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Résumé |
We study the Donaldson-Thomas invariants of the 2-dimensional stable sheaves in a smooth projective
threefold. The DT invariants are defined via integrating over the virtual fundamental class when it exists.
When the threefold is a K3 surface fibration we express the DT invariants of sheaves supported on the
fibers in terms of the the Euler characteristics of the Hilbert scheme of points on the K3 surface and the
Noether-Lefschetz numbers of the fibration. Using this we prove the modularity of the DT invariants
which was predicted in string theory. We develop a DT-theoretic conifold transition formula through
which we compute the generating series for the invariants of Hilbert scheme of points for singular
surfaces. We also use our geometric techniques to compute the generating series for DT invariants of
threefolds given as complete intersections such as the quintic threefold. Finally if time permits I explain a
connection between torsion DT invariants and higher dimensional Knot theory. |
