# Integrating hypercohomology classes

Let $$X$$ be a complex variety. By Poincare's lemma, its singular cohomology can be computed as hypercohomology of the holomorphic de Rham complex (viewing $$X$$ as a complex manifold) $$\text{H}^\cdot(X,\mathbf{C})\ \simeq\ \text{H}^\cdot(X,\Omega^\bullet).$$ By GAGA the right side is the same as the hypercohomology of the complex of algebraic de Rham complex.

In any case, you can compute the hypercohomology $$\text{H}^\cdot(X,\Omega^\bullet)$$ using Cech cohomology, in terms of an open cover $$\{U_\alpha\}$$ of $$X$$. For instance, if $$X$$ has dimension $$n$$, a representative of $$\text{H}^n(X,\Omega^\bullet)$$ is a collection of $$n$$ forms $$\omega_{n,\alpha}\ \in\ \text{H}^0(U_\alpha,\Omega^n)$$ which do not quite glue to a global $$n$$-form, but there are $$n-1$$ forms $$\omega_{n-1,\alpha,\beta}\ \in\ \text{H}^0(U_\alpha\cap U_\beta,\Omega^{n-1})$$ such that $$d\omega_{n-1,\alpha,\beta}=\omega_\alpha-\omega_\beta$$ on the intersection. Similarly, there are $$n-2$$ forms $$\omega_{n-2,\alpha,\beta,\gamma}$$ such that $$d\omega_{n-2,\alpha,\beta,\gamma}=\omega_{n-1,\alpha,\beta}-\omega_{n-1,\alpha,\gamma}+\omega_{n-1,\beta,\gamma}$$, and so on.

For example, on $$X=\mathbf{P}^1$$ with its standard cover by two affine lines, $$\text{H}^2(\mathbf{P}^1)$$ is generated by the meromorphic differential form $$dx/x\ \in\ \text{H}^0(\mathbf{A}^1_0\cap\mathbf{A}^1_\infty, \Omega^1).$$

Question: Assume that $$X$$ is smooth and proper. We then know that an element of $$\text{H}^\cdot(X,\mathbf{C}) \simeq \text{H}^\cdot(X,\Omega^\bullet)$$ can be integrated. What does integration look like in terms of these hypercohomology classes?

Looking at the $$\mathbf{P}^1$$ example, one might guess that the answer has something interesting to do with residues.

One way to see this is algebraically. Let $$\omega_X = \Omega^n_X$$. The map you seek to describe is $$\int_X \colon \mathrm{H}^n(X,\omega_X) \to \mathbb{C}$$ Abstractly, it arises as the counit of the Serre duality adjunction $$\mathrm{H}^n(X,-) \dashv -\otimes \omega_X$$ By the compatibility between local and global duality, there is a commutative diagram $$\require{AMScd}$$ $$\begin{CD} \otimes_P \mathrm{H}^n_P(\omega_X)@>can>> \mathrm{H}^n(X,\omega_X)\\ @V \otimes_P \mathrm{res}_P V V @VV\int_XV\\ \mathbb{C} @= \mathbb{C} \end{CD}$$ and the maps $$\mathrm{res}_P$$ that arise by local duality, are computed through higher dimensional residues. This is explained with great detail in J. Lipman's blue book:

Dualizing sheaves, differentials and residues on algebraic varieties. Astérisque, No. 117 (1984).

For the comparison between the algebraic and the analytic case and the $$\frac{1}{(2 \pi i)^n}$$ factor (up to a sign), see

Sastry & Tong, The Grothendieck Trace and the de Rham Integral, Canad. Math. Bull. Vol. 46 (3), 2003 pp. 429–440.

To make things more explicit, you may assume that your covering $$\{U_i\}_{i \in I}$$ is finite and each $$U_i$$ is the complement of a certain section $$f_i$$ of a line bundle. In this case a Czech cocycle is defined by a residual symbol $$\left[ \frac{ w } { f_0 \dots f_n } \right]$$ with $$w$$ a meromorphic $$n$$-form. This is described on page 195 of

Hartshorne, Residues and duality. Lecture Notes in Mathematics, 20, Springer-Verlag, 1966.

So the objective is to compute $$\int_X \left[ \frac{ w } { f_0 \dots f_n } \right]$$

The Czech-cocycle/symbol induces a certain element

$$\left[ \frac{ w } { f_0 \dots \hat{f_i} \dots f_n } \right]_P$$ in $$\mathrm{H}^n_P(\omega_X)$$ where $$p\in U_i$$ so $$(f_i)_P$$ is invertible in $$\mathcal{O}_{X,P}$$. This element is describable using a Koszul complex to compute $$\mathrm{H}^n_P(\omega_X)$$. Finally the computation of the integral is reduced to compute $$\mathrm{res}_P \left[ \frac{ w } { f_0 \dots \hat{f_i} \dots f_n } \right]_P$$. This can be done explicitly, over the points $$P$$ that are poles of the differential form $$w$$. Locally at $$P$$ $$w = \phi f_0 \wedge \hat{f_i} \wedge f_n$$ with $$\phi$$ in the fraction field of $$\mathcal{O}_{X,P}$$. Then, $$\mathrm{res}_P \left[ \frac{ w } { f_0 \dots \hat{f_i} \dots f_n } \right]_P = a_{(-1, \dots, -1)}$$ where $$a_{(-1, \dots, -1)} \in \mathbb{C}$$ denotes the corresponding coefficient in the Laurent expansion of $$\phi$$. This is described using the identification of $$\widehat{\mathcal{O}}_{X,P}$$ with the power series ring $$\mathbb{C}[[f_0 \dots \hat{f_i} \dots f_n]]$$.

• Sorry, I'm not sure if I understand - does this answer tell you explicitly how to integrate a Cech cocyle?
– Meow
Oct 26 '21 at 20:47
• Notice that you may represent $\mathrm{H}^n(X, \omega_X)$ as a certain quotient of $\check{C}^n(X,\omega_X)$ i.e. of $\prod_\alpha\Gamma(U_\alpha,\omega_X)$, and analogously for $H^n_P(\omega_X)$.Then, you use the diagram I mention. It is not immediate to unravel all definitions but it can be done in a relatively explicit way under the smoothness hypothesis. Oct 26 '21 at 21:13
• @Meow, I hope the edit makes all of this more explicit. Oct 27 '21 at 12:05
• Sorry, I don't think this is correct. We have $\omega_{n,\alpha}\vert_{U_\alpha\cap U_\beta}=\omega_{n,\beta}\vert_{U_\alpha\cap U_\beta}+d\omega_{n-1,\alpha,\beta}\vert_{U_\alpha\cap U_\beta}$, where here a Cech representative of a hypercohomology class of $\Omega^\bullet$ has the form $\{(\omega_{k,\alpha_1,...,\alpha_k}): k=0,...,n, \alpha_1,...,\alpha_k\in I\}$ where $I$ is the index set of the cover.
– Meow
Oct 28 '21 at 10:27
• Ok, you are right, but the difference is a closed form whose integral vanishes. Oct 28 '21 at 12:09