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Localizing virtual structure sheaves for almost perfect obstruction theories

Part of: Families, fibrations Cycles and subschemes Projective and enumerative geometry

Published online by Cambridge University Press:  07 December 2020

Young-Hoon Kiem  and
Michail Savvas
Young-Hoon Kiem
Affiliation:
Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul08826, Korea; E-mail: kiem@snu.ac.kr
Michail Savvas
Affiliation:
Department of Mathematics, University of California, San Diego, La Jolla CA92093, USA; E-mail: msavvas@ucsd.edu

Abstract

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Almost perfect obstruction theories were introduced in an earlier paper by the authors as the appropriate notion in order to define virtual structure sheaves and K-theoretic invariants for many moduli stacks of interest, including K-theoretic Donaldson-Thomas invariants of sheaves and complexes on Calabi-Yau threefolds. The construction of virtual structure sheaves is based on the K-theory and Gysin maps of sheaf stacks.

In this paper, we generalize the virtual torus localization and cosection localization formulas and their combination to the setting of almost perfect obstruction theory. To this end, we further investigate the K-theory of sheaf stacks and its functoriality properties. As applications of the localization formulas, we establish a K-theoretic wall-crossing formula for simple $\mathbb{C} ^\ast $ -wall crossings and define K-theoretic invariants refining the Jiang-Thomas virtual signed Euler characteristics.

MSC classification

Primary: 14N35: Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants
Secondary: 14C35: Applications of methods of algebraic $K$-theory 14D23: Stacks and moduli problems
Type
Mathematical Physics
Information
Forum of Mathematics, Sigma , Volume 8 , 2020 , e61
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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