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Formality conjecture for K3 surfaces

Part of: Local theory Homological methods Surfaces and higher-dimensional varieties Families, fibrations

Published online by Cambridge University Press:  23 April 2019

Nero Budur  and
Ziyu Zhang
Nero Budur
Affiliation:
KU Leuven, Celestijnenlaan 200B, B-3001 Leuven, Belgium email nero.budur@kuleuven.be https://www.kuleuven.be/wis/algebra/budur
Ziyu Zhang
Affiliation:
Institut für algebraische Geometrie, Leibniz Universität Hannover, Welfengarten 1, 30167 Hannover, Germany email zhangzy@math.uni-hannover.de https://ziyuzhang.github.io
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Abstract

We give a proof of the formality conjecture of Kaledin and Lehn: on a complex projective K3 surface, the differential graded (DG) algebra $\operatorname{RHom}^{\bullet }(F,F)$ is formal for any sheaf $F$ polystable with respect to an ample line bundle. Our main tool is the uniqueness of the DG enhancement of the bounded derived category of coherent sheaves. We also extend the formality result to derived objects that are polystable with respect to a generic Bridgeland stability condition.

Keywords

K3 surfacesstable sheavesformalityDG algebrasDG enhancements

MSC classification

Primary: 14D20: Algebraic moduli problems, moduli of vector bundles
Secondary: 16E45: Differential graded algebras and applications 14B05: Singularities 14J28: $K3$ surfaces and Enriques surfaces
Type
Research Article
Information
Compositio Mathematica , Volume 155 , Issue 5 , May 2019 , pp. 902 - 911
Copyright
© The Authors 2019 

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