Virtual fundamental classes via dg–manifolds

Ciocan-Fontanine, Ionuţ¹ ; Kapranov, Mikhail²

¹ Department of Mathematics, University of Minnesota, 127 Vincent Hall, 206 Church St SE, Minneapolis, MN 55455, USA
² Department of Mathematics, Yale University, 10 Hillhouse Avenue, New Haven, CT 06520, USA

Geometry & topology, Tome 13 (2009) no. 3, pp. 1779-1804.

Résumé

We construct virtual fundamental classes for dg–manifolds whose tangent sheaves have cohomology only in degrees 0 and 1. This condition is analogous to the existence of a perfect obstruction theory in the approach of Behrend and Fantechi [Invent. Math 128 (1997) 45-88] or Li and Tian [J. Amer. Math. Soc. 11 (1998) 119-174]. Our class is initially defined in $K$–theory as the class of the structure sheaf of the dg–manifold. We compare our construction with that of Behrend and Fantechi as well as with the original proposal of Kontsevich. We prove a Riemann–Roch type result for dg–manifolds which involves integration over the virtual class. We prove a localization theorem for our virtual classes. We also associate to any dg–manifold of our type a cobordism class of almost complex (smooth) manifolds. This supports the intuition that working with dg–manifolds is the correct algebro-geometric replacement of the analytic technique of“deforming to transversal intersection".

Détail

DOI : 10.2140/gt.2009.13.1779

Keywords: virtual class, dg-manifold, cobordism

Bibliographie

[1] K Akin, D A Buchsbaum, J Weyman, Schur functors and Schur complexes, Adv. in Math. 44 (1982) 207

[2] K Behrend, Differential graded schemes I: Perfect resolving algebras

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

[4] H Cartan, S Eilenberg, Homological algebra, Princeton Univ, Press (1956)

[5] I Ciocan-Fontanine, M Kapranov, Derived Quot schemes, Ann. Sci. École Norm. Sup. $(4)$ 34 (2001) 403

[6] I Ciocan-Fontanine, M M Kapranov, Derived Hilbert schemes, J. Amer. Math. Soc. 15 (2002) 787

[7] W Fulton, Intersection theory, Ergebnisse series (3) 2, Springer (1984)

[8] T Graber, R Pandharipande, Localization of virtual classes, Invent. Math. 135 (1999) 487

[9] M Kontsevich, Enumeration of rational curves via torus actions, from: "The moduli space of curves (Texel Island, 1994)" (editors R Dijkgraaf, C Faber, G v d Geer), Progr. Math. 129, Birkhäuser (1995) 335

[10] Y P Lee, Quantum $K$–theory. I. Foundations, Duke Math. J. 121 (2004) 389

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

[12] I G Macdonald, Symmetric functions and Hall polynomials, Oxford Math. Monogr., Oxford Sci. Publ., The Clarendon Press, Oxford Univ. Press (1995)

[13] D C Ravenel, Complex cobordism and stable homotopy groups of spheres, Pure and Applied Math. 121, Academic Press (1986)

[14] R E Stong, Relations among characteristic numbers. I, Topology 4 (1965) 267

[15] R W Thomason, Une formule de Lefschetz en $K$–théorie équivariante algébrique, Duke Math. J. 68 (1992) 447