12027220
Mirror symmetry and toric degenerations of partial flag manifolds
In this paper we propose and discuss a mirror construction for complete intersections in partial flag manifolds F (n1, . . . , nl, n). This construction includes our previous mirror construction for complete intersection in Grassmannians and the mirror construction of Givental for complete flag manifolds. The key idea of our construction is a degeneration of F (n1, . . . , nl, n) to a certain Gorenstein toric Fano variety P (n1, . . . , nl, n) which has been investigated by Gonciulea and Lakshmibai. We describe a natural small crepant desingularization of P (n1, . . . , nl, n) and prove a generalized version of a conjecture of Gonciulea and Lakshmibai on the singular locus of P (n1, . . . , nl, n). Mathematisches Institut, Eberhard-Karls-Universitat Tubingen, D-72076 Tubingen, Germany, email address: batyrev@bastau.mathematik.uni-tuebingen.de Department of Mathematics, Oklahoma State University, Stillwater, OK 74078, USA email address: ciocan@math.okstate.edu Department of Mathematics, University of California Davis, Davis, CA 95616, USA email address: bumsig@math.ucdavis.edu FB 17, Mathematik, Johannes Gutenberg-Universitat Mainz, D-55099 Mainz, Germany, email address: straten@mathematik.uni-mainz.de
18222916
Dual Polyhedra and Mirror Symmetry for Calabi-Yau Hypersurfaces in Toric Varieties
We consider families ${\cal F}(\Delta)$ consisting of complex $(n-1)$-dimensional projective algebraic compactifications of $\Delta$-regular affine hypersurfaces $Z_f$ defined by Laurent polynomials $f$ with a fixed $n$-dimensional Newton polyhedron $\Delta$ in $n$-dimensional algebraic torus ${\bf T} =({\bf C}^*)^n$. If the family ${\cal F}(\Delta)$ defined by a Newton polyhedron $\Delta$ consists of $(n-1)$-dimensional Calabi-Yau varieties, then the dual, or polar, polyhedron $\Delta^*$ in the dual space defines another family ${\cal F}(\Delta^*)$ of Calabi-Yau varieties, so that we obtain the remarkable duality between two {\em different families} of Calabi-Yau varieties. It is shown that the properties of this duality coincide with the properties of {\em Mirror Symmetry} discovered by physicists for Calabi-Yau $3$-folds. Our method allows to construct many new examples of Calabi-Yau $3$-folds and new candidats for their mirrors which were previously unknown for physicists. We conjecture that there exists an isomorphism between two conformal field theories corresponding to Calabi-Yau varieties from two families ${\cal F}(\Delta)$ and ${\cal F}(\Delta^*)$.
16401756
Generalized hypergeometric functions and rational curves on Calabi-Yau complete intersections in toric varieties
We formulate general conjectures about the relationship between the A-model connection on the cohomology of ad-dimensional Calabi-Yau complete intersectionV ofr hypersurfacesV1,...,Vr in a toric varietyPΣ and the system of differential operators annihilating the special generalized hypergeometric series Φ0 constructed from the fan Σ. Using this generalized hypergeometric series, we propose conjectural mirrorsV′ ofV and the canonicalq-coordinates on the moduli spaces of Calabi-Yau manifolds.In the second part of the paper we consider some examples of Calabi-Yau 3-folds having Picard number >1 in products of projective spaces. For conjectural mirrors, using the recurrent relation among coefficients of the restriction of the hypergeometric function Φ0 on a special line in the moduli space, we determine the Picard-Fuchs equation satisfied by periods of this special one-parameter subfamily. This allows to obtain some sequences of integers which can be conjecturally interpreted in terms of Gromov-Witten invariants. Using standard techniques from enumerative geometry, first terms of these sequence of integers are checked to coincide with numbers of rational curves on Calabi-Yau 3-folds.
122609279
Gröbner bases and convex polytopes
Grobner basics The state polytope Variation of term orders Toric ideals Enumeration, sampling and integer programming Primitive partition identities Universal Grobner bases Regular triangulations The second hypersimplex $\mathcal A$-graded algebras Canonical subalgebra bases Generators, Betti numbers and localizations Toric varieties in algebraic geometry Some specific Grobner bases Bibliography Index.
123375476
Degenerations of flag and Schubert varieties to toric varieties
In this paper, we prove the degenerations of Schubert varieties in a minusculeG/P, as well as the class of Kempf varieties in the flag varietySL(n)/B, to (normal) toric varieties. As a consequence, we obtain that determinantal varietes degenerate to (normal) toric varieties.
16493118
SCHUBERT VARIETIES , TORIC VARIETIES , AND LADDERDETERMINANTAL
We contruct certain normal toric varieties (associated to nite distribu-tive lattices) which are degenerations of the Grassmannians. We also determine the singular loci for certain normal toric varieties, namely the ones which are certain ladder determinantal varieties. As a consequence, we prove the conjecture of 17] on the components of the singular locus, for certain Schubert varieties in the ag variety.
16516514
Stationary Phase Integrals, Quantum Toda Lattices, Flag Manifolds and the Mirror Conjecture
A generalization of the mirror conjecture is proven for the manifolds of complete flags in C^n.
16250111
Toric Degenerations of Fano Varieties and Constructing Mirror Manifolds
For an arbitrary smooth n-dimensional Fano variety $X$ we introduce the notion of a small toric degeneration. Using small toric degenerations of Fano n-folds $X$, we propose a general method for constructing mirrors of Calabi-Yau complete intersections in $X$. Our mirror construction is based on a generalized monomial-divisor mirror correspondence which can be used for computing Gromov-Witten invariants of rational curves via specializations of GKZ-hypergeometric series.
2770458
On quantum cohomology rings of partial flag varieties
0. Introduction. The main goal of this paper is to give a unified description for the structure of the small quantum cohomology rings for all projective homogeneous spaces SLn(C)/P , where P is a parabolic subgroup. The quantum cohomology ring of a smooth projective variety, or more generally of a symplectic manifoldX, has been introduced by string theorists (see [Va] and [W1]). Roughly speaking, it is a deformation of the usual cohomology ring, with parameter space given byH ∗(X). The multiplicative structure of quantum cohomology encodes the enumerative geometry of rational curves on X. In the past few years, the highly nontrivial task of giving a rigorous mathematical treatment of quantum cohomology has been accomplished both in the realm of algebraic and symplectic geometry. In various degrees of generality, this can be found in [Beh], [BehM], [KM], [LiT1], [LiT2], [McS], and [RT], as well as in the surveys [FP] and [T]. If one restricts the parameter space to H 1,1(X), one gets the small quantum cohomology ring. This ring, in the case of partial flag varieties, is the object of the present paper. In order to state our main results, we first describe briefly the “classical” side of the story. We interpret the homogeneous space F := SLn(C)/P as the complex, projective variety, parametrizing flags of quotients of C of given ranks, say, nk > · · ·> n1. By a classical result of Ehresmann [E], the integral cohomology of F can be described geometrically as the free abelian group generated by the Schubert classes. These are the (Poincare duals of) fundamental classes of certain subvarieties w ⊂ F , one for each element of the subset S := S(n1, . . . ,nk) of the symmetric group Sn, consisting of permutations w with descents in {n1, . . . ,nk}. A description of the multiplicative structure is provided by yet another classical theorem, due to Borel [Bor], which gives a presentation forH ∗(F,Z). Specifically, let σ 1 1 , . . . ,σ 1 n1 ,σ 2 1 , . . . ,σ 2 n2−n1, . . . ,σ k+1 1 , . . . ,σ k+1 n−nk be n independent variables. Define An to be the block-diagonal matrix diag(D1,D2, . . . ,Dk+1), where
18043336
THEOREM FOR HOMOGENEOUS SPACES
We formulated a mirror-free approach to the mirror conjecture, namely, quantum hyperplane section conjecture, and proved it in the case of nonnegative complete intersections in homogeneous manifolds. For the proof we followed the scheme of Givental's proof of a mirror theorem for toric complete intersections.
15109303
On Hypergeometric Functions Connected with Quantum Cohomology of Flag Spaces
Abstract:The Givental recursion relations for hypergeometric series associated with equivariant quantum cohomology are computed for flags varieties G/B. A simple formula for these functions is obtained in the case G=SL(3).
2884104
A mirror theorem for toric complete intersections
We prove a generalized mirror conjecture for non-negative complete intersections in symplectic toric manifolds. Namely, we express solutions of the PDE system describing quantum cohomology of such a manifold in terms of suitable hypergeometric functions. Revision 03.03.97: we correct an error in Introduction.
119003295
Introduction to Toric Varieties.
Toric varieties are algebraic varieties arising from elementary geometric and combinatorial objects such as convex polytopes in Euclidean space with vertices on lattice points. Since many algebraic geometry notions such as singularities, birational maps, cycles, homology, intersection theory, and Riemann-Roch translate into simple facts about polytopes, toric varieties provide a marvelous source of examples in algebraic geometry. In the other direction, general facts from algebraic geometry have implications for such polytopes, such as to the problem of the number of lattice points they contain. In spite of the fact that toric varieties are very special in the spectrum of all algebraic varieties, they provide a remarkably useful testing ground for general theories.The aim of this mini-course is to develop the foundations of the study of toric varieties, with examples, and describe some of these relations and applications. The text concludes with Stanley's theorem characterizing the numbers of simplicies in each dimension in a convex simplicial polytope. Although some general theorems are quoted without proof, the concrete interpretations via simplicial geometry should make the text accessible to beginners in algebraic geometry.
119645049
Quantum cohomology of partial flag manifolds
$$F_{n_1 } \ldots _{n_k }$$

AbstractWe compute the quantum cohomology rings of the partial flag manifolds
$$F_{n_1 } \ldots _{n_k } = U(n)/(U(n_1 ) \times \cdots \times U(n_k ))$$
. The inductive computation uses the idea of Givental and Kim [1]. Also we define a notion of the vertical quantum cohomology ring of the algebraic bundle. For the flag bundle
$$F_{n_1 } \ldots _{n_k }$$
(E) associated with the vector bundleE this ring is found.
119680171
Quantum Cohomology of Partial Flag Manifolds and a Residue Formula for Their Intersection Parings
As a generalization of our previous paper [GK], we formulate a residue formula and some simple behaviors of equivariant quantum cohomology applying to compute the quantum cohomology of partial flag manifolds $F_{k_1,\cdots , k_l} $with a try to give a rigorous definition of equivariant quantum cohomology.