Kähler structure on Hurwitz spaces

Abstract

The classical Hurwitz spaces, that parameterize compact Riemann surfaces equipped with covering maps to P 1 of fixed numerical type with simple branch points, are extensively studied in the literature. We apply deformation theory, and present a study of the Kähler structure of the Hurwitz spaces, which reflects the variation of the complex structure of the Riemann surface as well as the variation of the meromorphic map. We introduce a generalized Weil-Petersson Kähler form on the Hurwitz space. This form turns out to be the curvature of a Quillen metric on a determinant line bundle. Alternatively, the generalized Weil-Petersson Kähler form can be characterized as the curvature form of the hermitian metric on the Deligne pairing of the relative canonical line bundle and the pull back of the anti-canonical line bundle on P 1 . Replacing the projective line by an arbitrary but fixed curve Y , we arrive at a generalized Hurwitz space with similar properties. The determinant line bundle extends to a compactification of the (generalized) Hurwitz space as a line bundle, and the Quillen metric yields a singular hermitian metric on the compactification so that a power of the determinant line bundle provides an embedding of the Hurwitz space in a projective space.2000 Mathematics Subject Classification. 32G15, 14H10, 53C55.

Kähler geometry on Hurwitz spaces

“…Definition 1.1. [ABS15] Let ω Y be a metric on Y of constant Ricci curvature equal to ǫ = 0 or ±1 depending on its genus. The Weil-Petersson inner product on the tangent space T S of the base S is defined by its norm…”

“…We fix a compact Riemann surface Y of arbitrary genus. We use the notation and definitions from [ABS15]. Definition 4.1.…”

“…Remark 4.3. The vector fields u s have already been constructed in the work [ABS15] without being aware of their deformation theoretic meaning. These ad hoc defined objects were used for defining the Weil-Petersson metric, see below.…”

“…Differential geometric setup and statement of results. The geometric study of the Hurwitz space using methods from differential geometry starts with the work of Axelsson, Biswas and Schumacher in [ABS15], where they had a deformation theoretic point of view. We first recall their setup and definitions.…”

The Weil–Petersson current on Douady spaces

“…In [25] it was shown that a the Weil–Petersson form extends as a positive current to a suitable compactification of the moduli space, and that the determinant line bundle can be extended as a holomorphic line bundle, whose curvature current is positive and extends the Weil–Petersson form. Similar constructions were carried out for moduli spaces of stable vector bundles [9], moduli spaces of Higgs bundles [11], and other cases such as Hurwitz spaces [3].…”