I had this simple question when formulating the [Todd class question](https://mathoverflow.net/questions/10630/why-todd-classes-appear-in-grothendieck-riemann-roch-formula).



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> 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?
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Homology class orthogonal to image of Chern characters?
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.


