轨形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,当一个紧辛轨形群胚(X,ω)沿着光滑点作加权涨开时,它的形如<α_1,…,α_m,[pt]>_(g,A)~X的轨形Gromov-Witten不变量的变化公式,其中[pt]∈H_(dR)~(2n)(X)是生成元,dimX=2n.我们证明了对于非零A∈H_2(|X|,Z),<α_1,…,α_m,[pt]>_(g,A)~X={_(g_1,pl(A)-e’)~xdimX=4,g≥0,∑((-1)g_1·2)/(2g_1+2)!_(g_2,pl(A)-e’)~xdimX=6,g≥0,_(g_1,pl(A)-e’)~xdimX≥8,g=0其中x是X沿一光滑点的权α=(α_1,…,α_n)的加权涨开,且α_1≥α_i,2≤i≤n.
We study the change of orbifold Gromov-Witten invariants of the form〈α_1,..., α_m,[pt]〉_(g,A)~X of a compact symplectic orbifold groupoid(X,ω) under weighted blow-up along smooth points. Here [pt]∈H_(dR)~(2 n)(X) is the generator, dimX=2n. We prove that for nonzero A∈H_2(|X|,Z),<α_1,…,α_m,[pt]>_(g,A)~X={_(g_1,pl(A)-e’)~xdimX=4,g≥0,∑((-1)g_1·2)/(2g_1+2)!_(g_2,pl(A)-e’)~xdimX=6,g≥0,_(g_1,pl(A)-e’)~xdimX≥8,g=0,where X is the weight a =(α_1,…,α_n) blow-up of X along a smooth point, and α_1≥α_i,2≤i≤n.
We study the change of orbifold Gromov-Witten invariants of the form〈α_1,..., α_m,[pt]〉_(g,A)~X of a compact symplectic orbifold groupoid(X,ω) under weighted blow-up along smooth points. Here [pt]∈H_(dR)~(2 n)(X) is the generator, dimX=2n. We prove that for nonzero A∈H_2(|X|,Z),<α_1,…,α_m,[pt]>_(g,A)~X={
引文
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[2]Abramovich D.,Fantechi B.,Orbifold techniques in Degeneration formulas,Ann.Sci Norm.Super.Pisa Cl.Sci.,2016,16(5):519-579.
[3]Adem A.,Leida J.,Ruan Y.,Orbifolds and Stringy Topology,Cambridge University Press,Cambridge,2007.
[4]Chen B.,Li A.M.,Sun S.,et al.,Relative orbifold Gromov-Witten theory and degeneration formula,arXiv:1110.6803.
[5]Chen B.,Li A.M.,Wang B.,Virtual neighborhood technique for pseudo-holomorphic spheres,arXiv:1306.3276.
[6]Chen W.,Ruan Y.,A new cohomology theory of orbifold,Comm.Math.Phys.,2004,248(1):1-31.
[7]Chen W.,Ruan Y.,Orbifold quantum cohomology,Cont.Math.,310:25-85.
[8]Godinho L.,Blowing up symplectic orbifolds,Ann.Global Anal.Geom.,2001,20(2):117-162.
[9]Graber T.,Pandharipande R.,Localization of virtual classes,Invent.Math.,1999,135(2):487-518.
[10]Graber T.,Vakil R.,Relative virtual localization and vanishing of tautological classes on moduli spaces of curves,Duke Math.J.,2005,130(1):1-37.
[11]Hu J.,Gromov-Witten invariants of blow-ups along points and curves,Math.Z.,2000,233:709-739.
[12]He W.,Hu J.,Orbifold Gromov-Witten invariants of weighted blow-up at smooth points,Acta Math.Sin.Engl.Ser.,2015,31(5):825-846.
[13]He W.,Hu J.,Ke H.Z.,et al.,Blowup formulae of high genus Gromov-Witten invariants in dimensional six,arXiv:1402.4221.
[14]Hu J.,Li T.J.,Ruan Y.,Birational cobordism invariance of uniruled symplectic manifolds,Invent.Math.,2008,172(2):231-275.
[15]Lerman E.,Symplectic cuts,Math.Research Lett.,1995,2:247-258.
[16]Li A.M.,Ruan Y.,Symplectic surgery and Gromov-Witten invariants of Calabi-Yau 3-folds,Invent.Math.,2001,145(1):151-218.
[17]Maulik D.,Pandharipande R.,A topological view of Gromov-Witten theory,Topology,2006,45:887-918.
[18]Mcduff D.,Hamiltonian S~1-manifolds are uniruled,Duke Math.J.,2009,146(3):449-506.
[19]Moerdijk I.,Pronk D.A.,Orbifolds,sheaves and groupoids,K-Theory,1997,12(1):3-21.
[20]Qi X.,A blow-up formula of high-genus Gromov-Witten invariants of symplectic 4-manifolds,Adv.Math.China,2014,43(4):603-607.
[21]Satake I.,On a generalization of the notion of manifold,Proc.Nat.Acad.Sci.U.S.A.,1956,42:359-363.