Mukai flop and Ruan cohomology
type: Item
instance of: scholarly article
title: Mukai flop and Ruan cohomology
zbMATH Open document ID: 1078.14081
author: Jianxun Hu
author: Wanchuan Zhang
published in: Mathematische Annalen
publication date: 2005-02-09T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0302022
review text: \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 t…
reviewed by: Mihai Halic
Mathematics Subject Classification ID: 14N35
Mathematics Subject Classification ID: 14E05
Mathematics Subject Classification ID: 14E30
zbMATH DE Number: 2132946
zbMATH Keywords: quantum corrected product
zbMATH Keywords: cohomological minimal model conjecture
MaRDI profile type: Publication
OpenAlex ID: W2013671815
arXiv ID: math/0302022
cites work: Birational Calabi--Yau n-folds have equal Betti numbers
cites work: Q4430436
cites work: Conditions for the analyticity of certain sets
cites work: Q3961493
cites work: Q2785535
cites work: Local mirror symmetry: Calculations and interpretations
cites work: Q3328651
cites work: Q4195061
cites work: Localization of virtual classes
cites work: Compact hyperkähler manifolds: Basic results
cites work: Q4858415
cites work: Mirror principle. I
cites work: Symplectic surgery and Gromov-Witten invariants of Calabi-Yau 3-folds
cites work: Blow ups and symplectic embeddings in dimension 4
cites work: Symplectic structure of the moduli space of sheaves on an abelian or K 3 surface
cites work: Quantum cohomology of projective bundles over ℙⁿ
cites work: Quantum cohomology and its application
cites work: Q4946346
cites work: Q5494031
cites work: A mathematical theory of quantum cohomology
cites work: On the topology of birational minimal models
DOI: 10.1007/S00208-004-0561-Y
description: scientific article