Kavli Affiliate: Yukinobu Toda
| First 5 Authors: Tudor Pădurariu, Yukinobu Toda, , ,
| Summary:
We introduce and begin the study of quasi-BPS categories for K3 surfaces,
which are a categorical version of the BPS cohomologies for K3 surfaces. We
construct semiorthogonal decompositions of derived categories of coherent
sheaves on moduli stacks of semistable objects on K3 surfaces, where each
summand is a categorical Hall product of quasi-BPS categories. We also prove
the wall-crossing equivalence of quasi-BPS categories, which generalizes
Halpern-Leistner’s wall-crossing equivalence of moduli spaces of stable objects
for primitive Mukai vectors on K3 surfaces.
We also introduce and study a reduced quasi-BPS category. When the weight is
coprime to the Mukai vector, the reduced quasi-BPS category is proper, smooth,
and its Serre functor is trivial ‘{e}tale locally on the good moduli space.
Moreover we prove that its topological K-theory recovers the BPS invariants of
K3 surfaces, which are known to be equal to the Euler characteristics of
Hilbert schemes of points on K3 surfaces. We regard reduced quasi-BPS
categories as noncommutative hyperk"ahler varieties which are categorical
versions of crepant resolutions of singular symplectic moduli spaces of
semistable objects on K3 surfaces.
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