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Nested Hilbert schemes on surfaces: Virtual fundamental class

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  • Amin Gholampour, University of Maryland
  • ,
  • Artan Sheshmani
  • Shing Tung Yau, Harvard University

We construct virtual fundamental classes on nested Hilbert schemes of points and curves in complex nonsingular projective surfaces. These classes recover the virtual classes of Seiberg-Witten theory as well as the (reduced) stable theory, and play a crucial role in the reduced Donaldson-Thomas theory of local-surface-threefolds that we study in [15]. We show that certain integrals against the virtual fundamental classes of punctual nested Hilbert schemes are expressed as integrals over the products of the Hilbert scheme of points. We are able to find explicit formulas for some of these integrals by relating them to Carlsson-Okounkov's vertex operator formulas.

Original languageEnglish
Article number107046
JournalAdvances in Mathematics
Publication statusPublished - 13 May 2020

    Research areas

  • Nested Hilbert scheme, Projective surfaces, Virtual fundamental class

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