Abstract
The Gauss-Manin connection for nonabelian cohomology spaces is the isomonodromy flow. We write down explicitly the vector fields of the isomonodromy flow and calculate its induced vector fields on the associated graded space of the nonabelian Hogde filtration. The result turns out to be intimately related to the quadratic part of the Hitchin map.
Highlights
The Gauss-Manin connection for nonabelian cohomology spaces is the isomonodromy flow
If s ∈ S and Xs the fiber of π over s, X → S can be viewed as a variation over S of complex structures on the underlying differentiable manifold of Xs
If we denote as H1(X, G) the first Cech cohomology of X with coefficients the constant sheaf in G, ConnX can be naturally identified with H1(X, G), by considering the gluing data of flat G-bundles
Summary
If we use σi ∈ H0(X, adP ⊗ Ω1X ) to denote the global section associated to σi, the pair ((ηij )i,j∈I , (σi)i∈I ) is a hyper Cech 1-cochain on X with coefficients in AP −[−,→s] adP ⊗ Ω1X where the map [ , s] is defined as: if s = s′ ⊗ ω, where s′ ∈ H0(X, AP ), ω ∈ H0(X, Ω1X ), [ , s] := [ , s′] ⊗ ω − s′ ⊗ [p∗( ), ω]. To any Dǫ family of triples (Xǫ,Pǫ,∇ǫ) is associated a hyper 1-cochain ((ηij )i,j∈I , (σi)i∈I ) by the above discussion It is closed because of three facts: first, (ηij )i,j∈I is a closed Cech 1-cochain with coefficients in AP - it’s closed again because it comes from the transition function φj−1 ◦ φi; second, because of (2); third, the complex AP −[−,→s] adP ⊗ Ω1X has only two nonzero terms.
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