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.


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 $H^{2i}(X,\mathbb{Q})$.

  • $\begingroup$ 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
  • 2
    $\begingroup$ Right, just use additivity. $\endgroup$ – Sasha Nov 13 '19 at 19:06

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.