Analog of holomorphic Lefschetz fixed point theorem for smooth algebraic varieties

Question

If $X$ is a compact complex manifold and $f: X \to X$ is a holomorphic map with isolated nondegenerate zeroes. Then there is a version of Lefschetz fixed point formula with traces on Dolbeaut cohomology $$ \sum_{f(z)=z} \frac{1}{\det(\operatorname{id}-df(z))}=\sum_q (-1)^q \operatorname{tr}(f^*|H^{0,q}_{\overline{\partial}}(X)) $$ Dolbeaut isomorphism $$ H^q(X, \Omega^p) \cong H^{p.q}_{\overline{\partial}}(X) $$ allow us to write the formula in the following way $$ \sum_{f(z)=z} \frac{1}{\det(\operatorname{id}-df(z))}=\sum_q (-1)^q \operatorname{tr}(f^*|H^{q}(X,\mathcal{O}_X)). $$ The last formula make sense for, say, smooth projective variety $X$ over algebraically closed field of characteristic zero. Two questions: is it still a valid formula for algebraic varieties? Where I can find a proof in such setting?

Answer 1

Please check Appendix A in the following notes of Lenny Taelman. He mentions the only published proof of Woods Hole formula: SGA 5, Illusie. But this proof is "rather convoluted," so he gives his own nice and short proof.

https://staff.fnwi.uva.nl/l.d.j.taelman/beijing.pdf

Here you will find some explanations about Grothendieck-Serre duality and much more!

Answer 2

This formula is called the Atiyah--Bott fixed point theorem, or the Woods Hole fixed point theorem.

My memory is that it is valid for smooth projective (or more generally proper) varieties in char. $p$ as well (if you think of it as a congruence mod $p$). But it does need a properness and smoothness assumption.

P.S. I think you can find a discussion of it in SGA 4 1/2 . (This is probably not the optimal reference; but I think it's where I first saw it. I forget if it is proved there, though, or just described.)