Multiplicativity of perverse filtration for Hilbert schemes of fibered surfaces, II

Zhang, Zili

doi:10.1090/tran/8461

Multiplicativity of perverse filtration for Hilbert schemes of fibered surfaces, II
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Trans. Amer. Math. Soc. 374 (2021), 8573-8602 Request permission

Abstract:

Let $S\to C$ be a smooth quasi-projective surface properly fibered onto a smooth curve. We prove that the multiplicativity of the perverse filtration on $H^*(S^{[n]},\mathbb {Q})$ associated with the natural map $S^{[n]}\to C^{(n)}$ implies that $S\to C$ is an elliptic fibration. The converse is also true when $S\to C$ is a Hitchin-type elliptic fibration.

References

W. P. Barth, K. Hulek, C. A. M. Peters, and A. Van de Ven, Compact complex surfaces, Second edition. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. 4. Springer-Verlag, Berlin, 2004.
A. A. Beĭlinson, J. Bernstein, and P. Deligne, Faisceaux pervers, Analysis and topology on singular spaces, I (Luminy, 1981) Astérisque, vol. 100, Soc. Math. France, Paris, 1982, pp. 5–171 (French). MR 751966
Mark Andrea A. de Cataldo, Tamás Hausel, and Luca Migliorini, Topology of Hitchin systems and Hodge theory of character varieties: the case $A_1$, Ann. of Math. (2) 175 (2012), no. 3, 1329–1407. MR 2912707, DOI 10.4007/annals.2012.175.3.7
M. de Cataldo, D. Maulik, and J. Shen, Hitchin fibrations, abelian surfaces, and the P=W conjecture, arXiv:1909.11885, 2019.
Mark Andrea A. de Cataldo and Luca Migliorini, Intersection forms, topology of maps and motivic decomposition for resolutions of threefolds, Algebraic cycles and motives. Vol. 1, London Math. Soc. Lecture Note Ser., vol. 343, Cambridge Univ. Press, Cambridge, 2007, pp. 102–137. MR 2385301, DOI 10.1017/CBO9780511721496.004
David Eisenbud and Joe Harris, 3264 and all that—a second course in algebraic geometry, Cambridge University Press, Cambridge, 2016. MR 3617981, DOI 10.1017/CBO9781139062046
Lothar Göttsche and Wolfgang Soergel, Perverse sheaves and the cohomology of Hilbert schemes of smooth algebraic surfaces, Math. Ann. 296 (1993), no. 2, 235–245. MR 1219901, DOI 10.1007/BF01445104
Ryoshi Hotta, Kiyoshi Takeuchi, and Toshiyuki Tanisaki, $D$-modules, perverse sheaves, and representation theory, Progress in Mathematics, vol. 236, Birkhäuser Boston, Inc., Boston, MA, 2008. Translated from the 1995 Japanese edition by Takeuchi. MR 2357361, DOI 10.1007/978-0-8176-4523-6
Manfred Lehn, Chern classes of tautological sheaves on Hilbert schemes of points on surfaces, Invent. Math. 136 (1999), no. 1, 157–207. MR 1681097, DOI 10.1007/s002220050307
Manfred Lehn and Christoph Sorger, The cup product of Hilbert schemes for $K3$ surfaces, Invent. Math. 152 (2003), no. 2, 305–329. MR 1974889, DOI 10.1007/s00222-002-0270-7
Wei-ping Li, Zhenbo Qin, and Weiqiang Wang, Vertex algebras and the cohomology ring structure of Hilbert schemes of points on surfaces, Math. Ann. 324 (2002), no. 1, 105–133. MR 1931760, DOI 10.1007/s002080200330
Hiraku Nakajima, Heisenberg algebra and Hilbert schemes of points on projective surfaces, Ann. of Math. (2) 145 (1997), no. 2, 379–388. MR 1441880, DOI 10.2307/2951818
Marc A. Nieper-Wisskirchen, Twisted cohomology of the Hilbert schemes of points on surfaces, Doc. Math. 14 (2009), 749–770. MR 2578804
Carlos T. Simpson, Higgs bundles and local systems, Inst. Hautes Études Sci. Publ. Math. 75 (1992), 5–95. MR 1179076
J. Shen and Z. Zhang, Perverse filtrations, Hilbert schemes, and the $P=W$ conjecture for parabolic Higgs bundles, arXiv:1810.05330, 2018.
Zili Zhang, Multiplicativity of perverse filtration for Hilbert schemes of fibered surfaces, Adv. Math. 312 (2017), 636–679. MR 3635821, DOI 10.1016/j.aim.2017.03.028
Z. Zhang, The $P=W$ identity for cluster varieties, arXiv:1903.07014, to appear in Math. Res. Lett.