Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 18 (2022), 076, 15 pages      arXiv:2201.03833
Contribution to the Special Issue on Enumerative and Gauge-Theoretic Invariants in honor of Lothar Göttsche on the occasion of his 60th birthday

Universality of Descendent Integrals over Moduli Spaces of Stable Sheaves on $K3$ Surfaces

Georg Oberdieck
Mathematisches Institut, Universität Bonn, Endenicher Allee 60, D-53115 Bonn, Germany

Received January 23, 2022, in final form October 06, 2022; Published online October 13, 2022

We interprete results of Markman on monodromy operators as a universality statement for descendent integrals over moduli spaces of stable sheaves on $K3$ surfaces. This yields effective methods to reduce these descendent integrals to integrals over the punctual Hilbert scheme of the $K3$ surface. As an application we establish the higher rank Segre-Verlinde correspondence for $K3$ surfaces as conjectured by Göttsche and Kool.

Key words: moduli spaces of sheaves; $K3$ surfaces; descendent integrals.

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