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Higher rank stable pairs on K3 surfaces

Bakker, Benjamin and Jorza, Andrei (2013) Higher rank stable pairs on K3 surfaces. Communications in Number Theory and Physics, 6 (4). pp. 805-847. ISSN 1931-4523.

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We define and compute higher rank analogs of Pandharipande– Thomas stable pair invariants in primitive classes for K3 surfaces. Higher rank stable pair invariants for Calabi–Yau threefolds have been defined by Sheshmani [26, 27] using moduli of pairs of the form O^n →F for F purely one-dimensional and computed via wall-crossing techniques. These invariants may be thought of as virtually counting embedded curves decorated with a (n − 1)- dimensional linear system. We treat invariants counting pairs O^n → E on a K3 surface for E an arbitrary stable sheaf of a fixed numerical type (“coherent systems” in the language of [16]) whose first Chern class is primitive, and fully compute them geometrically. The ordinary stable pair theory of K3 surfaces is treated by [22]; there they prove the KKV conjecture in primitive classes by showing the resulting partition functions are governed by quasimodular forms. We prove a “higher” KKV conjecture by showing that our higher rank partition functions are modular forms.

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Additional Information:© 2012 International Press. Received August 25, 2012. The authors would like to thank D. Maulik and J. Tsimerman for valuable conversations. The first author would like to thank R. Pandharipande in particular for introducing the authors to the subject matter and for many enlightening discussions which greatly improved the content and exposition of the paper. The second author is grateful for Dinakar Ramakrishnan for helpful comments. Both authors would finally like to thank the referee for many helpful suggestions. Part of the research reported here was completed while the authors were graduate students at Princeton University.
Issue or Number:4
Record Number:CaltechAUTHORS:20130806-102624527
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ID Code:39778
Deposited On:07 Aug 2013 17:50
Last Modified:03 Oct 2019 05:10

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