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Families of holomorphic bundles. (English) Zbl 1159.32011

The author’s motivation to investigate families of holomorphic bundles arose from the problem of proving the existence of curves on class VII surfaces. The main purpose of the present paper is to prove results that come from such investigations and to these ends the author employs an interesting technique. Given an arbitrary complex manifold \(X\), a “good one” \(Y\), and a family \({\mathcal E} \to Y\times X\) of holomorphic bundles on \(X\) parametrized by \(Y\), one can switch roles and consider \(\mathcal E\) as a family of bundles on \(Y\) parametrized by \(X\). One then applies the philosophical principle that a family of bundles on a “good space” is itself “good”. [“Good” has varying meanings depending on the situation; see section 1 of the paper.] In particular, one is interested in the sets \(X^{\text{st}}\) (resp. \(X^{\text{sst}}\)) of points in \(X\) where the corresponding bundles \({\mathcal E}^{x}\) are stable (resp. semistable).
If \((Y,g)\) is a connected compact Gauduchon manifold, then \(X^{\text{st}}\) is open in the classical topology. Further, if \(X\) is compact or the degree map \(\text{deg}_{g}: \text{Pic}(Y) \to \mathbb R\) is a topological invariant, then \(X^{\text{st}}\) and \(X^{\text{sst}}\) are both Zariski open.
If \((Y,g)\) is Kähler and \(X^{\text{st}}\) is non-empty, then the Petersson-Weil form extends to a positive closed current on \(X\).
The third result is in the setting where \((Y,g)\) is a compact Kähler manifold and \(X\) is a surface whose first Betti number is odd. Assume that the family of holomorphic bundles is generically stable. Then the family is degenerate at every point. In particular, the induced map \(X^{\text{st}}\) into the moduli space of stable bundles is either constant or generically has rank one and \(X\) has infinitely many compact curves in the second case.

MSC:

32L05 Holomorphic bundles and generalizations
32G13 Complex-analytic moduli problems
32G08 Deformations of fiber bundles
53C07 Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills)
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