Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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.

share|improve this question
    
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

2 Answers 2

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.

share|improve this answer

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.

share|improve this answer
3  
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

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.