Question

I had this simple question when formulating the Todd class question.

Does there exist an example of proper morphism $f:X\to Y$ together with nontrivial homology class $t\in H^*(X)$ such that for all coherent sheaves on $X$ the equality $f_*(\mathop{\text{ch}}(u)\cdot t) = 0$ holds?

Answer 1

Why not just take a class that is orthogonal to all the algebraic classes on $X$? For instance you can take $Y$ to be a point, and take $X$ to be a generic abelian surface over $\mathbb{C}$, i.e. and abelian surface with $NS(X) = \mathbb{Z}$. The Chern character of any coherent sheaf on $X$ is contained in $H^{0}(X)\oplus NS(X)\oplus H^{4}(X)$. Take now any $t \in NS(X)^{\perp} \subset H^{2}(X)$, i.e. a transcendental cohomology class of degree two.

Answer 2

As Tony Pantev points out, it is easy to make an example by taking non-algebraic classes.

If you impose that $t$ is algebraic, and take $X=Y$, you are very close to stating Grothendieck's Conjecture D. Let $X$ be smooth and projective. The conjecture is that any algebraic class in $H^*(X)$, which is orthogonal to all algebraic classes, is zero torsion.