I'm looking for an exact number and a reference and I searched papers about Gromov-Witten invariants but I failed to find an exact number of the degree zero Gromov-Witten invariant of quintic threefold. In Wikipedia, there are only positive degree Gromov-Witten invariants. It will be helpful for me to tell me about the reference and the exact number.
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2$\begingroup$ For every complex manifold $X$, the moduli stack of genus-$0$, $0$-pointed stable maps to $X$ is an empty stack. Thus, it is (tautologically) transverse, the virtual fundamental class equals the actual fundamental class, this is the zero class, and every pairing of this fundamental class against cohomology classes is zero. $\endgroup$– Jason StarrCommented Jan 9, 2018 at 18:13
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$\begingroup$ Typo correction: "... stable maps to $X$ is an empty stack." --> " ... stable maps to $X$ with zero homology class is an empty stack." $\endgroup$– Jason StarrCommented Jan 9, 2018 at 20:43
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1$\begingroup$ In addition to Jason's comments, when $g\geq 2$, $\overline M_{g,0}(X,0)=\overline M_{g}\times X$. The obstruction bundle is given by $\mathbb E^\vee \boxtimes TX$ where $\mathbb E$ is the hodge bundle. Then I believe a splitting principle computation plus a little intersection theory on $\overline M_g$ should give you the result. When $g=1$ you need one marking, and the computation follows from the similar idea. $\endgroup$– HongluCommented Jan 9, 2018 at 22:02
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$\begingroup$ Thank you for Jason and Honglu and I'm sorry for my late reply. I also found a good reference for degree zero Gromov-Witten invariant written by R. Pandharipande : arxiv.org/abs/math/0302077 $\endgroup$– keatonCommented Jan 31, 2018 at 7:04
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