Narrow Results

Let X be a smooth projective curve of genus g ≥ 2 over ℂ. Fix n ≥ 2, d ∈ ℤ. A pair (E, ϕ) over X consists of an algebraic vector bundle E of rank n and degree d over X and a section ϕ ∈ H⁰(E). There is a concept of stability for pairs which depends on a real parameter τ. Let be the moduli space of τ-semistable pairs of rank n and degree d over X. Here we prove that the cohomology groups of are Hodge structures isomorphic to direct summands of tensor products of the Hodge structure H¹(X). This implies a similar result for the moduli spaces of stable vector bundles over X.

The first goal of the article is to solve several fundamental problems in the theory of holomorphic bundles over non-algebraic manifolds. For instance, we prove that stability and semi-stability are Zariski open properties in families when the Gauduchon degree map is a topological invariant, or when the parameter manifold is compact. Second, we show that, for a generically stable family of bundles over a Kähler manifold, the Petersson–Weil form extends as a closed positive current on the whole parameter space of the family. This extension theorem uses classical tools from Yang–Mills theory (e.g., the Donaldson functional on the space of Hermitian metrics and its properties). We apply these results to study families of bundles over a Kählerian manifold Y parametrized by a non-Kählerian surface X, proving that such families must satisfy very restrictive conditions. These results play an important role in our program to prove existence of curves on class VII surfaces [22–24].