Author: Eiko

Tags: Mathematics, Algebraic Geometry, Connection, Gauss-Manin Connection, Differential Equation, Picard-Fuchs Equation, De-Rham Cohomology, Relative De-Rham Cohomology, Hodge Theory

Time: 2024-09-01 12:00:00 - 2025-01-16 17:19:21 (UTC)

Relative De Rham Cohomology

Let $\pi :X\to S$ be a smooth $k$-morphism of smooth $k$-schemes. We define the relative de Rham cohomology sheaf on $S$ is defined as the hypercohomology sheaf

$${\mathcal{H}}_{dR}^q(X/S):=R^q{\pi }_{\ast }{\mathrm{\Omega }}_{X/S}^{\bullet }$$

which is the cohomology of the derived complex $R{\pi }_{\ast }{\mathrm{\Omega }}_{X/S}^{\bullet }\in D^b(S)$, i.e. 

$${\mathcal{H}}_{dR}(X/S)=H(R{\pi }_{\ast }{\mathrm{\Omega }}_{X/S}^{\bullet })\in D^b(S).$$

Here ${\mathrm{\Omega }}_{X/S}^{\bullet }$ is the relative de Rham complex of sheaves on $X$.

The Gauss-Manin connection claims that there is a natural connection on ${\mathcal{H}}_{dR}^q(X/S)$, which connects its fibres $H_{dR}^q(X_t/K)$ as a connection on $S$.

Computing the Relative De Rham Cohomology

Typically the de Rham cohomology need to be computed with a Cech resolution.

Affine Case Has Acyclicity

If $Y$ is separated and $\pi :X\to Y$ is affine, then by Serre’s theorem, each sheaf ${\mathrm{\Omega }}_{X/Y}^i$ is acyclic for ${\pi }_{\ast }$, so ${\mathrm{\Omega }}_{X/Y}^{\bullet }$ actually becomes an acyclic resolution, no need to resolve it again. This means

$${\mathcal{H}}_{dR}^{\bullet }(X/Y)=H(R{\pi }_{\ast }{\mathrm{\Omega }}_{X/Y}^{\bullet })=H({\pi }_{\ast }{\mathrm{\Omega }}_{X/Y}^{\bullet }).$$

General Case Uses Cech Resolution

For the general separated morphism $\pi :X\to Y$, for simplicity let’s assume $Y$ to be affine. Then we need an affine cover of $U_i\to X$ such that $U_i\to Y$ is affine (which is automatic in our case). This means each $(i_J{)}_{\ast }{i}_J^{\ast }{\mathrm{\Omega }}_{X/Y}^p$ is acyclic for ${\pi }_{\ast }$. Here $i_J:U_J\hookrightarrow X$ is the inclusion of the open set $U_J={\cap }_{j\in J}{U}_j$.

We obtain a Cech complex (which is a ${\pi }_{\ast }$-acyclic resolution) for each ${\mathrm{\Omega }}_{X/Y}^p$,

$$E_0^{pq}={\check{C}}^q(U_{\bullet },{\mathrm{\Omega }}_{X/Y}^p)=\underset{|J|=q+1}{⨁}(i_J{)}_{\ast }{i}_J^{\ast }{\mathrm{\Omega }}_{X/Y}^p,$$

and the cohomology of this total complex

$$T^n=\underset{p+q=n}{⨁}{\check{C}}^q(U_{\bullet },{\mathrm{\Omega }}_{X/Y}^p),$$ $${\mathcal{H}}_{dR}^{\bullet }(X/Y)=H(T^{\bullet })$$

is the relative de Rham cohomology, which is also the spectral sequence associated to the double complex $E_0^{pq}$, whose first page is the vertical cohomologies, and is clearly

$$E_1^{pq}=H^q(R{\pi }_{\ast }{\mathrm{\Omega }}_{X/Y}^p)\Rightarrow {\mathcal{H}}_{dR}^{p+q}(X/Y).$$

Filtration on the total differentials ${\mathrm{\Omega }}_{X/k}^{\bullet }$

The smoothness implies the exact sequence $0\to {\pi }^{\ast }({\mathrm{\Omega }}_{S/k}^1)\to {\mathrm{\Omega }}_{X/k}^1\to {\mathrm{\Omega }}_{X/S}^1\to 0$. Which says the relative $1$-differentials are exactly those total $1$-differentials on $X$ quotient by those $1$-differentials on $S$. The ${\pi }^{\ast }$ functor is natural and necessary here because ${\mathrm{\Omega }}_S^1$ is not a sheaf on $X$.

The total differential complex ${\mathrm{\Omega }}_{X/k}^{\bullet }$ (which may also be viewed as an anti-commutative algebra) admits a filtration (of ideals! owo) given by counting the number of terms that essentially coming from $S$,

$$F^p{\mathrm{\Omega }}_{X/k}^{\bullet }=\mathrm{Image}({\mathrm{\Omega }}_{X/k}^{\bullet }[-p]{\otimes }_{{\mathcal{O}}_X}{\pi }^{\ast }({\mathrm{\Omega }}_{S/k}^p)\to {\mathrm{\Omega }}_{X/k}^{\bullet })$$

i.e. $F^p$ is the differentials where at least $p$-terms come from $S$. The grading shift $[-p]$ is necessary to match up the degrees of differentials, you can think of it as coming from ${\mathrm{\Omega }}_{S/k}^p[-p]$. This is a descending filtration

$${\mathrm{\Omega }}_{X/k}^{\bullet }=F^0{\mathrm{\Omega }}_{X/k}^{\bullet }\supset F^1{\mathrm{\Omega }}_{X/k}^{\bullet }\supset F^2{\mathrm{\Omega }}_{X/k}^{\bullet }\supset \cdots$$

where $m>n\Rightarrow F^m{\mathrm{\Omega }}_{X/k}^n=0$, whose graded terms are

$${\mathrm{Gr}}^p{\mathrm{\Omega }}_{X/k}^{\bullet }=F^p/F^{p+1}={\mathrm{\Omega }}_{X/S}^{\bullet }[-p]{\otimes }_{{\mathcal{O}}_X}{\pi }^{\ast }({\mathrm{\Omega }}_{S/k}^p),$$

for example

$$\begin{array}{rl}{\mathrm{Gr}}^0({\mathrm{\Omega }}_{X/k}^{\bullet })& ={\mathrm{\Omega }}_{X/k}^{\bullet }/F^1{\mathrm{\Omega }}_{X/k}^{\bullet }\\ & ={\mathrm{\Omega }}_{X/k}^{\bullet }/\{\text{Anything containing a term from }{\pi }^{\ast }{\mathrm{\Omega }}_{S/k}^{\bullet }\}\\ & \cong {\mathrm{\Omega }}_{X/S}^{\bullet }\end{array}$$

Note that the graded pieces ${\mathrm{Gr}}^p$ has cohomology starting at degree $p$. This hints us to use the spectral sequence associated to the filtration of $R{\pi }_{\ast }{\mathrm{\Omega }}_{X/k}^{\bullet }$, $E_0^{p\bullet }={\pi }_{\ast }{\mathrm{Gr}}^p{\mathrm{\Omega }}_{X/k}^{\bullet }[p]$, so

$$\begin{array}{rl}{E}_1^{pq}& =R^q({\pi }_{\ast }{\mathrm{Gr}}^p[p])\\ & =R^q({\pi }_{\ast }({\mathrm{\Omega }}_{X/S}^{\bullet }[-p][p]{\otimes }_{{\mathcal{O}}_X}{\pi }^{\ast }{\mathrm{\Omega }}_{S/k}^p))\\ & =R^q({\pi }_{\ast }({\mathrm{\Omega }}_{X/S}^{\bullet }{\otimes }_{{\mathcal{O}}_X}{\pi }^{\ast }{\mathrm{\Omega }}_{S/k}^p))\\ & =R^q({\pi }_{\ast }{\mathrm{\Omega }}_{X/S}^{\bullet }{\otimes }_{{\mathcal{O}}_S}{\mathrm{\Omega }}_{S/k}^p)\\ & =R^q({\pi }_{\ast }{\mathrm{\Omega }}_{X/S}^{\bullet }){\otimes }_{{\mathcal{O}}_S}{\mathrm{\Omega }}_{S/k}^p\\ & ={\mathcal{H}}_{dR}^q(X/S){\otimes }_{{\mathcal{O}}_S}{\mathrm{\Omega }}_{S/k}^p.\end{array}$$

Here we used the fact that the differential in ${\mathrm{\Omega }}_{X/S}^{\bullet }$ is ${\pi }^{-1}{\mathcal{O}}_S$-linear and that ${\mathrm{\Omega }}_{S/k}^p$ is locally free.

Exterior Product

There is a product

$$E_r^{pq}\times E_r^{p^{\prime }{q}^{\prime }}\to E_r^{p+p^{\prime },q+q^{\prime }}$$

$$(e,e^{\prime })\mapsto e\cdot e^{\prime }$$

with

$$\begin{array}{rl}e\cdot e^{\prime }& =(-1{)}^{(p+q)(p^{\prime }+q^{\prime })}{e}^{\prime }\cdot e,\\ d_r(e\cdot e^{\prime })& =d_r(e)\cdot e^{\prime }+(-1{)}^{p+q}e\cdot d_r(e^{\prime }).\end{array}$$

The Gauss-Manin Connection

Gauss-Manin connection is a natural connection associated with a family $X\to S$ on the ${\mathcal{H}}_{dR}^q(X/S)$ of relative de Rham cohomology sheaves. It can be obtained by the above spectral sequence associated to its filtrations induced by pulling back differentials from $S$.

The differential $d_1$ in $E_1$ actually gives the Gauss-Manin connection, writing it out we have

$$d_1^{pq}:E_1^{pq}=R^{p+q}{\pi }_{\ast }({\mathrm{Gr}}^p)\to R^{p+1+q}{\pi }_{\ast }{\mathrm{Gr}}^{p+1}=E_1^{p+1,q}$$

or

$${\mathcal{H}}_{dR}^q(X/S){\otimes }_{{\mathcal{O}}_S}{\mathrm{\Omega }}_{S/k}^p\to {\mathcal{H}}_{dR}^q(X/S){\otimes }_{{\mathcal{O}}_S}{\mathrm{\Omega }}_{S/k}^{p+1}.$$

And this is the connecting homomorphism of the exact sequence by applying the cohomology functors $R^q{\pi }_{\ast }\circ [p]$

$$0\to {\mathrm{Gr}}^{p+1}\to F^p/F^{p+2}\to {\mathrm{Gr}}^p\to 0$$

References

• Katz, Oda, On the differentiation of De Rham cohomology classes with respect to parameters.