Let K be a field of characteristic 0 and X a smooth projective variety over K of dimension n. Then it is well-known that there is a trace map $tr: H^{2n}(X) = H^n(X,\Omega^n_X) \longrightarrow K$ which is an isomorphism.

I would like to know whether there is an explicit formulation of this isomorphism when H stands for algebraic de Rham cohomology, in terms of hypercocycles afforded by a Cech resolution.

Let me explain in more detail my question and what kind of answer I expect. Let $\mathcal U = \{ U_0, ..., U_n \}$ be an open affine cover of X. Then

$H^n(X,\Omega^n_X) = \frac{\Omega^n(U_0 \cap ... \cap U_n)}{\sum^n_{i=0} \Omega^n(U_0 \cap ... \overset{\hat{}}{U_i} ... \cap U_n)}$

and therefore we can represent elements of $H^n(X,\Omega^n_X)$ by classes of differential n-forms on X which are regular on the intersection $U_0 \cap ... \cap U_n$ of the n+1 chosen open affine subsets.

Question: Is there an explicit formulation of the isomorphism tr on this space?

If n=1 the answer is clear: $U_0=X\setminus \{p\}$, $U_1=X\setminus \{q\}$ where p and q are points on X and an easy application of Riemann-Roch shows that any differential 1-form on $X\setminus \{p,q\}$ can be represented (up to adding to it 1-forms which have a single pole at either p or q) by a form $\omega$ which has log poles at p and q. By the residue formula we have $res_p(\omega)+res_q(\omega)=0$ and therefore we can define

$tr(\omega) = (1/2)ยท(res_p(\omega)-res_q(\omega))$

which yields an isomorphism.

Can you give me a similar formulation for higher-dimensional varieties? (Or better yet, so that I don't bother you much, just give me a precise reference for this, which should be very classical?)

My impression is that one could do the following (so I am asking if each of the steps is true):

1) Prove that the open subsets $U_i$ can be chosen as $U_i=X\setminus D_i$ where $\sum_i D_i$ is a divisor with normal crossings.

2) Prove that any n-form on $U_0 \cap ... \cap U_n$ can be represented by a n-form $\omega$ with log poles at the divisors $D_i$.

3) Prove that the recursive rule $tr(\omega) = (1/n+1) \sum_i (-1)^i tr(res_{D_i}(\omega))$ yields the desired isomorphism.

Can you prove it explicitly at least for algebraic surfaces, using Riemann-Roch?

quotientof H^2(X), so they don't capture the whole thing. For p>2 differentials of the second kind are less well-behaved. But ok, H^p(X) is explicitly algebraically described in terms of hypercocycles, which is fine. I mainly ask if my suggestion (3) for the trace map is correct. – vic Nov 3 '11 at 13:51