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Oberseminar SoSe 2010 Zusammenfassung des Vortrags von M. Guldbrandsen (Haugesund, NOR)

Obstruction theory for perfect complexes on abelian threefolds

To a sheaf E, or bounded complex, on an abelian threefold X, one may attach two line bundles: the determinant det(E) on X, and the determinant det(S(E)) of the Fourier-Mukai transform S(E) on the dual of X. We call the latter the codeterminant of E. The deformation and obstruction spaces of E split as direct sums of three pieces, where the "nontrivial" part controls deformations keeping both the determinant and the codeterminant fixed.

As a consequence, the attached virtual fundamental class of moduli spaces for sheaves is trivial if the determinant and/or codeterminant are allowed to move. Fixing the determinant and codeterminant, one obtains a moduli space with (in general) nontrivial virtual class, and an associated Donaldson-Thomas number. For example, the Hilbert scheme of points on X has trivial fundamental class, whereas the Hilbert scheme of points summing to zero under the group law, has nontrivial Donaldson-Thomas invariants.