Holonomy of affine surfaces

TL;DR

This paper studies the moduli space of complex affine surfaces, linking it to meromorphic connections, and analyzes the holonomy map's properties, including its derivatives and a special foliation structure.

Contribution

It establishes a universal property of the moduli space, computes derivatives of the holonomy map, and introduces the isoresidual foliation with a Hermitian metric, extending Veech's work.

Findings

Holonomy map is a submersion at generic affine surfaces.

Identified tangent space with hypercohomology of sheaves.

Introduced isoresidual foliation with leafwise Hermitian metric.

Abstract

We identify the moduli space of complex affine surfaces with the moduli space of regular meromorphic connections on Riemann surfaces and show that it satisfies a corresponding universal property. As a consequence, we identify the tangent space of the moduli space of affine surfaces, at an affine surface X, with the first hypercohomology of a two-term sequences of sheaves on X. In terms of this identification, we calculate the derivative and coderivative of the holonomy map, sending an affine surface to its holonomy character. Using these formulas, we show that the holonomy map is a submersion at every affine surface that is not a finite-area translation surface, extending work of Veech. Finally, we introduce a holomorphic foliation of some strata of meromorphic affine surfaces, which we call the isoresidual foliation, along whose leave holonomy characters and certain residues are constant. We equip this foliation with a leafwise indefinite Hermitian metric, again extending work of Veech.