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Universality of Descendent Integrals over Moduli Spaces of Stable Sheaves on $K3$ Surfaces
Symmetry, integrability and geometry: methods and applications, Tome 18 (2022) Cet article a éte moissonné depuis la source Math-Net.Ru

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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.
Keywords: moduli spaces of sheaves, $K3$ surfaces, descendent integrals. 
@article{SIGMA_2022_18_a75,
     author = {Georg Oberdieck},
     title = {Universality of {Descendent} {Integrals} over {Moduli} {Spaces} of {Stable} {Sheaves} on $K3$ {Surfaces}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2022},
     volume = {18},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2022_18_a75/}
}
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Georg Oberdieck. Universality of Descendent Integrals over Moduli Spaces of Stable Sheaves on $K3$ Surfaces. Symmetry, integrability and geometry: methods and applications, Tome 18 (2022). http://geodesic.mathdoc.fr/item/SIGMA_2022_18_a75/

[1] Ellingsrud G., Göttsche L., Lehn M., “On the cobordism class of the Hilbert scheme of a surface”, J. Algebraic Geom., 10 (2001), 81–100, arXiv: math.AG/9904095 | MR

[2] Frei S., “Moduli spaces of sheaves on $K3$ surfaces and Galois representations”, Selecta Math. (N.S.), 26 (2020), 6, 16 pp., arXiv: 1810.06735 | DOI | MR

[3] Göttsche L., Kool M., Virtual Segre and Verlinde numbers of projective surfaces, arXiv: 2007.11631

[4] Göttsche L., Nakajima H., Yoshioka K., “$K$-theoretic Donaldson invariants via instanton counting”, Pure Appl. Math. Q., 5 (2009), 1029–1111, arXiv: math.AG/0611945 | DOI | MR

[5] Gritsenko V., Hulek K., Sankaran G.K., “Moduli of $K3$ surfaces and irreducible symplectic manifolds”, Handbook of Moduli, v. I, Adv. Lect. Math. (ALM), 24, Int. Press, Somerville, MA, 2013, 459–526, arXiv: 1012.4155 | MR

[6] Huybrechts D., Lectures on $K3$ surfaces, Cambridge Stud. Adv. Math., 158, Cambridge University Press, Cambridge, 2016 | DOI | MR

[7] Huybrechts D., Lehn M., The geometry of moduli spaces of sheaves, Cambridge Math. Lib., 2nd ed., Cambridge University Press, Cambridge, 2010 | DOI | MR

[8] Lehn M., “Chern classes of tautological sheaves on Hilbert schemes of points on surfaces”, Invent. Math., 136 (1999), 157–207, arXiv: math.AG/9803091 | DOI | MR

[9] Marian A., Oprea D., Pandharipande R., “Segre classes and Hilbert schemes of points”, Ann. Sci. Éc. Norm. Supér., 50 (2017), 239–267, arXiv: 1507.00688 | DOI | MR

[10] Marian A., Oprea D., Pandharipande R., “The combinatorics of Lehn's conjecture”, J. Math. Soc. Japan, 71 (2019), 299–308, arXiv: 1708.08129 | DOI | MR

[11] Marian A., Oprea D., Pandharipande R., “Higher rank Segre integrals over the Hilbert scheme of points”, J. Eur. Math. Soc. (JEMS), 24 (2022), 2979–3015, arXiv: 1712.02382 | DOI | MR

[12] Markman E., “Generators of the cohomology ring of moduli spaces of sheaves on symplectic surfaces”, J. Reine Angew. Math., 544 (2002), 61–82, arXiv: math.AG/0009109 | DOI | MR

[13] Markman E., “On the monodromy of moduli spaces of sheaves on $K3$ surfaces”, J. Algebraic Geom., 17 (2008), 29–99, arXiv: math.AG/0305042 | DOI | MR

[14] Voisin C., “Segre classes of tautological bundles on Hilbert schemes of surfaces”, Algebr. Geom., 6 (2019), 186–195, arXiv: 1708.06325 | DOI | MR