Motivation:


Let $\pi\colon S \rightarrow B$ be smooth projective morphism of relative dimension 2 over a smooth projective scheme $B$. If the stucture sheaves of the fibres do not have higher cohomology, we have by GRR:


$$ \pi\_\* td(S/B) = ch(\mathcal{O}\_B) = 1$$


so push forwards of higher Todd classes are 0.


For trivial families of surfaces, we have obviously $td(S/B)\_i = 0$ for $i>2$.


Therefore, the following question is not entirely trivial:


Question:


For arbitrarily $i>0$, what are examples of smooth families of projective surfaces $\pi\colon S\rightarrow B$ with non-zero $\pi\_\* td(S/B)\_i$?



Example of a smooth family of projective surfaces with non-vanishing integrals of Todd classes
**Edit.** The odd Bernoulli numbers are zero, of course! So the first attempt below is wrong. The revised examples use complete subvarietes of moduli spaces of curves. These revised examples use nonvanishing of the **even** Bernoulli numbers.


**Revised examples from complete subvarieties of moduli spaces of curves.** Let $g\geq 2$ be an integer. Denote by $\mathcal{M}\_g$ the Deligne-Mumford stack over $\text{Spec}\ \mathbb{C}$ parameterizing families of smooth, projective, geometrically connected curves of genus $g$. Denote the universal curve as $$\pi:\mathcal{C}\_g \to \mathcal{M}\_g.$$ Denote by $\omega\_{\pi}$ the relative dualizing sheaf on $\mathcal{C}\_g$.


**Definition.** The **Hodge bundle** is the pushforward of the relative dualizing sheaf to $\mathcal{M}\_g$, $$\mathbf{E}\_g := \pi\_\* \omega\_{\pi}.$$ For every integer $r\geq 0$, the **kappa class of degree $r$** is the (cohomological) cycle class $$\kappa\_r = \pi\_\*(c\_1(\omega\_{\pi})^{r+1})\in \text{CH}^{\ r}(\mathcal{M}\_g).$$


The Hodge bundle is a rank $g$, locally free $\mathcal{O}\_{\mathcal{M}\_g}$-module (for the étale site with the usual structure sheaf). The kappa classes are elements in the Chow groups of the stack; the tensor product of these Chow groups with $\mathbb{Q}$ equal the associated $\mathbb{Q}$-Chow groups of the quasi-projective coarse moduli space $M\_g$ by work of Angelo Vistoli. Using the Grothendieck-Riemann-Roch formula, Mumford computed the Chern character of the Hodge bundle on the compactified moduli space $\overline{\mathcal{M}}\_g$. 


MR0717614 (85j:14046)   

Mumford, David   

Towards an enumerative geometry of the moduli space of curves.   

Arithmetic and geometry, Vol. II, 271–328,   

Progr. Math., 36, Birkhäuser Boston, Boston, MA, 1983.   

<http://www.dam.brown.edu/people/mumford/alg_geom/papers/1983b--EnumGeomModuli-NC.pdf>


Restricting to the open substack $\mathcal{M}\_g$ of $\overline{\mathcal{M}\_g}$ simplifies Mumford's formula.


**Theorem** [Mumford, Formula (5.2), p. 304, *Towards an enumerative geometry ...*] For every integer $s\geq 1$, $\text{ch}\_{2s}(\mathbf{E}\_g)$ is torsion, and $$\text{ch}\_{2s-1}(\mathbf{E}\_g) =(-1)^{s+1}\frac{|B\_{2s}|}{(2s)!}\kappa\_{2s-1}\in \text{CH}^{2s-1}(\mathcal{M}\_g)\otimes \mathbb{Q},$$ where $B\_{2s}$ is the Bernoulli number (nonzero since $2s$ is even). 


**Corollary**. The $\mathbb{Q}$-cycle class of the Chern character of $R\pi\_\*\mathcal{O}\_{\mathcal{C}\_g}$ equals $$(1-g) + \sum\_{s\geq 1}(-1)^{s+1} \frac{|B\_{2s}|}{(2s)!} \kappa\_{2s-1}.$$


**Proof.** By relative duality, the K-theory class of $R\pi\_\*\mathcal{O}\_{\mathcal{C}\_g}$ equals $1-[\mathbf{E}\_g^\vee]$. Thus, the Chern character of $R\pi\_\*\mathcal{O}\_{\mathcal{C}\_g}$ equals $$(1-g) + \sum\_{r\geq 1}(-1)^{r+1}\text{ch}\_r(\mathbf{E}\_g).$$ **QED**


Let $f:V\to \mathcal{M}\_g$ be a $1$-morphism that is generically finite to its image from an integral, projective $\mathbb{C}$-scheme of dimension $r$. The following theorem is discussed in the book of Harris and Morrison (one version of the theorem is attributed to Arakelov-Raynaud-Szpiro, but I have not been able to confirm this).


MR1631825 (99g:14031)   

Harris, Joe; Morrison, Ian   

Moduli of curves.   
 
Graduate Texts in Mathematics, 187. Springer-Verlag, New York, 1998. xiv+366 pp.   



**Theorem.** [cf. Harris-Morrison, Theorem 6.33, *Moduli of Curves*] For every irreducible, proper substack of $\mathcal{M}\_g$, the restriction of $\omega\_\pi$ to the inverse image under $\pi$ of the proper substack is "ample". In particular, the pullback to $V$ of $\pi\_\*(c\_1(\omega\_\pi)^{r+1})$ is a zero-cycle of positive (rational) degree. 


As always for Deligne-Mumford stacks, an invertible sheaf is "ample" if every sufficiently positive and divisible tensor power of the invertible sheaf is isomorphic to the pullback from the coarse moduli space of an ample invertible sheaf (in the usual sense). 


**Corollary.** If $r$ is odd, $r=2s-1$, then the pullback to $V$ of $\text{ch}\_{2s-1}(R\pi\_\*\mathcal{O}\_{\mathcal{C}\_g})$ is a zero-cycle of nonzero degree.


Kodaira constructed complete curves in $M\_g$. This was generalized to higher-dimensional complete subvarieties of $M\_g$ by E. Y. Miller in the 1980s. I found a very readable reference for higher-dimensional complete subvarieties written by Zaal.


Christiaan Zaal   

Complete subvarieties of moduli spaces of algebraic curves.   

Thesis, Universiteit von Amsterdam   

<https://pure.uva.nl/ws/files/3915897/36235_Thesis.pdf>


**Theorem.** [Kodaira, Miller, Zaal] For every integer $n\geq 0$, there exists an integer $g(n)\geq 2$ and a $1$-morphism $f:V\to \mathcal{M}\_{g(n)}$ that is generically finite to its image from an integral, projective, $\mathbb{C}$-scheme $V$ of dimension $n$.


(In the opposite direction, I believe that the theorem of Steven Diaz remains the best upper bound on the maximal dimension $n(g)$ of a complete subvariety of $M\_g$.)


Denote by $\pi\_f:\mathcal{C}\_f\to V$ the pullback by $f$ of the universal family of curves $\mathcal{C}\_g$. Then for every integer $s\geq 1$ with $2s-1\leq n$, $\text{ch}\_{2s-1}(R(\pi\_f)\_\*\mathcal{O}\_{\mathcal{C}\_f})$ is nonzero, and the even degree graded pieces of the Chern character (of degree $\geq 2$) are zero.


In particular, there exists an integer $g(1)\geq 2$ and there exists a generically finite $1$-morphism, $$e:W\to \mathcal{M}\_{g(1)},$$ from a smooth, projective $\mathbb{C}$-curve $W$. For $V$ as above of dimension $n$, denote the $(n+1)$-dimensional product $V\times W$ by $B$. Denote the product $\mathcal{C}\_f\times \mathcal{C}\_e$ by $S$. Denote by $\rho:S\to B$ the morphism $\pi\_f\times \pi\_e$. As a fiber product of smooth, projective morphisms of relative dimension $1$, the morphism $\rho$ is smooth and projective of relative dimension $2$. 


**Corollary.** For every integer $r$ with $1\leq r \leq n+1$, the cycle class $\text{ch}\_r(\rho\_\*\mathcal{O}\_S)$ has nonnegative degree on some irreducible, closed subvariety of $B$ of dimension $r$.


**Proof.** By Künneth's formula, $$\rho\_\*\mathcal{O}\_S = \text{pr}\_V^\*((\pi\_f)\_\*\mathcal{O}\_{\mathcal{C}\_f})\otimes \text{pr}\_W^\*((\pi\_e)\_\*\mathcal{O}\_{\mathcal{C}\_e}).$$ By the multiplicative property of the Chern character, for every integer $s\geq 1$, $$\text{ch}\_{2s-1}(\rho\_\*\mathcal{O}\_S) = (1-g(1))\text{pr}\_V^\*\text{ch}\_{2s-1}((\pi\_f)\_\*\mathcal{O}\_{\mathcal{C}\_f}),$$ and this has nonzero degree on irreducible closed subvarieties of the form $X\times\{w\_0\}$ of dimension $2s-1$. Also, $$ \text{ch}\_{2s}(\rho\_\*\mathcal{O}\_S) = \text{pr}\_V^\*\text{ch}\_{2s-1}((\pi\_f)\_\*\mathcal{O}\_{\mathcal{C}\_f})\cup \text{pr}\_W^\*\text{ch}\_1((\pi\_e)\_\*\mathcal{O}\_{\mathcal{C}\_e}).$$ This has nonzero degree on irreducible closed subvarieties of the form $X\times W$ of dimension $2s=(2s-1)+1$. **QED**


**First attempt with ridiculous error because the odd Bernoulli numbers equal zero.**
~~You can create many such examples with isotrivial families. The simplest isotrivial morphism is a smooth, projective morphism of relative dimension $1$ whose geometric fibers are isomorphic to the projective line, $$\rho:C\to X.$$ The pushforward of $T\_\rho =\textit{Hom}\_{\mathcal{O}\_C}(\Omega\_\rho,\mathcal{O}\_C)$ is a locally free $\mathcal{O}\_X$-module of rank $3$ whose total Chern class equals $1-a$ for a cycle class $a$ of degree $2$. For a locally free $\mathcal{O}\_X$-module $E$ of rank $2$, if $C/X$ represents the functor on $X$-schemes of invertible quotients of the pullback of $E$, then the class $a$ equals $$a = c\_1(E)^2 - 4c\_2(E).$$ Up to scaling and torsion, this is the unique polynomial expression in Chern classes of $E$ of cohomological degree $2$ that is compatible with arbitrary pullback and that is preserved by the operation of tensoring $E$ with invertible sheaves.~~


~~The first Chern class $c = c\_1(T\_\rho)$ has square equal to a pullback, $$c^2 = \rho^\* a.$$ Thus, the relative Todd class of $\rho$ equals $$\text{Td}(\rho) = [C]\cap\left(\sum\_{m\geq 0} \frac{B\_{2m}}{(2m)!}\rho^\*(a^m) - \sum\_{m\geq 0}\frac{B\_{2m+1}}{(2m+1)!}c\cup \rho^\*(a^m) \right). $$ Thus, the pushforward of the Todd class equals, $$\rho\_\*\text{Td}(\rho) = -2[X]\cap \sum\_{m\geq 0} \frac{B\_{2m+1}}{(2m+1)!} a^m.$$~~


~~In particular, for a second smooth, projective morphism of relative dimension $1$, $$\sigma:X\to B,$$ for $S$ defined to be $C$, and for the composition $\pi=\sigma\circ \rho$, the pushforward of the relative Todd class equals $$\pi\_\*\text{Td}(\pi) = \sigma\_\*\left(-2\text{Td}(\sigma)\cap \sum\_{m\geq 0}\frac{B\_{2m+1}}{(2m+1)!} a^m \right).$$ In the very simplest case that $X$ equals $\mathbb{P}^1\_B$, $\sigma$ is the natural projection, and for any choice of constant cross-section $s:B\to \mathbb{P}^1\_B$ (the $\infty$-section, for instance), this gives $$\pi\_\*\text{Td}(\pi) = -[B]\cap \left(\sum\_{m\geq 0} \frac{B\_{2m+1}}{(2m+1)!} (s^\*a)^m +2\sum\_{m\geq 0} \frac{B\_{2m+1}}{(2m+1)!}\sigma\_\*(a^m) \right).$$ To be very explict, if $E$ is the locally free sheaf $\sigma^\*\mathcal{L}\oplus \mathcal{O}\_{\mathbb{P}^1}(1)$ for some choice of invertible $\mathcal{O}\_B$-module $\mathcal{L}$ with first Chern class $c\_1(\mathcal{L})=\lambda$, this evaluates to $$\pi\_\*\text{Td}(\pi) = -[B]\cap \left( \sum\_{m\geq 0} \frac{B\_{2m+1}}{(2m+1)!}\lambda^{2m} + 4\sum\_{m\geq 1} \frac{B\_{2m+1}}{(2m+1)!} m\lambda^{2m-1}\right).$$ If $\mathcal{L}$ is ample, then $\lambda^r$ is nonzero for every integer $r=0,\dots,\text{dim}(B).$~~


