Homological mirror symmetry for curves of higher genus

By Alexander I. Efimov. 2009

78 citations recorded by the source corpus.

Abstract This paper is devoted to homological mirror symmetry conjecture for curves of higher genus. It was proposed by Katzarkov as a generalization of original Kontsevichʼs conjecture. A version of this conjecture in the case of the genus two curve was proved by Seidel [25] . Based on the paper of Seidel, we prove the conjecture (in the same version) for curves of genus g ⩾ 3 . Namely, we relate the Fukaya category of a genus g curve to the category of singularities of zero fiber in the mirror dual Landau–Ginzburg model. We also prove a kind of reconstruction theorem for hypersurface singularities. Namely, formal type of hypersurface singularity (i.e. a formal power series up to a formal change of variables) can be reconstructed, with some technical assumptions, from its D ( Z / 2 ) -G category of Landau–Ginzburg branes. The precise statement is Theorem 1.2.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/7a790d6b71abbc7f071168007876ccd59b4773e3

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Homological mirror symmetry for Calabi–Yau hypersurfaces in projective space (2011, unknown): “We construct a minimal model for this A∞ deformation of Λ ∗Cn, and prove that its deformation classes are given exactly by the coefficients of w (following [12]).”
Homological mirror symmetry for punctured spheres (2011, background): “Seidel’s argument was subsequently extended to higher genus curves [12], to pairs of pants and their higher-dimensional analogues [42], and to Calabi-Yau hypersurfaces in projective space [43].”
Lagrangian fibrations on blowups of toric varieties and mirror symmetry for hypersurfaces (2012, background): “A number of recent results [27, 40, 12, 1, 20] suggest that the phenomenon of mirror symmetry is not restricted to Calabi-Yau or Fano manifolds.”
Dynamical systems and categories (2013, background): “Let DπFuk(M) denote the triangulated hull, which is locally proper (in the Z/2-graded sense), and has a generator given by a collection of loops [47, 24].”
On the homological mirror symmetry conjecture for pairs of pants (2010, background): “One direction of this conjecture has been verified for X a curve of genus g ≥ 2, see [47, 14].”
Homological mirror symmetry for the genus two curve (2008, result): “The generalization of our results to curves of any genus ≥ 2 is carried out in [19].”