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Let $\mathcal{M}_g$ be the moduli space of curves of genus $g$. Consider the holomorphic bundle $\mathcal{H}^k\rightarrow\mathcal{M}_g$ whose fiber over a curve $C\in\mathcal{M}_g$ is the space of holomorphic $k$-differentials $H^0(C,K_C^k)$. This is often called the Hodge bundle.

My question is, is there a natural (flat or not) connection on $\mathcal{H}^k$?

Even for $k=1$, I don't find any reference by internet search. We do have a Gauss-Mannin connection on the bundle $E\rightarrow\mathcal{M}_g$ with fibers $E\,\big|_C=H^1(C,\mathbb{C})=H^{1,0}(C)\oplus H^{0,1}(C)=H^0(C,K_C)\oplus H^0(C,K_C)^*$, which however does not preserve the splitting.

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What do you mean by "connection". If you consider $\mathcal{M}_g$ as a complex orbifold with the "classical topology", then you can just choose a Riemannian metric and you have the associated Levi-Civita connection. Are you looking for a connection with some particular property? – Jason Starr Nov 20 '13 at 13:23
Yeah! I am looking for one with some particular property...but since this vector bundle is the daily issue for people working on moduli of curves, I just wanted to know if there is some connection well-known to them, but now it seems not. – Xin Nie Nov 21 '13 at 15:03
up vote 1 down vote accepted

The fourth power of the Hodge bundle is isomorphic to the $E_8$ conformal block bundle at level one, and for that bundle (or rather its projectivization) you have the Hitchin/KZ/WZW connection, which is projectively flat.

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There's no flat connection because the Chern classes are nonzero rationally. The Chern classes of the bundles $\mathcal H^k$ were actually discussed just yesterday.

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Just to add one observation: every holomorphic connection on a locally free sheaf over a smooth projective variety is automatically flat. Using the Satake compactification, there are complete curves in $\mathcal{M}_g$ containing a general point. So the restriction of the connection over such a curve would be flat. However, the first Chern class of $\mathcal{H}^k$ is a nonzero multiple of $\lambda$, which is nonzero on every complete curve in $\mathcal{M}_g$. Therefore, there can be no holomorphic connection on $\mathcal{H}^k$. – Jason Starr Nov 20 '13 at 13:21
@Jason, could you give some more explanations on the first sentence, preferably in differential-geometric terms? I can't see it locally.. – Xin Nie Nov 21 '13 at 15:14

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