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**Hitchin’s connection and differential operators with values in the determinant bundle.**
*(English)*
Zbl 1120.14008

Let \(M\) be a smooth variety over an algebraically closed field of characteristic zero and let \(S\to M\) be a smooth flat morphism. Let \(\pi:X\to S\) be a smooth family of curves and \(E\) a vector bundle over \(X\).

In this paper the authors prove that, under the assumption that Kodaira-Spencer maps of the families are isomorphisms, one can canonically produce locally free sheaves on X providing an identification theorem for the sheaf of differential operators of order \(\leq 1\) with values in the determinant bundle of \(E\) over \(S\). The authors give a new construction of the Hitchin’s connection on the push forward of the theta bundle on \(S\) as an example of their identification theorem. They fix \(C \to M\) a family of smooth curves of genus \(g\geq 2\) and \(f:S \to M\) the associated family of moduli spaces of stable bundles of rank \(r\geq 2\) with a fixed determinant. Hence, let \(X\) be the product \(C \times_{M} S\), and \(E\) the universal bundle on it. The assumption for the Kodaira-Spencer maps is satisfied for this particular input and the identification theorem applies and provide us with the required connection. The authors also generalise the construction to the logarithmic version of the Hitchin’s connection. They prove an analogous identification theorem when \(M\) is provided with a normal crossing divisor on it.

In this paper the authors prove that, under the assumption that Kodaira-Spencer maps of the families are isomorphisms, one can canonically produce locally free sheaves on X providing an identification theorem for the sheaf of differential operators of order \(\leq 1\) with values in the determinant bundle of \(E\) over \(S\). The authors give a new construction of the Hitchin’s connection on the push forward of the theta bundle on \(S\) as an example of their identification theorem. They fix \(C \to M\) a family of smooth curves of genus \(g\geq 2\) and \(f:S \to M\) the associated family of moduli spaces of stable bundles of rank \(r\geq 2\) with a fixed determinant. Hence, let \(X\) be the product \(C \times_{M} S\), and \(E\) the universal bundle on it. The assumption for the Kodaira-Spencer maps is satisfied for this particular input and the identification theorem applies and provide us with the required connection. The authors also generalise the construction to the logarithmic version of the Hitchin’s connection. They prove an analogous identification theorem when \(M\) is provided with a normal crossing divisor on it.

Reviewer: Marina Logares (Bonn)

### MSC:

14D21 | Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory) |

32G13 | Complex-analytic moduli problems |

58J52 | Determinants and determinant bundles, analytic torsion |

32C38 | Sheaves of differential operators and their modules, \(D\)-modules |