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SUMMARY:Motivic degree zero Donaldson-Thomas invariants - Kai Behrend (UBC
 )
DTSTART:20101110T141500Z
DTEND:20101110T151500Z
UID:TALK27375@talks.cam.ac.uk
CONTACT:Burt Totaro
DESCRIPTION:Given a smooth complex threefold X\, we define the virtual mot
 ive\nof the Hilbert scheme of n points on X. In the case when X is Calabi-
 Yau\,\nthis gives a motivic refinement of the n-point degree zero Donaldso
 n-Thomas\ninvariant of X. The key example is affine three-space\, where th
 e Hilbert\nscheme can be expressed as the critical locus of a regular func
 tion on a\nsmooth  variety\, and its virtual motive is defined in terms of
  the\nDenef-Loeser motivic nearby fiber. A crucial technical result assert
 s that\nif a function is equivariant with respect to a suitable torus acti
 on\, its\nmotivic nearby fiber is simply given by the motivic class of a g
 eneral\nfiber. This allows us to compute the generating function of the vi
 rtual\nmotives of the Hilbert schemes of affine three-space via a direct\n
 computation involving the motivic class of the commuting variety. We then\
 ngive a formula for the generating function for arbitrary X as a motivic\n
 exponential\, generalizing known results in lower dimensions. The weight\n
 polynomial specialization leads to a product formula in terms of deformed\
 nMacMahon functions\, analogous to Gottsche's formula for the Poincare\npo
 lynomials of the Hilbert schemes of points on surfaces.
LOCATION:MR13\, CMS
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