Virtual push-forwards

Manolache, Cristina¹

¹ Institut für Mathematik, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099 Berlin, Germany

Geometry & topology, Tome 16 (2012) no. 4, pp. 2003-2036.

Résumé

Let $p:F\to G$ be a morphism of DM stacks of positive virtual relative dimension $k$ and let $\gamma \in A^k\left(F\right)$. We give sufficient conditions for $p_{\ast }\left(\gamma \cdot {\left[F\right]}^{virt}\right)$ to be a multiple of ${\left[G\right]}^{virt}$. We show an analogue of the conservation of number for virtually smooth families. We show implications to Gromov–Witten invariants and give a new proof of a theorem of Marian, Oprea and Pandharipande which compares the virtual classes of moduli spaces of stable maps and moduli spaces of stable quotients.

Détail

DOI : 10.2140/gt.2012.16.2003

Classification : 14C17, 14N35
Keywords: virtual classes, Gromov–Witten invariants

Bibliographie

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

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

[3] T Coates, Givental's Lagrangian cone and $S^1$–equivariant Gromov–Witten theory, Math. Res. Lett. 15 (2008) 15

[4] K Costello, Higher genus Gromov–Witten invariants as genus zero invariants of symmetric products, Ann. of Math. 164 (2006) 561

[5] A J De Jong, The Stacks Project

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

[7] W Fulton, Intersection theory, Ergeb. Math. Grenzgeb. 2, Springer (1984)

[8] A Gathmann, Gromov–Witten invariants of hypersurfaces, habilitation thesis, University of Kaiserslautern (2003)

[9] B Kim, Some functoriality properties of Gromov–Witten invariants, personal communication

[10] B Kim, A Kresch, T Pantev, Functoriality in intersection theory and a conjecture of Cox, Katz, and Lee, J. Pure Appl. Algebra 179 (2003) 127

[11] B Kim, R Pandharipande, The connectedness of the moduli space of maps to homogeneous spaces, from: "Symplectic geometry and mirror symmetry (Seoul, 2000)", World Sci. Publ., River Edge, NJ (2001) 187

[12] A Kresch, Intersection theory on Deligne–Mumford stacks, manuscript

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

[14] H H Lai, Gromov–Witten invariants of blow-ups along submanifolds with convex normal bundles, Geom. Topol. 13 (2009) 1

[15] G Laumon, L Moret-Bailly, Champs algébriques, Ergeb. Math. Grenzgeb. 39, Springer (2000)

[16] J Li, G Tian, Virtual moduli cycles and Gromov–Witten invariants of general symplectic manifolds, from: "Topics in symplectic 4–manifolds (Irvine, CA, 1996)", First Int. Press Lect. Ser., I, Int. Press, Cambridge, MA (1998) 47

[17] M Lieblich, Remarks on the stack of coherent algebras, Int. Math. Res. Not. 2006 (2006) 12

[18] C Manolache, Virtual pull-backs, J. Algebraic Geom. 21 (2012) 201

[19] A Marian, D Oprea, R Pandharipande, The moduli space of stable quotients, Geom. Topol. 15 (2011) 1651

[20] M Olsson, Sheaves on Artin stacks, J. Reine Angew. Math. 603 (2007) 55

[21] M Popa, M Roth, Stable maps and Quot schemes, Invent. Math. 152 (2003) 625

[22] Y Toda, Moduli spaces of stable quotients and wall-crossing phenomena, Compos. Math. 147 (2011) 1479

[23] A Vistoli, Intersection theory on algebraic stacks and on their moduli spaces, Invent. Math. 97 (1989) 613