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Abstract: This is a survey of the book arXiv:0810.5645 with Yinan Song. Let X be aCalabi-Yau 3-fold over C. The Donaldson-Thomas invariants of X are integersDT^at which count stable sheaves with Chern character a on X, with respect toa Gieseker stability condition t. They are defined only for Chern characters afor which there are no strictly semistable sheaves on X. They have the goodproperty that they are unchanged under deformations of X. Their behaviour underchange of stability condition t was not understood until now.We discuss -generalized Donaldson-Thomas invariants- \bar{DT}^at. These arerational numbers, defined for all Chern characters a, and are equal to DT^atif there are no strictly semistable sheaves in class a. They aredeformation-invariant, and have a known transformation law under change ofstability condition. We conjecture they can be written in terms of integral-BPS invariants- \hat{DT}^at when the stability condition t is -generic-.We extend the theory to abelian categories of representations of a quiverwith relations coming from a superpotential, and connect our ideas withSzendroi-s -noncommutative Donaldson-Thomas invariants- and work by Reineke andothers. There is significant overlap between arXiv:0810.5645 and theindependent paper arXiv:0811.2435 by Kontsevich and Soibelman.



Autor: Dominic Joyce

Fuente: https://arxiv.org/



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