Semisimple Quantum Cohomology and Blow-ups
By Arend Bayer. 2004
62 citations recorded by the source corpus.
Using results of Gathmann, we prove the following theorem: If a smooth projective variety X has generically semisimple (p,p)-quantum cohomology, then the same is true for the blow-up of X at any number of points. This a successful test for a modified version of Dubrovin's conjecture from the ICM 1998.
Subfields: Algebra and Number Theory, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/9e5ef5ff388608b917c6f2ab87702d3ae1a5f396
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In Algebra and Number Theory
- Michael Entov
- P. Seidel
- Yuri I. Manin
- Michael Usher
- C. Hertling
- L. Polterovich
- C. Teleman
- V. Golyshev
In Geometry and Topology
- Michael Usher
- L. Polterovich
- Michael Entov
- Yong-Geun Oh
- P. Seidel
- C. Teleman
- E. Shelukhin
- Yuri I. Manin
Citations in the corpus
- Symplectic quasi-states and semi-simplicity of quantum homology (2007, background): “Theorem 5.1 below states that this assumption is equivalent to semi-simplicity of QH2n(M) in the sense of the present paper.”
- The structure of 2D semi-simple field theories (2007, background): “Examples of varieties with generically semi-simple quantum cohomology include: toric manifolds, most Fano threefolds with no odd Betti numbers [6], as well as blow-ups of such varieties at any number of points [3].”
- An update on semisimple quantum cohomology and F-manifolds (2008, background): “Bayer [4], who established that blowing up points on such a variety does not destroy semisimplicity.”
- Deformed Hamiltonian Floer theory, capacity estimates and Calabi quasimorphisms (2010, methodology): “Following Bayer [2], we formally adjoin to this ring an invertible element Z D q .”
- Fukaya A∞-structures Associated to Lefschetz Fibrations. II (2014, methodology): “jA<0, then hi A= 0 unless Alies in the image of H 2(Ej) !H 2(E). Similar (actually more general) properties of Gromov-Witten invariants of blowups were derived in the algebro-geometric context in [19, 25, 6]. When summing over classes A, we use a formal parameter qas before, and allow a bulk term”