Reference for Gauss-Manin connection
Some basic references, in no particular order:
- Atiyah, Hirzebruch - Integrals of the second kind on an algebraic variety
- Grothendieck - On the de Rham cohomology of algebraic varieties
- Katz - On the differential equations satisfied by period matrices
- Katz, Oda - On the differentiation of De Rham cohomology classes with respect to parameters
- Manin - Algebraic curves over fields with differentiation (in Russian)
- Griffiths - Periods on integrals on algebraic manifolds
The basic idea behind the Gauss-Manin connection is actually very simple. Suppose that $f: X \to B$ is a proper map between manifolds, with $\dim X > \dim B$. Then generically, the fibers $X_b := f^{-1}(b)$ are smooth compact manifolds, and moreover by the Ehresmann fibration theorem they will be diffeomorphic (provided the set of regular values $B_{reg} \subseteq B$ is connected). In particular, they have isomorphic homology and cohomology.
Now suppose that $\alpha \in \Omega^k(X)$ such that the restriction $i_b^\ast \alpha \in \Omega^k(X_b)$ is closed. Then this gives a family of cohomology classes: $[i_b^\ast \alpha] \in H^k(X_b)$. Let $b_1, \ldots, b_n$ be a set of local coordinates in $B$. Then consider the classes of the form $$ \left[ i_b^\ast \left(\frac{\partial^{i_1+\cdots+i_n} \alpha}{\partial b_1^{i_1} \cdots \partial b_n^{i_n} } \right)\right] \in H^k(X_b) $$ By taking higher and higher derivatives if necessary, eventually the number of classes of this form will exceed the $k$th Betti number of $X_b$. Then necessarily, some linear combination of these classes must equal zero, i.e. the family of classes $[i_b^\ast \alpha]$ satisfies a linear PDE. This is the PDE encoded by the Gauss-Manin connection.
Let $π : X \to T$ be a smooth algebraic family of complex projective manifolds of dimension d such that the parameter space $T$ is a nonsingular variety. Consider the local systems $R^kπ_∗\mathbb C$, $0 ≤ k ≤ 2d$, and the associated vector bundles $\mathcal H^k:= (R^kπ_∗\mathbb C)\otimes_\mathbb C \mathcal O_T$ over $T$. These vector bundles are equipped with the Gauss–Manin connection.
See this Master thesis written in French