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This article deals with two topics: the first, which has a general character, is a variation formula for the determinant line bundle in non-Kahlerian geometry. This formula, which is a consequence of the non-Kahlerian version of the Grothendieck–Riemann–Roch theorem proved recently by Bismut [Bi], 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.

A Variation Formula for the Determinant Line Bundle. Compact Subspaces of Moduli Spaces of Stable Bundles over Class VII Surfaces