Mukai flop and Ruan cohomology
\textit{Y.~Ruan} [Cohomology ring of crepant resolutions of orbifolds, preprint, math.AG/0108195] formulated the cohomological minimal model conjecture, which asserts that \(K\)-equivalent projective manifolds have isomorphic quantum corrected rings. The quantum corrected product corresponding to a birational map is defined similarly to the usual quantum product, with the difference that one takes into account only contributions coming from exceptional effective curves. The authors answer in affirmative the conjecture in the case of simple Mukai flops: \[ \mathbb P^n\cong Z\subset X \overset{\pi}{\dasharrow} X'\supset Z'\cong (\mathbb P^n)^\vee. \] In this case, the quantum corrected product is defined by \[ \langle \alpha *_{\pi}\beta,\gamma \rangle=\sum_{d\geq 0} \Psi^X_{d\ell}(\alpha,\beta,\gamma)q^d, \] where \(\Psi^X_{d\ell}\) is the \(3\)-point Gromov-Witten invariant of \(X\), and \(\ell\in H_2(Z)\) is the class of a line. It can be written as \(\alpha *_\pi\beta=\alpha\cup\beta+\alpha *_{qc} \beta\). For two projective varieties \(X\) and \(X'\) related by a simple Mukai flop, the authors give an explicit isomorphism between their cohomology rings. Secondly, they observe that the correction terms in the quantum corrected product depends only on the local geometry of the exceptional locus, and reduce the problem to computations on the moduli space of stable maps to \(\mathbb P^n\): \[ \Psi^X_{d\ell}(\alpha,\beta,\gamma)=\int_{\overline{\mathcal M}_{0,3} (\mathbb P^n,d)}ev_1^*(\alpha)\cdot ev_2^*(\beta)\cdot ev_3^*(\gamma)\cdot\Phi, \] where the \(ev\)'s are the evaluation morphisms, and \(\Phi\) is an obstruction class corresponding to the embedding \(Z\subset X\). Using the localization formula for the virtual fundamental class of \textit{T.~Graber} and \textit{R.~Pandharipande} [Invent. Math. 135, 487--518 (1999; Zbl 0953.14035)], the authors prove that \(\Phi\) always contributes with a factor of zero, and therefore all invariants (of arbitrary genus) of the form above vanish.
scientific article