Relative Gromov-Witten invariants of projective completions of vector bundles
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摘要
It was proved by Fan and Lee(2016) and Fan(2017) that the absolute Gromov-Witten invariants of two projective bundles P(V_i) → X are identified canonically when the total Chern classes c(V_1) and c(V_2) satisfy c(V_1) = c(V_2) for two bundles V_1 and V_2 over a smooth projective variety X. In this paper, we show that the relative Gromov-Witten invariants of(P(V_i ⊕ O), P(V_i)), i = 1, 2 are identified canonically when c(V_1) = c(V_2),where P(V_i ⊕ O) are the projective completions of the bundles V_i → X, and the projective bundles P(V_i) are the exceptional divisors in P(V_i ⊕ O).
It was proved by Fan and Lee(2016) and Fan(2017) that the absolute Gromov-Witten invariants of two projective bundles P(V_i) → X are identified canonically when the total Chern classes c(V_1) and c(V_2) satisfy c(V_1) = c(V_2) for two bundles V_1 and V_2 over a smooth projective variety X. In this paper, we show that the relative Gromov-Witten invariants of(P(V_i ⊕ O), P(V_i)), i = 1, 2 are identified canonically when c(V_1) = c(V_2),where P(V_i ⊕ O) are the projective completions of the bundles V_i → X, and the projective bundles P(V_i) are the exceptional divisors in P(V_i ⊕ O).
It was proved by Fan and Lee(2016) and Fan(2017) that the absolute Gromov-Witten invariants of two projective bundles P(V_i) → X are identified canonically when the total Chern classes c(V_1) and c(V_2) satisfy c(V_1) = c(V_2) for two bundles V_1 and V_2 over a smooth projective variety X. In this paper, we show that the relative Gromov-Witten invariants of(P(V_i ⊕ O), P(V_i)), i = 1, 2 are identified canonically when c(V_1) = c(V_2),where P(V_i ⊕ O) are the projective completions of the bundles V_i → X, and the projective bundles P(V_i) are the exceptional divisors in P(V_i ⊕ O).
引文
1 Chen B,Du C-Y,Hu J.Weighted blowup correspondence for orbifold Gromov-Witten theory and applications.ArX-iv:1712.01478,2017
2 Chen B,Du C-Y,Wang R.Orbifold Gromov-Witten theory of weighted blowups.Https://www.math.uci.edu/～ruiw10/pdf/weightedGW.pdf,2017
3 Coates T,Givental A.Quantum Riemann-Roch,Lefschetz and Serre.Ann of Math(2),2007,165:15-53
4 Fan H.Chern classes and Gromov-Witten theory of projective bundles.ArXiv:1705.07421,2017
5 Fan H,Lee Y-P.On Gromov-Witten theory for projective bundles.ArXiv:1607.00740,2016
6 Graber T,Pandharipande R.Localization of virtual classes.Invent Math,1999,135:487-518
7 Graber T,Vakil R.Relative virtual localization and vanishing of tautological classes on moduli spaces of curves.Duke Math J,2005,130:1-37
8 Hu J,Li T,Ruan Y.Birational cobordism invariance of uniruled symplectic manifolds.Invent Math,2008,172:231-275
9 Ionel E,Parker T.The symplectic sum formula for Gromov-Witten invariants.Ann of Math(2),2004,159:935-1025
10 Li A-M,Ruan Y.Symplectic surgery and Gromov-Witten invariants of Calabi-Yau 3-folds.Invent Math,2001,145:151-218
11 Li J.A degeration formula of GW-invariants.J Differential Geom,2002,60:199-293
12 Li T-J,Ruan Y.Symplectic birational geometry.In:New Perspectives and Challenges in Symplectic Field Theory.CRM Proceedings&Lectures Notes,vol.49.Providence:Amer Math Soc,2009,307-326
13 Liu C-C M.Localization in(orbifold)Gromov-Witten theory.In:Handbook of Moduli,Volume II.Advanced Lectures in Mathematics,vol.25.Beijing:International Press and Higher Education Press,2013,353-425
14 Maulik D,Pandharipande R.A topological view of Gromov-Witten theory.Topology,2006,45:887-918
2 Chen B,Du C-Y,Wang R.Orbifold Gromov-Witten theory of weighted blowups.Https://www.math.uci.edu/～ruiw10/pdf/weightedGW.pdf,2017
3 Coates T,Givental A.Quantum Riemann-Roch,Lefschetz and Serre.Ann of Math(2),2007,165:15-53
4 Fan H.Chern classes and Gromov-Witten theory of projective bundles.ArXiv:1705.07421,2017
5 Fan H,Lee Y-P.On Gromov-Witten theory for projective bundles.ArXiv:1607.00740,2016
6 Graber T,Pandharipande R.Localization of virtual classes.Invent Math,1999,135:487-518
7 Graber T,Vakil R.Relative virtual localization and vanishing of tautological classes on moduli spaces of curves.Duke Math J,2005,130:1-37
8 Hu J,Li T,Ruan Y.Birational cobordism invariance of uniruled symplectic manifolds.Invent Math,2008,172:231-275
9 Ionel E,Parker T.The symplectic sum formula for Gromov-Witten invariants.Ann of Math(2),2004,159:935-1025
10 Li A-M,Ruan Y.Symplectic surgery and Gromov-Witten invariants of Calabi-Yau 3-folds.Invent Math,2001,145:151-218
11 Li J.A degeration formula of GW-invariants.J Differential Geom,2002,60:199-293
12 Li T-J,Ruan Y.Symplectic birational geometry.In:New Perspectives and Challenges in Symplectic Field Theory.CRM Proceedings&Lectures Notes,vol.49.Providence:Amer Math Soc,2009,307-326
13 Liu C-C M.Localization in(orbifold)Gromov-Witten theory.In:Handbook of Moduli,Volume II.Advanced Lectures in Mathematics,vol.25.Beijing:International Press and Higher Education Press,2013,353-425
14 Maulik D,Pandharipande R.A topological view of Gromov-Witten theory.Topology,2006,45:887-918