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-K\"ahlerian
geometry. This formula, which is a consequence of the non-K\"ahlerian 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 E on a product B×X by a unitary flat line
bundle on the fiber X. When this fiber is a complex surface and 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=[A] corresponding to the canonical extension 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∈Ya​⊂Y such that
Ya​∖{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 b2​=2 we obtain a new way to complete the proof of a
theorem in a previous article: any minimal class VII surface with b2​=2 has a
cycle of curves. Applications to class VII surfaces with higher b2​ will be
be discussed in a forthcoming article.Comment: 25 pages. Comments, suggestions are most welcome Revised version:
minor correction