轨形Gromov-Witten不变量沿光滑点的加权涨开公式

Weighted Blow-up of Orbifold Gromov-Witten Invariants Along Smooth Points

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={<p~*a_1,...,p~*a_m,1_x_((-1/a_1))>_(g_1,pl(A)-e’)~xdimX=4,g≥0,∑((-1)g_1·2)/(2g_1+2)!<p~*a_1,...,p~*a_m,1_x_((-1/a_1))>_(g_2,pl(A)-e’)~xdimX=6,g≥0,<p~*a_1,...,p~*a_m,1_x_((-1/a_1))>_(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.

Abstract We study the change of orbifold Gromov-Witten invariants of the form 〈α1…,am,[pt]g,A^x〉of a compact symplectic orbifold groupoid （X, w） under weighted blow-up along smooth points. Here [pt]∈HdR^2n（X）is the generator, dimX=2n. We prove that for nonzero A∈H2（｜X｜,Z）,〈a1…,am,[pt]g,A^x〉……where X is the weight a = （al,..., an） blow-up of X along a smooth point, and a1≥ai,2≤i≤n.