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### K-theoretic Donaldson-Thomas theory and the Hilbert scheme of points on a surface

File | Description | Size | Format | |
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KDTHilbrevision1-alggeom.pdf | Working paper | 533.07 kB | Adobe PDF | View/Open |

Title: | K-theoretic Donaldson-Thomas theory and the Hilbert scheme of points on a surface |

Authors: | Arbesfeld, N |

Item Type: | Working Paper |

Abstract: | Integrals of characteristic classes of tautological sheaves on the Hilbert scheme of points on a surface frequently arise in enumerative problems. We use the K-theoretic Donaldson-Thomas theory of certain toric Calabi-Yau threefolds to study K-theoretic variants of such expressions. We study limits of the K-theoretic Donaldson-Thomas partition function of a toric Calabi-Yau threefold under certain one-parameter subgroups called slopes, and formulate a condition under which two such limits coincide. We then explicitly compute the limits of components of the partition function under so-called preferred slopes, obtaining explicit combinatorial expressions related to the refined topological vertex of Iqbal, Kos\c{c}az and Vafa. Applying these results to specific Calabi-Yau threefolds, we deduce dualities satisfied by a generating function built from tautological bundles on the Hilbert scheme of points on $\mathbb{C}^2$. We then use this duality to study holomorphic Euler characteristics of exterior and symmetric powers of tautological bundles on the Hilbert scheme of points on a general surface. |

Issue Date: | 8-Oct-2020 |

URI: | http://hdl.handle.net/10044/1/87682 |

Publisher: | arXiv |

Copyright Statement: | © 2020 The Author(s). |

Keywords: | math.AG math.AG hep-th math-ph math.MP 14C05, 14N35, 14C35 math.AG math.AG hep-th math-ph math.MP 14C05, 14N35, 14C35 |

Notes: | 56 pages, 10 figures, exposition expanded and section on rank two vector bundles added |

Publication Status: | Published |

Appears in Collections: | Pure Mathematics Mathematics |