For any scheme M with a perfect obstruction theory, Jiang and Thomas associated a scheme N with a symmetric perfect obstruction theory. The scheme N is a cone over M given by the dual of the obstruction sheaf of M, and contains M as its zero section. Locally, N is the critical locus of a regular function. In this note we prove that N is a d-critical scheme in the sense of Joyce. There exists a global motive for N locally given by the motive of the vanishing cycle of the local regular function. We prove a motivic localization formula under the good and circle compact C*-action for N. When taking the Euler characteristic, the weighted Euler characteristic of N weighted by the Behrend function is the signed Euler characteristic of M by motivic method. As applications, using the main theorem we study the motivic generating series of the motivic Vafa-Witten invariants for K3 surfaces.
Yunfeng JIANG
. MOTIVIC VIRTUAL SIGNED EULER CHARACTERISTICS AND THEIR APPLICATIONS TO VAFA-WITTEN INVARIANTS[J]. Acta mathematica scientia, Series B, 2022
, 42(6)
: 2279
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DOI: 10.1007/s10473-022-0605-y
[1] Behrend K. Donaldson-Thomas invariants via microlocal geometry. Ann of Math, 2009, 170: 1307-1338
[2] Behrend K, Bryan J, Szendroi B. Motivic degree zero Donaldson-Thomas invariants. Invent Math, 2013, 192(1): 111-160
[3] Behrend K, Fantechi B. The intrinsic normal cone. Invent Math, 1997, 128: 45-88
[4] Bussi V, Joyce D, Meinhardt S. On motivic vanishing cycles of critical loci. J Alg Geo, 2019, 28(3): 405-438
[5] Chang H, Li J. Semi-perfect obstruction theory and DT invariants of derived objects. Communications in Analysis and Geometry, 2011, 19(4): 807-830
[6] Costello K. Notes on supersymmetric and holomorphic field theories in dimensions 2 and 4. Pure Appl Math Quart, 2013, 9: 73-165
[7] Davison B. The critical CoHA of a self dual quiver with potential. arXiv:1311.7172
[8] Davison B, Meinhardt S. The motivic Donaldson-Thomas invariants of (-2)-curves. Algebra and Number Theory, 2017, 11(6): 1243-1286
[9] Göttsche L. On the motive of the Hilbert scheme of points on a surface. Math Res Lett, 2001, 8: 613-627
[10] Göttsche L, Kool M. Virtual refinements of the Vafa-Witten formula. arXiv:1703.07196
[11] Graber T, Pandharipande R. Localization of virtual classes. Invent Math, 1999, 135: 487-518
[12] Gusein-Zade S M, Luengo I, Melle-Hernández A. Power structure over the Grothendieck ring of varieties and generating series of Hilbert schemes of points. Mich Math J, 2006, 54: 353-359
[13] Jiang Y. Motivic Milnor fiber of cyclic L-infinity algebras. Acta Mathematica Sinica, 2017, 33(7): 933-950
[14] Jiang Y. The moduli space of stable coherent sheaves via non-archimedean geometry. arXiv:1703.00497
[15] Jiang Y. Note on MacPherson’s local Euler obstruction. Michigan Mathematical Journal, 2019, 68: 227-250
[16] Jiang Y. Symmetric semi-perfect obstruction theory revisited. Acta Mathematica Sinica, to appear, arXiv:1811.08480
[17] Jiang Y. Counting twisted sheaves and S-duality. Adv Math, 2022, 400: Art 108332
[18] Jiang Y, Thomas R. Virtual signed Euler characteristics. Journal of Algebraic Geometry, 2017, 26: 379-397
[19] Jiang Y, Kundu P. The Tanaka-Thomas’s Vafa-Witten invariants for surface Deligne-Mumford stacks. Pure and Applied Math Quarterly, 2021, 17(1): 503-573
[20] Jiang Y, Tseng H -H. A proof of all rank S-duality conjecture for K3 surfaces. arXiv:2003.09562
[21] Jiang Y, Kool M. Twisted sheaves and SU(r)/Zr Vafa-Witten theory. Mathematische Annalen, 2022, 382: 719-743
[22] Joyce D. A classical model for derived critical locus. Journal of Differential Geometry, 2015, 101: 289-367
[23] Joyce D, Upmeier M. Orientation data for moduli spaces of coherent sheaves over Calabi-Yau 3-folds. Adv Math, 2021, 381: Art 107627
[24] Kiem Y -H, Li J. Localizing virtual cycles by cosections. J Amer Math Soc, 2013, 26: 1025-1050
[25] Li J, Tian G. Virtual moduli cycles and Gromov-Witten invariants of algebraic varieties. J Amer Math Soc, 1998, 11: 119-174
[26] Looijenga E. Motivic measures. Asterisque, 2002, 276: 267-297
[27] Maulik D. Motivic residues and Donaldson-Thomas theory. In preparation
[28] Maulik D, Toda Y. Gopakumar-Vafa invariants via vanishing cycles. Invent Math, 2018, 213: 1017-1097
[29] Maulik D, Thomas R P. Sheaf counting on local K3 surfaces. Pure Appl Math Quar, 2018, 14(3/4): 419-441
[30] Nekrasov N, Okounkov A. Membranes and Sheaves. Alg Geo, 2016, 3(3): 320-369
[31] Nicaise J. A trace formula for rigid varieties, and motivic Weil generating series for formal schemes. Math Ann, 2008, 343(2): 285-349
[32] Nicaise J, Payne S. A tropical motivic Fubini theorem with applications to Donaldson-Thomas theory. Duke Math J, 2019, 168(10): 1843-1886
[33] Pantev T, Toën B, Vaquie M, Vezzosi G. Shifted symplectic structures. Publ Math IHES, 2013, 117: 271-328
[34] Tanaka Y, Thomas R. Vafa-Witten invariants for projective surfaces I: stable case. J Alg Geom, 2020, 29: 603-668
[35] Tanaka Y, Thomas R. Vafa-Witten invariants for projective surfaces II: semistable case. Pure Appl Math Quart, 2017, 13: 517-562
[36] Thomas R P. Equivariant K-theory and refined Vafa-Witten invariants. Comm Math Phys, 2020, 378(2): 1451-1500
[37] Toën B, Vezzosi G. Homotopical algebraic geometry II: Geometric stacks and applications. Mem Amer Math Soc, 2008, 193(902)
[38] Vafa C, Witten E. A strong coupling test of S-duality. Nucl Phys, 1994, 431B: 3-77
[39] Yoshioka K. Some examples of Mukai’s reflections on K3 surfaces. J Reine Angew Math, 1999, 515: 97-123
