Author: Eiko

Time: 2024-10-14 11:29:31 - 2024-10-27 06:17:14 (UTC)

Section To A Connection, With Gauss Manin

Fibrewise

$$0\to H_{dR}^1(X_t/K{)}^{\ast }\otimes {\mathcal{O}}_{X_t}\to \mathcal{E}\to {\mathcal{O}}_{X_t}\to 0$$

These are just extensions for trivial by trivial (no connections on the first and third objects).

This encodes the Coleman integral structures on $X_t$, as when you choose a basis $\underset{―}{\omega }=({\omega }_i)$ for $H_{dR}^1(X_t/K)$ (giving a basis for $V$ is the same as giving a basis for $V^{\ast }$), define a connection on $\mathcal{E}$ as

$$\left(\begin{array}{cc}0& \underset{―}{\omega }\\ 0& 0\end{array}\right)$$

and the flat section represents integrations. Say $(0,1{)}^t$ is a vector in the fibre of $\mathcal{E}$ at point $b$, then the flat section is $({\int }_b^x\underset{―}{\omega },1{)}^t$.

Globalize

We want to find the global version of this,

the global version of de-Rham is the relative de-Rham with Gauss-Manin connection $(\mathcal{V},\mathrm{\nabla })$

$$0\to {\pi }^{\ast }(\mathcal{V},\mathrm{\nabla }{)}^{\ast }\to \mathcal{E}\to {\mathcal{O}}_X\to 0.$$

Whose fibrewise version is the above.

The connection defined on $\mathcal{E}$ is given by

$$\left(\begin{array}{cc}\mathrm{\Theta }& {\pi }^{\ast }\underset{―}{\omega }\\ 0& 0\end{array}\right)$$

where $\mathrm{\Theta }$ is the dual (transpose) of the Gauss-Manin connection.

Flat Connection

Recall that a connection $\mathrm{\nabla }:\mathcal{E}\to {\mathrm{\Omega }}_X^1\otimes \mathcal{E}$ given by matrix $d\cdot I_n+\mathrm{\Lambda }\in \mathrm{Hom}({\mathcal{O}}^n,({\mathrm{\Omega }}^1{)}^n)$

For a connection to be flat, it means ${\mathrm{\nabla }}^2:\mathcal{E}\to {\mathrm{\Omega }}_X^2\otimes \mathcal{E}$ is zero, i.e. the curvature is zero.

• The connection is flat/integrable iff the map ${\mathrm{SDer}}_k({\mathcal{O}}_S)\to {\mathrm{SEnd}}_k(\mathcal{E}):Y\mapsto {\mathrm{\nabla }}_Y$ is a Lie algebra homomorphism.

In terms of matrix $\mathrm{\nabla }=d\cdot I_n+\mathrm{\Lambda }$, as we have seen in coordinate form of connections, the flatness condition $\mathrm{\nabla }\circ \mathrm{\nabla }=0$ expands to $d\mathrm{\Lambda }+\mathrm{\Lambda }\wedge \mathrm{\Lambda }=0$.