Cobordism invariants of the moduli space of stable pairs

For a quasi-projective scheme M which carries a perfect obstruction theory, we construct the virtual cobordism class of M. If M is projective, we prove that the corresponding Chern numbers of the virtual cobordism class are given by integrals of the Chern classes of the virtual tangent bundle. Further, we study cobordism invariants of the moduli space of stable pairs introduced by Pandharipande-Thomas. Rationality of the partition function is conjectured together with a functional equation, which can be regarded as a generalization of the rationality and 1/q q symmetry of the Calabi-Yau case. We prove rationality for nonsingular projective toric 3-folds by the theory of descendents.

[1]  R. Pandharipande,et al.  Gromov-Witten/pairs correspondence for the quintic 3-fold , 2012, 1206.5490.

[2]  A. Givental,et al.  Quantum Riemann–Roch, Lefschetz and Serre , 2001, math/0110142.

[3]  R. Pandharipande,et al.  Descendent theory for stable pairs on toric 3-folds , 2010, 1011.4054.

[4]  T. Bridgeland.  Hall algebras and curve-counting invariants , 2010, 1002.4374.

[5]  Richard P. Thomas,et al.  Curve counting via stable pairs in the derived category , 2007, 0707.2348.

[6]  B. Fantechi,et al.  Riemann-Roch theorems and elliptic genus for virtually smooth schemes , 2007, 0706.0988.

[7]  R. Pandharipande,et al.  Descendents on local curves: Stationary theory , 2011, 1109.1258.

[8]  R. Pandharipande,et al.  Hodge integrals and Gromov-Witten theory , 1998 .

[9]  Virtual fundamental classes, global normal cones and Fulton’s canonical classes , 2005, math/0509076.

[10]  Jun Li,et al.  Virtual moduli cycles and Gromov-Witten invariants of algebraic varieties , 1996 .

[11]  Yukinobu Toda.  Generating functions of stable pair invariants via wall-crossings in derived categories , 2008, 0806.0062.

[12]  Richard P. Thomas,et al.  The 3-fold vertex via stable pairs , 2007, 0709.3823.

[13]  Algebraic cobordism revisited , 2006, math/0605196.

[14]  R. Pandharipande,et al.  Gromov–Witten theory and Donaldson–Thomas theory, I , 2003, Compositio Mathematica.

[15]  A. Okounkov,et al.  Gromov-Witten/Donaldson-Thomas correspondence for toric 3-folds , 2008, 0809.3976.

[16]  B. Fantechi,et al.  The intrinsic normal cone , 1996, alg-geom/9601010.

[17]  Virtual fundamental classes via dg-manifolds , 2007, math/0703214.

[18]  M. Levine.  Algebraic Cobordism , 2003, math/0304206.

[19]  R. Pandharipande,et al.  Gromov–Witten/pairs descendent correspondence for toric 3–folds , 2012, 1203.0468.

[20]  Timo Schürg,et al.  DERIVED ALGEBRAIC COBORDISM , 2012, Journal of the Institute of Mathematics of Jussieu.

[21]  Gromov-Witten invariants in algebraic geometry , 1996, alg-geom/9601011.

[22]  Richard P. Thomas,et al.  Curves on K3 surfaces and modular forms , 2010, 1001.2719.

[23]  R. Pandharipande,et al.  Localization of virtual classes , 1997, alg-geom/9708001.

[24]  Richard P. Thomas,et al.  Stable pairs and BPS invariants , 2007, 0711.3899.

[25]  Cumrun Vafa,et al.  Mirror Symmetry , 2000, hep-th/0002222.

[26]  Richard P. Thomas,et al.  Hilbert schemes and stable pairs: GIT and derived category wall crossings , 2009, 0903.1444.

[27]  R. Pandharipande,et al.  Descendents on local curves: rationality , 2010, Compositio Mathematica.

[28]  W. Fulton,et al.  Riemann-roch for singular varieties , 1975 .

[29]  Johan P. Hansen,et al.  INTERSECTION THEORY , 2011 .

[30]  Yukinobu Toda.  Curve counting theories via stable objects I. DT/PT correspondence , 2009, 0902.4371.