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A variation formula for the determinant line bundle. Compact subspaces of moduli spaces of stable bundles over class VII surfaces

This article deals with two topics: the first, which has a general character, is a variation formula for the the determinant line bundle in non-Kählerian geometry. This formula, which is a consequence of the non-Kählerian version of the Grothendieck-Riemann Roch theorem proved recently by Bismut, gives the variation of the determinant line bundle corresponding to a perturbation of a Fourier-Mukai kernel ${\cal E}$ on a product $B\times X$ by a unitary flat line bundle on the fiber $X$. When this fiber is a complex surface and ${\cal E}$ is a holomorphic 2-bundle, the result can be interpreted as a Donaldson invariant. The second topic concerns a geometric application of our variation formula, namely we will study compact complex subspaces of the moduli spaces of stable bundles considered in our program for proving existence of curves on minimal class VII surfaces. Such a moduli space comes with a distinguished point $a=[{\cal A}]$ corresponding to the canonical extension ${\cal A}$ of $X$. The compact subspaces $Y\subset {\cal M}^\st$ containing this distinguished point play an important role in our program. We will prove a non-existence result: there exists no compact complex subspace of positive dimension $Y\subset {\cal M}^\st$ containing $a$ with an open neighborhood $a\in Y_a\subset Y$ such that $Y_a\setminus\{a\}$ consists only of non-filtrable bundles. In other words, within any compact complex subspace of positive dimension $Y\subset {\cal M}^\st$ containing $a$, the point $a$ can be approached by filtrable bundles. Specializing to the case $b_2=2$ we obtain a new way to complete the proof of a theorem in a previous article: any minimal class VII surface with $b_2=2$ has a cycle of curves. Applications to class VII surfaces with higher $b_2$ will be be discussed in a forthcoming article.