轨形Riemann面的Gromov-Witten不变量沿光滑点的加权涨开公式
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摘要
本文考虑当一个紧辛轨形(orbifold)Riemann面(X,ω)沿着光滑点作加权涨开时,它的轨形Gromov-Witten不变量的变化情形和辛一致规则(uniruledness)性质的变化情形;证明了如下的结果:<α_1,...,α_m>~X _g,A=~X _g,A=I_A·■.第一个公式表明,当(X,ω)是辛一致规则的(uniruled)时,它的沿光滑点的加权涨开■也是辛一致规则的.
In this paper, we study the change of orbifold Gromov-Witten invariants, and the change of the symplectic uniruledness of a compact symplectic orbifold Riemannian surface under weighted blow-up along smooth points. We prove that<α_1,...,α_m>~X _g,A=~X _g,A=I_A·■.The first formula shows that when(X, ω) is symplectic uniruled, then so is( X, ω), the weighted blow up of■along smooth point.
In this paper, we study the change of orbifold Gromov-Witten invariants, and the change of the symplectic uniruledness of a compact symplectic orbifold Riemannian surface under weighted blow-up along smooth points. We prove that<α_1,...,α_m>~X _g,A=
引文
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12 Qi X.A blow-up formula of high-genus Gromov-Witten invariants of symplectic 4-manifolds.Adv Math(China),2014,43:603-607
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14 He W,Hu J.Orbifold Gromov-Witten invariants of weighted blow-up at smooth points.Acta Math Sin Engl Ser,2015,31:825-846
15 Adem A,Leida J,Ruan Y.Orbifolds and Stringy Topology.Cambridge:Cambridge University Press,2007
16 Chen B,Hu S.A de Rham model for Chen-Ruan cohomology ring of abelian orbifolds.Math Ann,2006,336:51-71
17 Satake I.On a generalization of the notion of manifold.Proc Natl Acad Sci USA,1956,42:359-363
18 Godinho L.Blowing up symplectic orbifolds.Ann Global Anal Geom,2001,20:117-162
19 Chen W,Ruan Y.A new cohomology theory of orbifold.Comm Math Phys,2004,248:1-31
20 Chen W,Ruan Y.Orbifold quantum cohomology.Contemp Math,2000,310:25-85
21 Chen B,Li A M,Wang B.Virtual neighborhood technique for pseudo-holomorphic spheres.ArXiv:1306.3276,2013
2 Chen B,Tian G.Virtual manifolds and localization.Acta Math Sin Engl Ser,2010,26:1-24
3 Chen B,Li A M.Symplectic virtual localization of Gromov-Witten invariants.Ar Xiv:math/0610370,2006
4 Graber T,Vakil R.Relative virtual localization and vanishing of tautological classes on moduli spaces of curves.Duke Math J,2005,130:1-37
5 Li A M,Ruan Y.Symplectic surgery and Gromov-Witten invariants of Calabi-Yau 3-folds.Invent Math,2001,145:151-218
6 Li J.A degeration formula of GW-invariants.J Differential Geom,2002,60:199-293
7 Chen B,Li A M,Sun S,et al.Relative orbifold Gromov-Witten theory and degeneration formula.Ar Xiv:1110.6803,2011
8 Abramovich D,Fantechi B.Orbifold techniques in degeneration formulas.Ann Sc Norm Super Pisa Cl Sci(5),2016,16:519-579
9 Hu J.Gromov-Witten invariants of blow-ups along points and curves.Math Z,2000,233:709-739
10 Hu J.Gromov-Witten invariants of blow-ups along surfaces.Compos Math,2001,125:345-352
11 Hu J.Local Gromov-Witten invariants of blowups of Fano surfaces.J Geom Phys,2011,61:1051-1060
12 Qi X.A blow-up formula of high-genus Gromov-Witten invariants of symplectic 4-manifolds.Adv Math(China),2014,43:603-607
13 He W,Hu J,Ke H Z,et al.Blowup formulae of high genus Gromov-Witten invariants in dimensional six.Ar Xiv:1402.4221,2014
14 He W,Hu J.Orbifold Gromov-Witten invariants of weighted blow-up at smooth points.Acta Math Sin Engl Ser,2015,31:825-846
15 Adem A,Leida J,Ruan Y.Orbifolds and Stringy Topology.Cambridge:Cambridge University Press,2007
16 Chen B,Hu S.A de Rham model for Chen-Ruan cohomology ring of abelian orbifolds.Math Ann,2006,336:51-71
17 Satake I.On a generalization of the notion of manifold.Proc Natl Acad Sci USA,1956,42:359-363
18 Godinho L.Blowing up symplectic orbifolds.Ann Global Anal Geom,2001,20:117-162
19 Chen W,Ruan Y.A new cohomology theory of orbifold.Comm Math Phys,2004,248:1-31
20 Chen W,Ruan Y.Orbifold quantum cohomology.Contemp Math,2000,310:25-85
21 Chen B,Li A M,Wang B.Virtual neighborhood technique for pseudo-holomorphic spheres.ArXiv:1306.3276,2013