Virtual pull-backs

Manolache, Cristina

doi:10.1090/S1056-3911-2011-00606-1

Virtual pull-backs

Author: Cristina Manolache
Journal: J. Algebraic Geom. 21 (2012), 201-245
DOI: https://doi.org/10.1090/S1056-3911-2011-00606-1
Published electronically: December 15, 2011
MathSciNet review: 2877433
Full-text PDF

Abstract | References | Additional Information

Abstract: We propose a generalization of Gysin maps for DM-type morphisms of stacks $F\to G$ that admit a perfect relative obstruction theory $E_{F/G}^{\bullet }$, which we call a “virtual pull-back”. We prove functoriality properties of virtual pull-backs. As applications, we analyze Gromov-Witten invariants of blow-ups and projective bundles.

References

K. Behrend, Gromov-Witten invariants in algebraic geometry, Invent. Math. 127 (1997), no. 3, 601–617. MR 1431140, DOI https://doi.org/10.1007/s002220050132
K. Behrend, The product formula for Gromov-Witten invariants, J. Algebraic Geom. 8 (1999), no. 3, 529–541. MR 1689355
K. Behrend and B. Fantechi, The intrinsic normal cone, Invent. Math. 128 (1997), no. 1, 45–88. MR 1437495, DOI https://doi.org/10.1007/s002220050136
Izzet Coskun, Gromov-Witten invariants of jumping curves, Trans. Amer. Math. Soc. 360 (2008), no. 2, 989–1004. MR 2346480, DOI https://doi.org/10.1090/S0002-9947-07-04284-5
Kevin Costello, Higher genus Gromov-Witten invariants as genus zero invariants of symmetric products, Ann. of Math. (2) 164 (2006), no. 2, 561–601. MR 2247968, DOI https://doi.org/10.4007/annals.2006.164.561
Artur Elezi, Mirror symmetry and quantum cohomology for projective bundles, Int. J. Pure Appl. Math. 36 (2007), no. 1, 75–86. MR 2311715
Artur Elezi, A mirror conjecture for projective bundles, Int. Math. Res. Not. 55 (2005), 3445–3458. MR 2204641, DOI https://doi.org/10.1155/IMRN.2005.3445
W. Fulton and R. Pandharipande, Notes on stable maps and quantum cohomology, Algebraic geometry—Santa Cruz 1995, Proc. Sympos. Pure Math., vol. 62, Amer. Math. Soc., Providence, RI, 1997, pp. 45–96. MR 1492534, DOI https://doi.org/10.1090/pspum/062.2/1492534
William Fulton, Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 2, Springer-Verlag, Berlin, 1984. MR 732620
Andreas Gathmann, Gromov-Witten invariants of blow-ups, J. Algebraic Geom. 10 (2001), no. 3, 399–432. MR 1832328
Sergei I. Gelfand and Yuri I. Manin, Methods of homological algebra, Springer-Verlag, Berlin, 1996. Translated from the 1988 Russian original. MR 1438306
J. Hu, Gromov-Witten invariants of blow-ups along points and curves, Math. Z. 233 (2000), no. 4, 709–739. MR 1759269, DOI https://doi.org/10.1007/s002090050495
Jianxun Hu, Gromov-Witten invariants of blow-ups along surfaces, Compositio Math. 125 (2001), no. 3, 345–352. MR 1818985, DOI https://doi.org/10.1023/A%3A1002694704429
Jianxun Hu and Hou-Yang Zhang, Elliptic GW invariants of blowups along curves and surfaces, Int. J. Math. Math. Sci. 1 (2005), 81–90. MR 2146017, DOI https://doi.org/10.1155/IJMMS.2005.81
Bumsig Kim, Andrew Kresch, and Tony Pantev, Functoriality in intersection theory and a conjecture of Cox, Katz, and Lee, J. Pure Appl. Algebra 179 (2003), no. 1-2, 127–136. MR 1958379, DOI https://doi.org/10.1016/S0022-4049%2802%2900293-1
B. Kim and R. Pandharipande, The connectedness of the moduli space of maps to homogeneous spaces, Symplectic geometry and mirror symmetry (Seoul, 2000) World Sci. Publ., River Edge, NJ, 2001, pp. 187–201. MR 1882330, DOI https://doi.org/10.1142/9789812799821_0006
Andrew Kresch, Canonical rational equivalence of intersections of divisors, Invent. Math. 136 (1999), no. 3, 483–496. MR 1695204, DOI https://doi.org/10.1007/s002220050317
Andrew Kresch, Cycle groups for Artin stacks, Invent. Math. 138 (1999), no. 3, 495–536. MR 1719823, DOI https://doi.org/10.1007/s002220050351

A. Kresch: Intersection theory on Deligne-Mumford stacks, manuscript.

Hsin-Hong Lai, Gromov-Witten invariants of blow-ups along submanifolds with convex normal bundles, Geom. Topol. 13 (2009), no. 1, 1–48. MR 2469512, DOI https://doi.org/10.2140/gt.2009.13.1

Gérard Laumon and Laurent Moret-Bailly, Champs algébriques, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 39, Springer-Verlag, Berlin, 2000 (French). MR 1771927

Jun Li, Stable morphisms to singular schemes and relative stable morphisms, J. Differential Geom. 57 (2001), no. 3, 509–578. MR 1882667

Jun Li and Gang Tian, Virtual moduli cycles and Gromov-Witten invariants of general symplectic manifolds, Topics in symplectic $4$-manifolds (Irvine, CA, 1996) First Int. Press Lect. Ser., I, Int. Press, Cambridge, MA, 1998, pp. 47–83. MR 1635695

F. Nironi, Moduli spaces of stable sheaves on DM-stacks, Ph.D. Thesis, SISSA, 2008.

Martin Olsson, Sheaves on Artin stacks, J. Reine Angew. Math. 603 (2007), 55–112. MR 2312554, DOI https://doi.org/10.1515/CRELLE.2007.012

Zhenbo Qin and Yongbin Ruan, Quantum cohomology of projective bundles over $\mathbf P^n$, Trans. Amer. Math. Soc. 350 (1998), no. 9, 3615–3638. MR 1422617, DOI https://doi.org/10.1090/S0002-9947-98-01968-0

Angelo Vistoli, Intersection theory on algebraic stacks and on their moduli spaces, Invent. Math. 97 (1989), no. 3, 613–670. MR 1005008, DOI https://doi.org/10.1007/BF01388892

Additional Information

Cristina Manolache
Affiliation: International School of Advanced Studies (SISSA/ISAS), Via Beirut, 2-4, 34014 Trieste, Italy
Address at time of publication: Humboldt-Universität zu Berlin, Unter den Linden 6, 10099 Berlin, Germany
MR Author ID: 818812

Received by editor(s): November 3, 2008
Received by editor(s) in revised form: March 19, 2009
Published electronically: December 15, 2011
Additional Notes: The author was partially supported by an ENIGMA grant.