Let C be a smooth curve embedded in a smooth quasi-projective three-fold Y, and let QnC = Quotn(IC) be the Quot scheme of length n quotients of its ideal sheaf. We show the identity ?(QnC ) = (-1)n?(QnC ), where ? is the Behrend weighted Euler characteristic. When Y is a projective CalabiYau three-fold, this shows that the Donaldson Thomas (DT) contribution of a smooth rigid curve is the signed Euler characteristic of the moduli space. This can be rephrased as a DT/PT wall-crossing type formula, which can be formulated for arbitrary smooth curves. In general, such wall-crossing formula is shown to be equivalent to a certain Behrend function identity.
Ricolfi A.T. (2018). Local Contributions to Donaldson-Thomas Invariants. INTERNATIONAL MATHEMATICS RESEARCH NOTICES, 2018(19), 5995-6025 [10.1093/imrn/rnx046].
Local Contributions to Donaldson-Thomas Invariants
Ricolfi A. T.
2018
Abstract
Let C be a smooth curve embedded in a smooth quasi-projective three-fold Y, and let QnC = Quotn(IC) be the Quot scheme of length n quotients of its ideal sheaf. We show the identity ?(QnC ) = (-1)n?(QnC ), where ? is the Behrend weighted Euler characteristic. When Y is a projective CalabiYau three-fold, this shows that the Donaldson Thomas (DT) contribution of a smooth rigid curve is the signed Euler characteristic of the moduli space. This can be rephrased as a DT/PT wall-crossing type formula, which can be formulated for arbitrary smooth curves. In general, such wall-crossing formula is shown to be equivalent to a certain Behrend function identity.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
