Virtual classes via vanishing cycles

Kinjo, Tasuki¹

¹ Research Institute for Mathematical Sciences, Kyoto University, Kyoto, Japan

Geometry & topology, Tome 29 (2025) no. 1, pp. 349-397.

Résumé

We develop a new method to construct the virtual fundamental classes for quasismooth derived schemes (and more generally, derived $1$-Artin stacks) using the perverse sheaves of vanishing cycles on their $-1$-shifted cotangent spaces. It is based on the author’s previous work that can be regarded as a version of the Thom isomorphism for $-1$-shifted cotangent spaces. We use the Fourier–Sato transform to prove that our classes coincide with the virtual fundamental classes introduced by the work of Behrend–Fantechi and Li–Tian, under the schematic and quasiprojectivity assumption. We also discuss an approach to construct DT4 virtual classes for $-2$-shifted symplectic derived schemes using the perverse sheaves of vanishing cycles.

Détail

DOI : 10.2140/gt.2025.29.349

Keywords: virtual fundamental classes, vanishing cycles, derived algebraic geometry, shifted symplectic geometry, cohomological Donaldson–Thomas theory

Bibliographie

[1] L Amorim, O Ben-Bassat, Perversely categorified Lagrangian correspondences, Adv. Theor. Math. Phys. 21 (2017) 289 | DOI

[2] D Arinkin, D Gaitsgory, Singular support of coherent sheaves and the geometric Langlands conjecture, Selecta Math. 21 (2015) 1 | DOI

[3] K Behrend, Gromov–Witten invariants in algebraic geometry, Invent. Math. 127 (1997) 601 | DOI

[4] K Behrend, Donaldson–Thomas type invariants via microlocal geometry, Ann. of Math. 170 (2009) 1307 | DOI

[5] K Behrend, B Fantechi, The intrinsic normal cone, Invent. Math. 128 (1997) 45 | DOI

[6] O Ben-Bassat, C Brav, V Bussi, D Joyce, A “Darboux theorem” for shifted symplectic structures on derived Artin stacks, with applications, Geom. Topol. 19 (2015) 1287 | DOI

[7] D Borisov, D Joyce, Virtual fundamental classes for moduli spaces of sheaves on Calabi–Yau four-folds, Geom. Topol. 21 (2017) 3231 | DOI

[8] C Brav, V Bussi, D Dupont, D Joyce, B Szendrői, Symmetries and stabilization for sheaves of vanishing cycles, J. Singul. 11 (2015) 85 | DOI

[9] C Brav, V Bussi, D Joyce, A Darboux theorem for derived schemes with shifted symplectic structure, J. Amer. Math. Soc. 32 (2019) 399 | DOI

[10] Y Cao, N C Leung, Donaldson–Thomas theory for Calabi–Yau 4-folds, preprint (2014)

[11] I Ciocan-Fontanine, M Kapranov, Virtual fundamental classes via dg-manifolds, Geom. Topol. 13 (2009) 1779 | DOI

[12] B Davison, The critical CoHA of a quiver with potential, Q. J. Math. 68 (2017) 635 | DOI

[13] D Edidin, W Graham, Characteristic classes and quadric bundles, Duke Math. J. 78 (1995) 277 | DOI

[14] A I Efimov, Cyclic homology of categories of matrix factorizations, Int. Math. Res. Not. 2018 (2018) 3834 | DOI

[15] B Fantechi, L Göttsche, Riemann–Roch theorems and elliptic genus for virtually smooth schemes, Geom. Topol. 14 (2010) 83 | DOI

[16] W Fulton, Intersection theory, 2, Springer (1984) | DOI

[17] D Joyce, A classical model for derived critical loci, J. Differential Geom. 101 (2015) 289

[18] D Joyce, Shifted symplectic derived algebraic geometry and generalizations of Donaldson–Thomas theory, lecture notes (2018)

[19] D Joyce, M Upmeier, Orientation data for moduli spaces of coherent sheaves over Calabi–Yau 3-folds, Adv. Math. 381 (2021) 107627 | DOI

[20] M Kashiwara, P Schapira, Sheaves on manifolds, 292, Springer (1990) | DOI

[21] T Kinjo, Dimensional reduction in cohomological Donaldson–Thomas theory, Compos. Math. 158 (2022) 123 | DOI

[22] A Kresch, Cycle groups for Artin stacks, Invent. Math. 138 (1999) 495 | DOI

[23] J Li, G Tian, Virtual moduli cycles and Gromov–Witten invariants of algebraic varieties, J. Amer. Math. Soc. 11 (1998) 119 | DOI

[24] D Massey, Natural commuting of vanishing cycles and the Verdier dual, Pacific J. Math. 284 (2016) 431 | DOI

[25] J Oh, R P Thomas, Counting sheaves on Calabi–Yau 4-folds, I, Duke Math. J. 172 (2023) 1333 | DOI

[26] T Pantev, B Toën, M Vaquié, G Vezzosi, Shifted symplectic structures, Publ. Math. Inst. Hautes Études Sci. 117 (2013) 271 | DOI

[27] A Preygel, Ind-coherent complexes on loop spaces and connections, from: "Stacks and categories in geometry, topology, and algebra" (editors T Pantev, C Simpson, B Toën, M Vaquié, G Vezzosi), Contemp. Math. 643, Amer. Math. Soc. (2015) 289 | DOI

[28] K Schefers, An equivalence between vanishing cycles and microlocalization, preprint (2022)

[29] J Schürmann, Topology of singular spaces and constructible sheaves, 63, Birkhäuser (2003) | DOI

[30] R P Thomas, A holomorphic Casson invariant for Calabi–Yau 3-folds, and bundles on K3 fibrations, J. Differential Geom. 54 (2000) 367

[31] Y Toda, Categorical Donaldson–Thomas theory for local surfaces, 2350, Springer (2024)

[32] J L Verdier, Spécialisation de faisceaux et monodromie modérée, from: "Analysis and topology on singular spaces, II–III", Astérisque 101–102, Soc. Math. France (1983) 332