin is by Argello Vistolo, working with non-complete ojira spaces like keeping a change in a pocket with holes.  This golden thing is often quoted by Abramovic in his talks.  For degeneration, the king observation by Pakesh Saito is that in the story of beauty and beast, Argelli beast became a nice man by the love of the girl.  In the designate object, Argelli objects in algebraic geometry became a nice object by the power of the log and the power of love of the girl and the power of love are similar.  And then here is the mysterious coincidence of the letters.  Who is the first one?  In the case of modular space, these two things are connected in this way.  It is a nice object in the usual geometry, but it is not compact.  It can be ugly in the usual geometry, but it becomes a nice object in the log geometry.  The space of Argelli and Varaitis is TEL-tropical.  Today I give the information of the joint work with Kajiwara and Nakayama in which the territorial compactification is understood as the space of log-Aberian Varaitis with TEL structure.  Usually, the designate-Aberian Varaitis have no group structure and other Argelli, but it is nice here in the log geometry and they have group structure.  And here the PEL is the polarization, the endomorphism ring, and the L is the level structure.  So the endomorphism ring is fixed or something like that.  The ring of this endomorphism preserves the structure of the Argelli and Varaitis.  Now we can have a ring because two endomorphisms can be added because log-Aberian Varaitis can have a group structure.  So without a group structure, it is very hard to consider the endomorphism of the degenerate-Aberian Varaitis.  But we can have such a nice object in the logarithmic world.  So it becomes very natural and nice.  The formulation becomes very transparent and nice with the log-Aberian Varaitis.  And at present, our work is only the reinterpretation of the work of Ran, but I hope that in the future the log-Aberian Varaitis will have good applications because the theory becomes so transparent.  And then in the case of the endomorphism, the theory was made by Farting and Chai in 1990.  And also I hope to say that Fujiwara had also some nice work on the case of PEL type structure for Argelli Varaitis, for the Tore Dalcon Motivation.  But unfortunately this work was unpublished around 1990.  And I just introduced the rough idea of log-Aberian Varaitis.  So log-Aberian Varaitis over FES-log-log-scheme is a contravariant factor from the category of FES-log-scheme to the category of Arbelian Groups.  And so it is naturally the structure of the group structure.  And if K is a complete discrete variation field and A K is Arbelian Varaitis over K with semi-stable reduction, then A K extends to log-Aberian Varaitis over OK.  Here OK has the standard is endowed with the standard log structure and A K extends to log-Aberian Varaitis over OK uniquely.  And this semi-stable reduction, how to treat semi-stable reduction nicely was the starting point, motivation of the creation of the log geometry.  So then in the usual Arge-Brick geometry, in this case of the theory of Arbelian Varaitis, in the usual Arge-Brick geometry, A K extends to a robust scheme with semi-stable reduction, but it doesn't have the group structure.  So the group structure cannot extend to such schemes, but the Arbelian, log-Aberian Varaitis has a group structure.  And for example, if A K is a teiton elliptical curve with Q invariant Q, then A is if we restrict the phonkta A to the FES log schemes over OK module Mk to the N, over this Arltinian quotient of the OK, then this restriction can be written in the  very simple way. It is a quotient of GM log Q divided by the power of Q. Here Q is a QE invariant.  And this presentation is very similar to the presentation of the teiton elliptical curve, an analytic presentation of the teiton elliptical curve in the analytic geometry over the field K.  But that can be treated in an algebraic way, not an analytic way, an algebraic way in this log geometry.  And GM log Q is, no, no, no, I made a mistake. So GM log Q is this, and sorry GM log Q is a part of GM log, such as Q to the A device T, and T device Q to the B in the log structure for some AB.  And then we divide this part of the Mk group by the power of QZ. And this is very similar to the presentation of the teiton elliptical curve as a multiplicative group by the powers of Q in the analytic geometry over QK.  And here the Q is the Q is the element of Mk minus 0. And it is regarded as the section of the log structure M of the spec of K, called the standard log structure.  And then for log structure, the log structure is the log structure, the C of monoid of the log structure is regarded as the extension of the unit group. So we have a bigger unit group.  And this is a monoid, but this monoid is embedded in the group associated with the monoid. And you have also a map, multiplicative map from MT to OT. And then here for these schemes over OK, Mojiro, Mk to the N, Q is nilpotent.  And if n is 0, then Q is 0 here. But you can, in this case, so but Q is put in the log structure and then Q becomes invertible. That is, we are making an nilpotent element invertible here.  And in the case n is 1, we are making 0 invertible here. So we can make 0 invertible in the log geometry. And by such such thing, then we can have this group structure of this degenerate of the Arbillian variety.  So log geometry is such a theory to make 0 invertible. Yeah, so now I put one comment that a log Arbillian variety is a log algebraic space in the second sense.  And the usual Arbillian space is something which is covered by a usual scheme by a termorphism.  The morphism here is relatively etal. X is regarded as a functor, but this is relatively it is representable. This morphism is represented by a termorphism.  And Q is also representable and you have such a suggestion. That is the algebraic space. Arbillian space is something which is covered by by by representable to put object by etal morphism.  And log algebraic space.  The first sense is a is a similar to a functor covered by a representable object by log etal morphisms. So it's our morphism is replaced by log with our most log Arbillian varieties is such a object.  And log algebraic space in the first sense is is such X, unctah X, covered by a representable object by the classical etal morphism.  And this is also equal to the algebraic space with with usual algebraic space with an infuse of structure.  So there are two things, first sense and second sense and log Arbillian varieties log algebraic space in the second sense.  Yeah.  Now, now I, I, and it's so I don't explain more about log with Arbillian variety if I explain too much then I will lose time so.  And then now I will go to the problem of Mojira with PL structure for polarization and endomorphism and level structure.  So, so then in such theory, we fix a semi semi semi semi simple algebra over Q and with such a operator operator, it's two to four locations identity and it is changes the order of the multiplication and and then.  So, when you say log etal, you mean the common log etal or the general the more general log etal.  Yeah, yeah, I think so yeah.  So, I think we have also another question from Bruno.  Sorry.  We have a second question.  It was a mistake I apologize.  So, so then the module I, so I don't explain all things because I, it is, there are too many things to be put to just I, I explain the main part and the satisfying certain conditions, including the this, this size skew symmetric non-degenerate  by linear form and then another the group of similitudes for this pair of the outside.  This is algebraic group over Q, algebraic group over Q, deductive algebraic group over Q and and then by some I don't explain the some condition but under some condition.  Then we have a similar similar data for this algebraic group and then we have associated reflex fields and then we have we can start the module.  But this is the usual things in the, in the theory of PL structure.  This is something we don't know.  And then, then in the following the, following the usual story, we fix the structure of B, which is the order in B subring B, which is stable under this star.  And the VZ is an integral version of V, VZ stable Z lattice in V, satisfying this condition.  And then we fix a set of prime numbers here from now on, I follow the notation of, of, of, of long in his work.  And this square is a set of prime numbers without bad primes. So bad prime is a contents of primes for which this integral version of site.  The site for this integral structure is not perfect pairings at P, at the prime.  Such, such prime is a bad prime. Maybe we also we remove some, some bad primes.  So then, then, now the notation is that this local, localization of Z around this square is the localization of Z by inverting the all prime numbers feature outside the square.  And we have the localization of this integer ring of the, the, the reflex field by, by, by this square is just a product.  And Z hat delta, delta, z hat square means the product of all the L for L prime numbers L, which do not belong, belong to, to, to, to this square.  So that is, this is not, not the for L for in delta.  This is the usual notation for people, people in, in the study of Shimura varieties. We put to, we put to such notation primes in upper place, then we are removing the, the, the section.  And if you put to lower place the prime set of prime numbers, then we are keeping these, then, then, like this, like this.  And we fix the open compact subgroup H in, in, in this group because following this integral structure we can consider define the, the such Z hat to square set points of G.  And we assume that it is neat. I don't explain the meaning of neat here.  By, by, by this neat condition we have nice modular space. Yeah. And so then our modularity function.  We have a modularity function.  So, I think to, is it to 20 minutes, 20.  I forgot the time, the end of the time, time of, of, of this talk. It is until 20 or it's okay.  20 or yeah. Yeah. So, so now we can start the modularity function.  We can start three modularity functions. And the first one is just a modularity function for Arbelian varieties.  And these six second and third are functors for, for, for modularity functions, modularity functions for log Arbelian varieties.  They are all functors from the, the Fs, category of Fs log schemes over this localization of the integer ring of the reflex field and to the category of sets.  And the Fs and the Fs. So here, we first define F1, F bar and F bar, sigma is, comes later.  And this, this sigma will become the, the toroidal compactification.  But, and then I have to explain sigma, conical position. This sigma is conical position.  But, but F1 and F bar can be defined first in a very, very simple way.  That is, we can connect, we can use the same, same presentation of the definition for F1 and F bar just connecting by less.  Because, because we, we have a logarithmic Arbelian varieties have, have group structure.  And, and then, so we can define the endomorphism and the level structure in the, in the simple way.  So, so then Arbelian, so the Fs and F bar, this F bar is such such a, a and, and u and a and p.  So, for the yotais, endomorphism, it is a level structure, p is a polarization. All those things can be defined nicely by using the group structure.  And, and then, so the a is a, a is a Arbelian scheme over s and this logarithmic Arbelian variety over s and yotais ring homomorphism from bz to end a.  Here we are using the, the group structure of a in the quarter.  Because, because endomorphism means, means homomorphism for the group structure. So, so we are using the such group structure to, to, to, to, for this.  And, and then, we, a, a is a level structure.  So, his isomorphism between this standard object tensor.  So, this is a square part of, square, square, removing the square part of the state module.  So, state module, upper with upper square is a, is a, a, a, a, a, not in, in square.  And it is the inverse limit of the kernel of the multiplication by n only, where n ranges over all int, integers feature prime to, to, to square.  So, here we are using the group structure and, and the multiplication by n on the log Arbelian variety.  And, and so just like the case of Arbelian variety. And, and then the, the, the level structure is the isomorphism module h.  So, so we, we, we, we, h is acting here. And so then, then h is acting here.  Then, if two isomorphisms are connected by, by h, then we regard them as the same, same element, same level structure.  And this is, and then this is a busy isomorphism on the pro etal side.  This is more busy isomorphism on the pro etal, this pro log etal side.  And finally, p is a polarization. It is a homomorphism from a to the, to the x to sheep of, of a and gm log.  This is also defined by using the, the, the, the group structure.  And so, this polarization, I don't give the precise definition of the polarization. It is such homomorphism, satisfying some, some condition.  And, and then these three things, you turn the etal and p should satisfy some compatibility.  Yeah, yeah. So, this already defines the, so the, the, if the definition of this is well known, that, that is a, a very unvariety with the structure.  And here the, the log, log, a very unvariety for the log, a very unvariety with the, the, the structure is defined in the exactly the same style.  So, the definition is very simple. So, so, then, then, actually, our, our, one of the main result is that, if f bar is, is f, this, I will explain this sigma part of the, of f bar.  But if we, we don't think sigma part, then f bar is f bar sigma is coincides with coincide with the, the toroidal compactification of run.  But f bar is bigger as a factor. And it is a log algebraic space in the second sense.  So, this, this is the fact this is a log algebraic space in the second sense is one, one, one means, means around here.  So, so, now, so, then, the remaining part, I still have a big, still a big time, but I'm all, we must finish my story because the definitions are so simple.  And, and then, but still, for this sigma part, I need some, some complicated story. So, then, this part will have time, need time.  Yeah, I think, yeah. So, but roughly speaking, this sigma part of f bar is element of f bar, such that all element point of s, the local monodrome of a, at s, belongs to sigma.  And I have, I hope to explain this more precisely.  But I am afraid that my talk finishes soon.  I, I, I, I, I, because, so, it's, it's, it's, it's so simple.  Story is so simple. But yeah, here, the sigma part, story of sigma part is, is not so simple. Yeah. So, so the local, what is the local monodrome?  So, this part is a part of the, we are discussing the local monodrome type of local monodrome.  And then the local monodrome of the Aberian variety, a local Aberian varieties is as follows.  Usually, Aberian variety doesn't have local monodrome. This local monodrome appears only for local Aberian varieties. And, and then for, for, for, for, for point of s, small s, for the point of s, small s, we, we consider the separable closure s bar of the, of the point s.  And so, with the inverse image of the log structure base. And so, this is why now we consider the log Aberian variety over s bar to, to, to think about the local monodrome of A over s.  And then, then we consider the log Aberian variety over our spec of separable closed feed. This has some FS log structure. Then the state module, state module, this, no, no square part of the state module has a filtration, weight filtration,  W0, W-1, W-2, and W-3 is 0, such that, and then this, this quotient of W0 by W-1, even as group W0 is, is, has some integral structure.  And the group W-2 has also integral structure and is, and then group W is the tensor product of the, of this, this integral structure times this, this, the, the square and the group minus W-2 is similar.  And then we, such, things then, then for element of the pi-1 log, log fundamental group of s bar, which acts on the state module, then we have for element of the log fundamental group,  E-1 square is 0. And, and then the action of the E-1 factor in this way, it goes to group 0 by the canonical projection.  And then, then it goes to group, group minus 2. And then group minus 2 is embedded in state module. And so, so the map is, goes like this, E-1 goes like this.  And, and then, and furthermore, this pi-1 log s bar is, is a completion of this integral structure.  And it is, if s bar is characteristic of the field s, field of s is, of characteristic 0, it is a, this is a profiling completion of this. And if we, if the ratio characteristic of s is p, then this is a profiling completion of this for the non-p part, in the case of characteristic p.  And so this is a dense subspace of this subgroup of this, which has is a finitely generated Arbillian group. And if we use this, then, then this action E-1 keeps, respect this, this structure.  And such thing happens in the designation, in the photo log Arbillian variety. And in the case, in the case, K is a complete discrete variation field.  And K, and here, as before, so, K is a Arbillian variety over K of semi-stable reduction, and is a unique extension log Arbillian variety over K. And if s is a closed point and s bar is a separable closure of the closed point, then this pi-1 log fundamental group, pi-1 log s bar is the, the, the lower group of the,  the lower group of the maximal term extension of K over the maximal undamaged extension of K. And in the, so, then such stories very well known for the, for the Arbillian variety of semi-stable reduction over, over, over, over, over, over, over complete discrete variation field.  And then this state module is, is, can be taken over, is, for AK and for AR, essentially the same, they have the same state module. And, and then such stories well known for the Arbillian variety with semi-stable reduction.  And, and then, for example, if A is a Q-tate curve, then this, this, we are considering, we are considering the, the, the story, here, the story of, of the special fiber of the log Arbillian variety.  Yeah. And then, then, then the state module is the same as the state module of the, of the unique fiber. Yeah. And, and if A is a Q-tate curve, that is, this special fiber is, is, is written in this way.  Then, then the group zero is, is the power of this QZ, Q, QZ, which is isomorph to Z, and group minus two is, is such, is also isomorph to Z. These integral structures are understood in this way.  Yeah. And I, I am, I am just, maybe I am omitting the, the, the, I am sorry that I, some, some thing like this is culturally necessary here, but I am omitting this, this thing to make things description simpler.  We have to fix some, some, yeah, yeah, I am omitting that, yeah, yeah, yeah.  And then, then for this story of, of Sigma, we fix, fix Sigma, the, the, the, the family of condi compositions. This is such thing as follows exist. This is written in the paper work of Lang.  This is, this is written in the work of Lang, but it is, it is used from the old work of Manford and other people on the, on the territorial compactification, and they consider condi compositions over complex number field.  And, and they, and they consider the condi compositions.  But in the work of Lang, because he did not update module, in the not have the group structure of the designate of a new variety, so the combination becomes becomes a little complicated, more complicated.  Here, the combination becomes simpler. Essentially the same as, as Fat Lang did. Yeah.  Condi compositions and W ranges over, I naturally generated BZ module and, and he is a surjective homomorphism of this standard one to two to W after tensor Z hat square. This is a surjective BZ module.  Such that the, I don't explain this part so well the kernel, kernel, the annihilator of the, of the kernel for this side kills each, each seven such conditions should be put.  And then, then, and then for each such a pair of W and D, the, the condi, we should have the condi composition of the, the space of positive semi definite symmetric by linear forms W tensor W tensor to R with rational kernels are such positive semi, semi, semi, semi, semi,  dynamic partner when it comes to symmetry before moreข value belongs to甜 bolt P dot directly to elsewhere in this measurement Everything becomes in T Our standard in this measurement is gonna we are not going to have to use the figures  that is this condi-composition for W and GH,  H in the open compact subgroup,  H is the same of the condi-composition for W and G.  And for a subjective homomorphism  from W to another W prime,  then the condi-composition given for W  should be related to the condi-composition  for W in some simple way.  Yeah.  And then the factor F bar, F bar, F bar sigma, S,  is the, is the such a yota, ah, sorry, I forgot.  This is in F bar S.  So vice.  And then the conditions are that,  for all S point S for the special fiber  for the restriction of A to S bar,  the separable closure of S.  And there exists for each point, there is a sigma,  and then here the, the, the, the,  you have a map introduced by yota, yota, ah, yeah.  So other, other, other, other, other,  other, other, other, other,  the, the level structure is the,  is the isomorphism from, from this to, to, to,  to, to R log t, t, t, r, e,  t t square e with, with this.  This is the level structure.  And then, then we can have, we have a,  a map to grew W, zero.  And it is a grew W, zero.  z times z hat.  So then we have a map g for this integral object  and then we are using this group as w here.  So then we have a composition associated on the space of the space of semi-definite  in our forms on this space and then the condition is that for each point s,  we have some element of cone on the index sigma such that for all a in this sub-space of,  so we have now monoid inside the integral structure of the log fundamental group.  And then we have a prime from group for a here, we have a prime from group w0 to group minus 2.  And then the condition is that we then we have by this a prime associated to a minus action of a minus 1,  then it preserves the z structure and then we have a map from group 0w times group 0z  to group 0z times group w minus 2z by this prime.  And then we have a pairing to z by this polarization.  Yeah, polarization gives such a map and then this belongs to this.  The condition is that this map is a Pyrenean form on this space  and this condition is that this map for all a, this map should belong to sigma.  So that is we have a map from this monoid to this sigma by in such a way.  So this is the definition of the sigma part and the main theorem is that this f bar,  so we have the f bar sigma and f bar and f is a usual monoid fangota for a Pyrenean varieties.  So this f is actually the other is a monoid space of a Pyrenean varieties is PEL type and this is a classical object.  And this f bar sigma is log algebraic space.  So f bar is a log algebraic space in the second sense.  And if sigma is a log algebraic space in the first sense, it is an algebraic space with an Fs log structure.  And this algebraic space is actually the toroidal compactification of run.  And like run did sometime we can, we can have also not only algebraic space,  but a projective, projective modular space, representable modular compactification.  Such thing can be proved, but it is already proved.  It is proved by run and such thing is, yeah.  But the main good thing about the proof is, the main thing is that the definition becomes so simple at least for f bar,  the definition is perfectly the same as the case of usual Aperian varieties.  And the sigma part is a little bit complicated, but we can still use the group structure.  And the search, yeah.  So a group structure makes all things simpler.  And the proof of this main theorem is also becomes easy.  And using the algorithm criterion or log algorithm criterion.  And for example, the case, so from the partings try to run without endomorphism ring to the case with endomorphism ring was a hard process.  And that was a very, very hard process.  But this becomes very simple, because this is by just we can prove the space that the home space from between log Aperian varieties A and A prime is representative, representable over the base.  And so then if by using this is proved by some arching criteria.  And then this is relatively, so if we take as the base the modular space of without endomorphism as the base and A and A prime, the universal objects over it, then such thing can give the representative such theorem.  And so, because, because we use that this is a space then this case becomes algebraic space.  This becomes, if we know that this is algebraic space, then we can deduce that this to the  PL case is also algebraic space because it is relatively the home space is one is relatively the home space over the, over the, over the case because q, because we are, because we are consigned, endomorphism ring here.  So, so just, just this is relatively home space over this, and the home space is representable so, so we can deduce the representability of this from this.  So, from the type to run is becomes simpler.  Yeah, so, so I had to present my this method gives only a new interpretation of the theory of longer, but I hope that in the future, the, the, this theory of longer varieties will be useful in the theory of similar varieties and some other other theories.  Yeah, I, I have still several minutes but I finished the, the stories. So I finish here. Yeah, yeah. Thank you very much.  Thank you very much.  So, we will take a few questions. So is there any question in the.  So, maybe Sophie.  Can you see the, can you see the stratification of the toroidal classification in a simple way.  Oh, thank you.  How, how, how is this, this W.  Yeah, yeah, so I, yeah, yeah, yeah, so I, that is like, yes, so I, that is, so if you consider the local Archinian case, then the log of the varieties is like, like this, so this is, this is generalized to some G, G log.  Y, y, y, y, y are, for y is only G to ZN and such he is a semi-Abelian scheme and so over Archinian based, we have always over, in the case of,  Yeah, yeah, yeah. So the type of the log-Aberian varieties is like, is log-Aberian varieties  over such log point is Warlog, Arachnian, Arachnian. This is classified by using the  dimension of this Arbillian part, Arbillian skin part and this Taurus part and this rank Y and  such rank this is, I think, and T is, I think, GM to R and these are Arbillian skin. And such  thing appears if we consider semi-stabilization over complete risk variation. So such R and  such dimension of display becomes the, gives the stratification of the proidal  compatibility, I think. Yeah, yeah. Thank you. So Luke has a question and you have two questions  from the chat. So maybe your question is, we finish with your question. So we take two questions  from the chat. So I have a question about, in which category do you take your X in your,  for your slides when you define your F, I think F sigma, you take some text.  In the pluralization, this is a category of sheaves of, or sheaves of Arbillian groups on the,  on the, on the, on the, if yes, if yes, if we consider log-Aberian varieties over S  and, and this sheave means a, a tartopology in the classical sense. Yeah, yeah, yeah, yeah.  There's another question that, okay, I don't understand what I am reading. So do you expect  to generalize from log-Aberian varieties with the best structures to a notion of log-Aberian  varieties with odd cycles and give a reformulated moduli function of the real  commodification of the general structure varieties in the sense of,  there are similar varieties of, of Arbillian type or some general,  more general, similar varieties than the similar varieties for PEA type and, and your question  is about it, the possibility of the generalization to, to more general similar varieties.  Yeah, actually it's not my question, the question is in the chat, but  you can, you can answer the question you think too, yeah?  Yeah, yeah, yeah. So, yeah, we have not yet considered such a problem and, and so,  so there are some interesting things which should be considered by using log-Aberian varieties,  maybe a level structures for PEA not invertible, for N level structure for N not invertible.  Oh yeah, but this is already difficult for Arbillian varieties, but,  yeah, yeah, yeah. In the such more general similar varieties should be considered related,  the similar varieties related to Arbillian varieties, but, but not, not included in the,  in the similar varieties of PEA type, so maybe it is nice to think about them, yeah, yeah, yeah,  yeah, yeah, we have not yet considered such a thing.  Thank you. So, Luke has a question?  Yes, some time ago I think it's Nakayama Chikara who asked me the following question,  suppose you're over a CDVR and you have a commutative flat group scheme,  locally of an ant type, over this complete DVR, and suppose that on the, on the special fiber,  it is similar billion, then is it similar billion everywhere?  Ah, the question, and the answer I, so at the time it seems that you didn't know the answer,  and then Erstedrik Petmar would have found some, some arguments for that,  so showing roughly that if you have something really potent in the general fiber,  then taking some, some closure, then you get something also  unique potent in the special fiber, so we'll come back to that. And I wonder whether this  had something to do with your work and the criterion, the acting criterion you mentioned at the end?  Ah, so I hope Nakayama knows this well, I hope in these days Nakayama is much cleverer than me,  and so Nakayama, but is it in the, in the, in the, in the, in this work or not? Is it in different  direction? Is this my useful for, what is my useful or what you discussed or not? Is it different  thing? Ah, I am sorry, I, I cannot catch what you explained so well, so, so maybe we, I will ask  Nakayama. Because the seemingly obedient thing is stable and originalization, so that was the  property, is an open property somehow, almost. So, so yeah, I could not follow  this thing so, how do you are saying so well, I, I hope that you, you explained this later slowly,  because I have a second question, so maybe a much older question, so now you have these  log-in billion schemes, so in the log, the group structure, so is there a way to better or  better understand, you understand in a simpler way, gotentic monotony pairing in SGS 7, 9?  Ah, yeah, yeah, in the SGS 9 is confronted prime by prime and I think there, so of course you  only have the, the Nehromodern, you don't have this, this, you have any, I think it should come out from  what you, yeah, about it, about that conjecture, the my student, Suzukuma,  I was kidding Takashi, yes. Takashi, Suzuki, yes.  Well, he's a good expert but I, I am, I am not the good expert about it and so, ah,  yeah, I'm sorry that, that Suzuki is much better than me.  You have the third question, so, and the last one, so very long ago you wrote a beautiful  print, unpublished and maybe you're finished, maybe incomplete, so log, log due to the NACRI,  so here, log due to the NACRI for these long and billion schemes.  Oh, yeah, yeah, yeah, that should be, I hope to, to complete it soon, because here are the  two points of the Abelian varieties, log Abelian varieties are so important and, and I hope that  log crystalline theory of crystalline convolution of log Abelian varieties are important and,  and the two points are important and, and then such things should be, should be considered well,  yeah, yeah, so I, I have, I have, I hope to complete that paper soon, yeah, thank you very much, thank you very much.  Okay, so I think after you've had the question, so after you can ask your question, if you unmute your microphone.  Ah, what was my question? Oh, yeah, so, um, yeah, the question was, what's the nature of the embedding  of this F bar sigma in F bar, is it the closed immersion and open immersion? Ah, that is,  that is like a log smooth, or, or some, or some, brewing up, so that it is inclusion, but in the if we  consider it up, yes, yes, yes, okay, in the category of schemes then it is like a brewing up, yeah, yeah, yeah, yeah, yeah, yeah, yeah.  Okay, thank you very much. So I think we have no other questions. So let's thank the speaker again.
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Thank you very much for the invitation. I hope that he will become more and more active and younger and younger. I respect him, also like my father, and I am thankful to Ruku very much. Today I am the founder of the Logarithmic Geometry and I talk about the Logarithmic Geometry and the Logarithmic Geometry. I use the document camera to work. I hope I myself cannot see so well, but I hope you can see this. This is a space with a she-hub ring and a log-she-hub ring and a she-hub monolith. This scheme was started by the great teacher, Urotan Dik of Ruku. The theory of the log scheme was started. Ruku was very much excited and probably he was happy that he was making a new world of the algebraic geometry like his teacher did. We discussed Ruku and I shared many dreams on the logarithmic geometry. I think the log-Aberian variety was already in the dream from the early days of the geometry. I am happy that today I can talk about the logarithmic-Aberian varieties after 30 years. Here is a nice textbook on the logarithmic geometry written by Oga. In the first line of this book, it is written that the logarithmic geometry was developed to deal with two fundamental and related problems in algebraic geometry. One is compactification, one is degeneration. How compactification is important and how degeneration is important. There are two famous golden things and famous observations. For the compactification, Golden Thin is by Argello Vistolo, working with non-complete ojira spaces like keeping a change in a pocket with holes. This golden thing is often quoted by Abramovic in his talks. For degeneration, the king observation by Pakesh Saito is that in the story of beauty and beast, Argelli beast became a nice man by the love of the girl. In the designate object, Argelli objects in algebraic geometry became a nice object by the power of the log and the power of love of the girl and the power of love are similar. And then here is the mysterious coincidence of the letters. Who is the first one? In the case of modular space, these two things are connected in this way. It is a nice object in the usual geometry, but it is not compact. It can be ugly in the usual geometry, but it becomes a nice object in the log geometry. The space of Argelli and Varaitis is TEL-tropical. Today I give the information of the joint work with Kajiwara and Nakayama in which the territorial compactification is understood as the space of log-Aberian Varaitis with TEL structure. Usually, the designate-Aberian Varaitis have no group structure and other Argelli, but it is nice here in the log geometry and they have group structure. And here the PEL is the polarization, the endomorphism ring, and the L is the level structure. So the endomorphism ring is fixed or something like that. The ring of this endomorphism preserves the structure of the Argelli and Varaitis. Now we can have a ring because two endomorphisms can be added because log-Aberian Varaitis can have a group structure. So without a group structure, it is very hard to consider the endomorphism of the degenerate-Aberian Varaitis. But we can have such a nice object in the logarithmic world. So it becomes very natural and nice. The formulation becomes very transparent and nice with the log-Aberian Varaitis. And at present, our work is only the reinterpretation of the work of Ran, but I hope that in the future the log-Aberian Varaitis will have good applications because the theory becomes so transparent. And then in the case of the endomorphism, the theory was made by Farting and Chai in 1990. And also I hope to say that Fujiwara had also some nice work on the case of PEL type structure for Argelli Varaitis, for the Tore Dalcon Motivation. But unfortunately this work was unpublished around 1990. And I just introduced the rough idea of log-Aberian Varaitis. So log-Aberian Varaitis over FES-log-log-scheme is a contravariant factor from the category of FES-log-scheme to the category of Arbelian Groups. And so it is naturally the structure of the group structure. And if K is a complete discrete variation field and A K is Arbelian Varaitis over K with semi-stable reduction, then A K extends to log-Aberian Varaitis over OK. Here OK has the standard is endowed with the standard log structure and A K extends to log-Aberian Varaitis over OK uniquely. And this semi-stable reduction, how to treat semi-stable reduction nicely was the starting point, motivation of the creation of the log geometry. So then in the usual Arge-Brick geometry, in this case of the theory of Arbelian Varaitis, in the usual Arge-Brick geometry, A K extends to a robust scheme with semi-stable reduction, but it doesn't have the group structure. So the group structure cannot extend to such schemes, but the Arbelian, log-Aberian Varaitis has a group structure. And for example, if A K is a teiton elliptical curve with Q invariant Q, then A is if we restrict the phonkta A to the FES log schemes over OK module Mk to the N, over this Arltinian quotient of the OK, then this restriction can be written in the very simple way. It is a quotient of GM log Q divided by the power of Q. Here Q is a QE invariant. And this presentation is very similar to the presentation of the teiton elliptical curve, an analytic presentation of the teiton elliptical curve in the analytic geometry over the field K. But that can be treated in an algebraic way, not an analytic way, an algebraic way in this log geometry. And GM log Q is, no, no, no, I made a mistake. So GM log Q is this, and sorry GM log Q is a part of GM log, such as Q to the A device T, and T device Q to the B in the log structure for some AB. And then we divide this part of the Mk group by the power of QZ. And this is very similar to the presentation of the teiton elliptical curve as a multiplicative group by the powers of Q in the analytic geometry over QK. And here the Q is the Q is the element of Mk minus 0. And it is regarded as the section of the log structure M of the spec of K, called the standard log structure. And then for log structure, the log structure is the log structure, the C of monoid of the log structure is regarded as the extension of the unit group. So we have a bigger unit group. And this is a monoid, but this monoid is embedded in the group associated with the monoid. And you have also a map, multiplicative map from MT to OT. And then here for these schemes over OK, Mojiro, Mk to the N, Q is nilpotent. And if n is 0, then Q is 0 here. But you can, in this case, so but Q is put in the log structure and then Q becomes invertible. That is, we are making an nilpotent element invertible here. And in the case n is 1, we are making 0 invertible here. So we can make 0 invertible in the log geometry. And by such such thing, then we can have this group structure of this degenerate of the Arbillian variety. So log geometry is such a theory to make 0 invertible. Yeah, so now I put one comment that a log Arbillian variety is a log algebraic space in the second sense. And the usual Arbillian space is something which is covered by a usual scheme by a termorphism. The morphism here is relatively etal. X is regarded as a functor, but this is relatively it is representable. This morphism is represented by a termorphism. And Q is also representable and you have such a suggestion. That is the algebraic space. Arbillian space is something which is covered by by by representable to put object by etal morphism. And log algebraic space. The first sense is a is a similar to a functor covered by a representable object by log etal morphisms. So it's our morphism is replaced by log with our most log Arbillian varieties is such a object. And log algebraic space in the first sense is is such X, unctah X, covered by a representable object by the classical etal morphism. And this is also equal to the algebraic space with with usual algebraic space with an infuse of structure. So there are two things, first sense and second sense and log Arbillian varieties log algebraic space in the second sense. Yeah. Now, now I, I, and it's so I don't explain more about log with Arbillian variety if I explain too much then I will lose time so. And then now I will go to the problem of Mojira with PL structure for polarization and endomorphism and level structure. So, so then in such theory, we fix a semi semi semi semi simple algebra over Q and with such a operator operator, it's two to four locations identity and it is changes the order of the multiplication and and then. So, when you say log etal, you mean the common log etal or the general the more general log etal. Yeah, yeah, I think so yeah. So, I think we have also another question from Bruno. Sorry. We have a second question. It was a mistake I apologize. So, so then the module I, so I don't explain all things because I, it is, there are too many things to be put to just I, I explain the main part and the satisfying certain conditions, including the this, this size skew symmetric non-degenerate by linear form and then another the group of similitudes for this pair of the outside. This is algebraic group over Q, algebraic group over Q, deductive algebraic group over Q and and then by some I don't explain the some condition but under some condition. Then we have a similar similar data for this algebraic group and then we have associated reflex fields and then we have we can start the module. But this is the usual things in the, in the theory of PL structure. This is something we don't know. And then, then in the following the, following the usual story, we fix the structure of B, which is the order in B subring B, which is stable under this star. And the VZ is an integral version of V, VZ stable Z lattice in V, satisfying this condition. And then we fix a set of prime numbers here from now on, I follow the notation of, of, of, of long in his work. And this square is a set of prime numbers without bad primes. So bad prime is a contents of primes for which this integral version of site. The site for this integral structure is not perfect pairings at P, at the prime. Such, such prime is a bad prime. Maybe we also we remove some, some bad primes. So then, then, now the notation is that this local, localization of Z around this square is the localization of Z by inverting the all prime numbers feature outside the square. And we have the localization of this integer ring of the, the, the reflex field by, by, by this square is just a product. And Z hat delta, delta, z hat square means the product of all the L for L prime numbers L, which do not belong, belong to, to, to, to this square. So that is, this is not, not the for L for in delta. This is the usual notation for people, people in, in the study of Shimura varieties. We put to, we put to such notation primes in upper place, then we are removing the, the, the section. And if you put to lower place the prime set of prime numbers, then we are keeping these, then, then, like this, like this. And we fix the open compact subgroup H in, in, in this group because following this integral structure we can consider define the, the such Z hat to square set points of G. And we assume that it is neat. I don't explain the meaning of neat here. By, by, by this neat condition we have nice modular space. Yeah. And so then our modularity function. We have a modularity function. So, I think to, is it to 20 minutes, 20. I forgot the time, the end of the time, time of, of, of this talk. It is until 20 or it's okay. 20 or yeah. Yeah. So, so now we can start the modularity function. We can start three modularity functions. And the first one is just a modularity function for Arbelian varieties. And these six second and third are functors for, for, for modularity functions, modularity functions for log Arbelian varieties. They are all functors from the, the Fs, category of Fs log schemes over this localization of the integer ring of the reflex field and to the category of sets. And the Fs and the Fs. So here, we first define F1, F bar and F bar, sigma is, comes later. And this, this sigma will become the, the toroidal compactification. But, and then I have to explain sigma, conical position. This sigma is conical position. But, but F1 and F bar can be defined first in a very, very simple way. That is, we can connect, we can use the same, same presentation of the definition for F1 and F bar just connecting by less. Because, because we, we have a logarithmic Arbelian varieties have, have group structure. And, and then, so we can define the endomorphism and the level structure in the, in the simple way. So, so then Arbelian, so the Fs and F bar, this F bar is such such a, a and, and u and a and p. So, for the yotais, endomorphism, it is a level structure, p is a polarization. All those things can be defined nicely by using the group structure. And, and then, so the a is a, a is a Arbelian scheme over s and this logarithmic Arbelian variety over s and yotais ring homomorphism from bz to end a. Here we are using the, the group structure of a in the quarter. Because, because endomorphism means, means homomorphism for the group structure. So, so we are using the such group structure to, to, to, to, for this. And, and then, we, a, a is a level structure. So, his isomorphism between this standard object tensor. So, this is a square part of, square, square, removing the square part of the state module. So, state module, upper with upper square is a, is a, a, a, a, a, not in, in square. And it is the inverse limit of the kernel of the multiplication by n only, where n ranges over all int, integers feature prime to, to, to square. So, here we are using the group structure and, and the multiplication by n on the log Arbelian variety. And, and so just like the case of Arbelian variety. And, and then the, the, the level structure is the isomorphism module h. So, so we, we, we, we, h is acting here. And so then, then h is acting here. Then, if two isomorphisms are connected by, by h, then we regard them as the same, same element, same level structure. And this is, and then this is a busy isomorphism on the pro etal side. This is more busy isomorphism on the pro etal, this pro log etal side. And finally, p is a polarization. It is a homomorphism from a to the, to the x to sheep of, of a and gm log. This is also defined by using the, the, the, the group structure. And so, this polarization, I don't give the precise definition of the polarization. It is such homomorphism, satisfying some, some condition. And, and then these three things, you turn the etal and p should satisfy some compatibility. Yeah, yeah. So, this already defines the, so the, the, if the definition of this is well known, that, that is a, a very unvariety with the structure. And here the, the log, log, a very unvariety for the log, a very unvariety with the, the, the structure is defined in the exactly the same style. So, the definition is very simple. So, so, then, then, actually, our, our, one of the main result is that, if f bar is, is f, this, I will explain this sigma part of the, of f bar. But if we, we don't think sigma part, then f bar is f bar sigma is coincides with coincide with the, the toroidal compactification of run. But f bar is bigger as a factor. And it is a log algebraic space in the second sense. So, this, this is the fact this is a log algebraic space in the second sense is one, one, one means, means around here. So, so, now, so, then, the remaining part, I still have a big, still a big time, but I'm all, we must finish my story because the definitions are so simple. And, and then, but still, for this sigma part, I need some, some complicated story. So, then, this part will have time, need time. Yeah, I think, yeah. So, but roughly speaking, this sigma part of f bar is element of f bar, such that all element point of s, the local monodrome of a, at s, belongs to sigma. And I have, I hope to explain this more precisely. But I am afraid that my talk finishes soon. I, I, I, I, I, because, so, it's, it's, it's, it's so simple. Story is so simple. But yeah, here, the sigma part, story of sigma part is, is not so simple. Yeah. So, so the local, what is the local monodrome? So, this part is a part of the, we are discussing the local monodrome type of local monodrome. And then the local monodrome of the Aberian variety, a local Aberian varieties is as follows. Usually, Aberian variety doesn't have local monodrome. This local monodrome appears only for local Aberian varieties. And, and then for, for, for, for, for point of s, small s, for the point of s, small s, we, we consider the separable closure s bar of the, of the point s. And so, with the inverse image of the log structure base. And so, this is why now we consider the log Aberian variety over s bar to, to, to think about the local monodrome of A over s. And then, then we consider the log Aberian variety over our spec of separable closed feed. This has some FS log structure. Then the state module, state module, this, no, no square part of the state module has a filtration, weight filtration, W0, W-1, W-2, and W-3 is 0, such that, and then this, this quotient of W0 by W-1, even as group W0 is, is, has some integral structure. And the group W-2 has also integral structure and is, and then group W is the tensor product of the, of this, this integral structure times this, this, the, the square and the group minus W-2 is similar. And then we, such, things then, then for element of the pi-1 log, log fundamental group of s bar, which acts on the state module, then we have for element of the log fundamental group, E-1 square is 0. And, and then the action of the E-1 factor in this way, it goes to group 0 by the canonical projection. And then, then it goes to group, group minus 2. And then group minus 2 is embedded in state module. And so, so the map is, goes like this, E-1 goes like this. And, and then, and furthermore, this pi-1 log s bar is, is a completion of this integral structure. And it is, if s bar is characteristic of the field s, field of s is, of characteristic 0, it is a, this is a profiling completion of this. And if we, if the ratio characteristic of s is p, then this is a profiling completion of this for the non-p part, in the case of characteristic p. And so this is a dense subspace of this subgroup of this, which has is a finitely generated Arbillian group. And if we use this, then, then this action E-1 keeps, respect this, this structure. And such thing happens in the designation, in the photo log Arbillian variety. And in the case, in the case, K is a complete discrete variation field. And K, and here, as before, so, K is a Arbillian variety over K of semi-stable reduction, and is a unique extension log Arbillian variety over K. And if s is a closed point and s bar is a separable closure of the closed point, then this pi-1 log fundamental group, pi-1 log s bar is the, the, the lower group of the, the lower group of the maximal term extension of K over the maximal undamaged extension of K. And in the, so, then such stories very well known for the, for the Arbillian variety of semi-stable reduction over, over, over, over, over, over, over complete discrete variation field. And then this state module is, is, can be taken over, is, for AK and for AR, essentially the same, they have the same state module. And, and then such stories well known for the Arbillian variety with semi-stable reduction. And, and then, for example, if A is a Q-tate curve, then this, this, we are considering, we are considering the, the, the story, here, the story of, of the special fiber of the log Arbillian variety. Yeah. And then, then, then the state module is the same as the state module of the, of the unique fiber. Yeah. And, and if A is a Q-tate curve, that is, this special fiber is, is, is written in this way. Then, then the group zero is, is the power of this QZ, Q, QZ, which is isomorph to Z, and group minus two is, is such, is also isomorph to Z. These integral structures are understood in this way. Yeah. And I, I am, I am just, maybe I am omitting the, the, the, I am sorry that I, some, some thing like this is culturally necessary here, but I am omitting this, this thing to make things description simpler. We have to fix some, some, yeah, yeah, I am omitting that, yeah, yeah, yeah. And then, then for this story of, of Sigma, we fix, fix Sigma, the, the, the, the family of condi compositions. This is such thing as follows exist. This is written in the paper work of Lang. This is, this is written in the work of Lang, but it is, it is used from the old work of Manford and other people on the, on the territorial compactification, and they consider condi compositions over complex number field. And, and they, and they consider the condi compositions. But in the work of Lang, because he did not update module, in the not have the group structure of the designate of a new variety, so the combination becomes becomes a little complicated, more complicated. Here, the combination becomes simpler. Essentially the same as, as Fat Lang did. Yeah. Condi compositions and W ranges over, I naturally generated BZ module and, and he is a surjective homomorphism of this standard one to two to W after tensor Z hat square. This is a surjective BZ module. Such that the, I don't explain this part so well the kernel, kernel, the annihilator of the, of the kernel for this side kills each, each seven such conditions should be put. And then, then, and then for each such a pair of W and D, the, the condi, we should have the condi composition of the, the space of positive semi definite symmetric by linear forms W tensor W tensor to R with rational kernels are such positive semi, semi, semi, semi, semi, dynamic partner when it comes to symmetry before moreข value belongs to甜 bolt P dot directly to elsewhere in this measurement Everything becomes in T Our standard in this measurement is gonna we are not going to have to use the figures that is this condi-composition for W and GH, H in the open compact subgroup, H is the same of the condi-composition for W and G. And for a subjective homomorphism from W to another W prime, then the condi-composition given for W should be related to the condi-composition for W in some simple way. Yeah. And then the factor F bar, F bar, F bar sigma, S, is the, is the such a yota, ah, sorry, I forgot. This is in F bar S. So vice. And then the conditions are that, for all S point S for the special fiber for the restriction of A to S bar, the separable closure of S. And there exists for each point, there is a sigma, and then here the, the, the, the, you have a map introduced by yota, yota, ah, yeah. So other, other, other, other, other, other, other, other, other, the, the level structure is the, is the isomorphism from, from this to, to, to, to, to R log t, t, t, r, e, t t square e with, with this. This is the level structure. And then, then we can have, we have a, a map to grew W, zero. And it is a grew W, zero. z times z hat. So then we have a map g for this integral object and then we are using this group as w here. So then we have a composition associated on the space of the space of semi-definite in our forms on this space and then the condition is that for each point s, we have some element of cone on the index sigma such that for all a in this sub-space of, so we have now monoid inside the integral structure of the log fundamental group. And then we have a prime from group for a here, we have a prime from group w0 to group minus 2. And then the condition is that we then we have by this a prime associated to a minus action of a minus 1, then it preserves the z structure and then we have a map from group 0w times group 0z to group 0z times group w minus 2z by this prime. And then we have a pairing to z by this polarization. Yeah, polarization gives such a map and then this belongs to this. The condition is that this map is a Pyrenean form on this space and this condition is that this map for all a, this map should belong to sigma. So that is we have a map from this monoid to this sigma by in such a way. So this is the definition of the sigma part and the main theorem is that this f bar, so we have the f bar sigma and f bar and f is a usual monoid fangota for a Pyrenean varieties. So this f is actually the other is a monoid space of a Pyrenean varieties is PEL type and this is a classical object. And this f bar sigma is log algebraic space. So f bar is a log algebraic space in the second sense. And if sigma is a log algebraic space in the first sense, it is an algebraic space with an Fs log structure. And this algebraic space is actually the toroidal compactification of run. And like run did sometime we can, we can have also not only algebraic space, but a projective, projective modular space, representable modular compactification. Such thing can be proved, but it is already proved. It is proved by run and such thing is, yeah. But the main good thing about the proof is, the main thing is that the definition becomes so simple at least for f bar, the definition is perfectly the same as the case of usual Aperian varieties. And the sigma part is a little bit complicated, but we can still use the group structure. And the search, yeah. So a group structure makes all things simpler. And the proof of this main theorem is also becomes easy. And using the algorithm criterion or log algorithm criterion. And for example, the case, so from the partings try to run without endomorphism ring to the case with endomorphism ring was a hard process. And that was a very, very hard process. But this becomes very simple, because this is by just we can prove the space that the home space from between log Aperian varieties A and A prime is representative, representable over the base. And so then if by using this is proved by some arching criteria. And then this is relatively, so if we take as the base the modular space of without endomorphism as the base and A and A prime, the universal objects over it, then such thing can give the representative such theorem. And so, because, because we use that this is a space then this case becomes algebraic space. This becomes, if we know that this is algebraic space, then we can deduce that this to the PL case is also algebraic space because it is relatively the home space is one is relatively the home space over the, over the, over the case because q, because we are, because we are consigned, endomorphism ring here. So, so just, just this is relatively home space over this, and the home space is representable so, so we can deduce the representability of this from this. So, from the type to run is becomes simpler. Yeah, so, so I had to present my this method gives only a new interpretation of the theory of longer, but I hope that in the future, the, the, this theory of longer varieties will be useful in the theory of similar varieties and some other other theories. Yeah, I, I have still several minutes but I finished the, the stories. So I finish here. Yeah, yeah. Thank you very much. Thank you very much. So, we will take a few questions. So is there any question in the. So, maybe Sophie. Can you see the, can you see the stratification of the toroidal classification in a simple way. Oh, thank you. How, how, how is this, this W. Yeah, yeah, so I, yeah, yeah, yeah, so I, that is like, yes, so I, that is, so if you consider the local Archinian case, then the log of the varieties is like, like this, so this is, this is generalized to some G, G log. Y, y, y, y, y are, for y is only G to ZN and such he is a semi-Abelian scheme and so over Archinian based, we have always over, in the case of, Yeah, yeah, yeah. So the type of the log-Aberian varieties is like, is log-Aberian varieties over such log point is Warlog, Arachnian, Arachnian. This is classified by using the dimension of this Arbillian part, Arbillian skin part and this Taurus part and this rank Y and such rank this is, I think, and T is, I think, GM to R and these are Arbillian skin. And such thing appears if we consider semi-stabilization over complete risk variation. So such R and such dimension of display becomes the, gives the stratification of the proidal compatibility, I think. Yeah, yeah. Thank you. So Luke has a question and you have two questions from the chat. So maybe your question is, we finish with your question. So we take two questions from the chat. So I have a question about, in which category do you take your X in your, for your slides when you define your F, I think F sigma, you take some text. In the pluralization, this is a category of sheaves of, or sheaves of Arbillian groups on the, on the, on the, on the, if yes, if yes, if we consider log-Aberian varieties over S and, and this sheave means a, a tartopology in the classical sense. Yeah, yeah, yeah, yeah. There's another question that, okay, I don't understand what I am reading. So do you expect to generalize from log-Aberian varieties with the best structures to a notion of log-Aberian varieties with odd cycles and give a reformulated moduli function of the real commodification of the general structure varieties in the sense of, there are similar varieties of, of Arbillian type or some general, more general, similar varieties than the similar varieties for PEA type and, and your question is about it, the possibility of the generalization to, to more general similar varieties. Yeah, actually it's not my question, the question is in the chat, but you can, you can answer the question you think too, yeah? Yeah, yeah, yeah. So, yeah, we have not yet considered such a problem and, and so, so there are some interesting things which should be considered by using log-Aberian varieties, maybe a level structures for PEA not invertible, for N level structure for N not invertible. Oh yeah, but this is already difficult for Arbillian varieties, but, yeah, yeah, yeah. In the such more general similar varieties should be considered related, the similar varieties related to Arbillian varieties, but, but not, not included in the, in the similar varieties of PEA type, so maybe it is nice to think about them, yeah, yeah, yeah, yeah, yeah, we have not yet considered such a thing. Thank you. So, Luke has a question? Yes, some time ago I think it's Nakayama Chikara who asked me the following question, suppose you're over a CDVR and you have a commutative flat group scheme, locally of an ant type, over this complete DVR, and suppose that on the, on the special fiber, it is similar billion, then is it similar billion everywhere? Ah, the question, and the answer I, so at the time it seems that you didn't know the answer, and then Erstedrik Petmar would have found some, some arguments for that, so showing roughly that if you have something really potent in the general fiber, then taking some, some closure, then you get something also unique potent in the special fiber, so we'll come back to that. And I wonder whether this had something to do with your work and the criterion, the acting criterion you mentioned at the end? Ah, so I hope Nakayama knows this well, I hope in these days Nakayama is much cleverer than me, and so Nakayama, but is it in the, in the, in the, in the, in this work or not? Is it in different direction? Is this my useful for, what is my useful or what you discussed or not? Is it different thing? Ah, I am sorry, I, I cannot catch what you explained so well, so, so maybe we, I will ask Nakayama. Because the seemingly obedient thing is stable and originalization, so that was the property, is an open property somehow, almost. So, so yeah, I could not follow this thing so, how do you are saying so well, I, I hope that you, you explained this later slowly, because I have a second question, so maybe a much older question, so now you have these log-in billion schemes, so in the log, the group structure, so is there a way to better or better understand, you understand in a simpler way, gotentic monotony pairing in SGS 7, 9? Ah, yeah, yeah, in the SGS 9 is confronted prime by prime and I think there, so of course you only have the, the Nehromodern, you don't have this, this, you have any, I think it should come out from what you, yeah, about it, about that conjecture, the my student, Suzukuma, I was kidding Takashi, yes. Takashi, Suzuki, yes. Well, he's a good expert but I, I am, I am not the good expert about it and so, ah, yeah, I'm sorry that, that Suzuki is much better than me. You have the third question, so, and the last one, so very long ago you wrote a beautiful print, unpublished and maybe you're finished, maybe incomplete, so log, log due to the NACRI, so here, log due to the NACRI for these long and billion schemes. Oh, yeah, yeah, yeah, that should be, I hope to, to complete it soon, because here are the two points of the Abelian varieties, log Abelian varieties are so important and, and I hope that log crystalline theory of crystalline convolution of log Abelian varieties are important and, and the two points are important and, and then such things should be, should be considered well, yeah, yeah, so I, I have, I have, I hope to complete that paper soon, yeah, thank you very much, thank you very much. Okay, so I think after you've had the question, so after you can ask your question, if you unmute your microphone. Ah, what was my question? Oh, yeah, so, um, yeah, the question was, what's the nature of the embedding of this F bar sigma in F bar, is it the closed immersion and open immersion? Ah, that is, that is like a log smooth, or, or some, or some, brewing up, so that it is inclusion, but in the if we consider it up, yes, yes, yes, okay, in the category of schemes then it is like a brewing up, yeah, yeah, yeah, yeah, yeah, yeah, yeah. Okay, thank you very much. So I think we have no other questions. So let's thank the speaker again.
that he was making a new world of the algebraic give the information of the joint work with Kajiwara and Nakayama in which the territorial in the log geometry and they have group log structure is the log structure, the C of all the L for L prime numbers L, which do not This is the usual notation for people, people in, square. So, here we are using the group structure composition associated on the space of the space stratification of the toroidal classification in