I wish to understand the notion of ``Gass-Manin connection''. I have some understanding of differential geometry, topology and algebraic geometry. Where should I begin? IF the sources are freely available, that will be good. Even better will be if someone can give some little motivation for the concept. My aim is to understand it in the context of moduli of curves.


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.

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    $\begingroup$ By $\frac{\partial \alpha}{\partial b}$ do you mean lie derivative w.r.t. a horizontal lift of $\frac{\partial}{\partial b}$? Why is this independent of choice of connection? Is it the case that the cohomology class obtained is independent of the connection? $\endgroup$ – Saal Hardali Feb 1 '18 at 9:55

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


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