Connection on line bundle on projective curve - MathOverflow

Connection on line bundle on projective curve

2010-04-23T17:13:27Z

Let $C$ be a smooth projective curve. It is known that a line bundle on $C$ is of degree 0, if we can impose a connection structure on it.

Now my question is: Given a line bundle $L$ of degree 0, if there exist connection structure on $L$.

When $C$ is projective line, the question is obvious, how about when the genus is greater than 0?

Answer by David Ben-Zvi for Connection on line bundle on projective curve

2010-04-23T18:04:19Z

Yes (we're over C yes?)- the space of line bundles with connection forms a torsor for the cotangent bundle over the Jacobian. The class of this torsor (as an element in $H^1(Jac,\Omega^1)=H^{1,1}$) is the Chern class of the theta line bundle. In fact one can construct canonical connections on generic line bundles of degree zero on a curve (in fact line bundles outside the theta divisor), once you give yourself the choice of a theta characteristic on the curve -- this is the theory of the "prime form" or Szego kernel (it's much much older, but I think a fairly easy "modern" exposition of this and the nonabelian version is in here).

[EDIT: Nonalgebraically it is very easy to see this assertion from the Hodge theorem: the space of line bundles with a flat connection is $H^1(X,C^\times)$, which can be identified with the product $H^1(X,O^\times)^{\circ}\times H^0(X,\Omega^1)$ just by exponentiating the Hodge theorem for H^1. Note that ANY holomorphic connection on a Riemann surface/algebraic curve is flat for dimension reason - there are no holomorphic 2-forms that could serve as curvature.. For higher rank bundles the analogous result is the topic of the Corlette-Simpson nonabelian Hodge theorem.]

More generally Andre Weil proved that a rank n bundle on a curve /C admits a connection if and only if every indecomposable summand is a vector bundle of degree zero. There's a nice proof of this by Atiyah as an application of the Atiyah class (which is the canonical obstruction to the existence of a connection on a vector bundle). In particular over the moduli of stable degree zero bundles we again have a nontrivial torsor for the cotangent bundle parametrizing bundles with connection, and the class of this torsor is again the Chern class of the determinant line bundle.

Answer by Sebastian for Connection on line bundle on projective curve

2010-04-23T18:04:47Z

Yes, you can: Every holomorphic line bundle of degree $0$ has a flat connection which is compatible with the holomorphic structure, i.e. $\nabla s=w_s s$ for local holomorphic sections, where $w_s\in\Gamma(M,K).$ But flatness implies $d w_s=0.$ Thus $/nabla$ is a holomorphic connection.

To see the existence of a flat connection compatible with the holomorphic structure, take any connection $\tilde\nabla$ compatible with the holomorphic structure. The curvature $F$ satisfies $\int_MF=0,$ so Serre duality implies the existence of a section $\eta\in\Gamma(M;K)$ with $d\eta=F.$ Then $\nabla=\tilde\nabla-\eta$ is flat and also compatible with the holomorphic structure.