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The 'usual' Hirzebruch-Riemann-Roch theorem gives a topological expression for $\chi(E)$, where $E$ is a coherent sheaf on a smooth projective variety. Is there a generalisation of this giving a topological expression for $$\chi(E,F)=\sum_i (-1)^i \mathrm{Ext}^i(E,F)?$$ By definition $\chi(E)=\chi(\mathcal{O},E)$, and if $E$ is locally free, then of course $\chi(E,F)=\chi(\mathcal{Hom}(E,F))$, but in my case of interest $E$ is not locally free.

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If the variety $X$ is smooth, one has $$ \chi(E,F) = \deg\Big( \mathrm{ch}(E)^\vee \cdot \mathrm{ch}(F) \cdot \mathrm{td}_X \Big), $$ where $(-)^\vee$ is the involution of $\bigoplus H^{2i}(X,\mathbb{Q})$ that acts by $(-1)^i$ on the summand $H^{2i}(X,\mathbb{Q})$.

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  • $\begingroup$ Thanks! I guess the proof should be similar to the one for the 'usual' Riemann-Roch theorem, using that $\chi(E,F)$ is additive in $E$ as well? Is there a reference for this? $\endgroup$ – Bernie Nov 13 '19 at 14:50
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    $\begingroup$ Right, just use additivity. $\endgroup$ – Sasha Nov 13 '19 at 19:06

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