Homological Mirror Symmetry for manifolds of general type

Kapustin, Anton; Katzarkov, Ludmil; Orlov, Dmitri; Yotov, Mirroslav

doi:10.2478/s11533-009-0056-x

Cited by 36 publications

(51 citation statements)

References 18 publications

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“…Let us assume that both X and X ′ are smooth. Then, as explained for example in [3,13], every complex…”

Section: 2mentioning

confidence: 99%

“…Arguments of this sort may prove useful in the study of some instances of mirror symmetry featuring pairs (X , f ) -called in this context Landau-Ginzburg models -as one of the mirror partners. For example, the construction of Landau-Ginzburg models in [1,6,10,13,27] involves arbitrary choices (including resolutions of singularities), and one then has to check that invariants of (X , f ) of interest in mirror symmetry are independent of the choices. One such invariant is the triangulated category of singularities (of the singular fibers of f ).…”

Section: Introductionmentioning

confidence: 99%

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Exponentially Twisted Cyclic Homology

Shklyarov

2016

Int Math Res Notices

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ABSTRACT. By a theorem of Bernhard Keller the de Rham cohomology of a smooth variety is isomorphic to the periodic cyclic homology of the differential graded category of perfect complexes on the variety. Both the de Rham cohomology and the cyclic homology can be twisted by the exponential of a regular function on the variety. We explain that the isomorphism holds true in the twisted setting and draw some conclusions on derived invariance of the algebraic Gauss-Manin systems associated with regular functions.

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“…Let us assume that both X and X ′ are smooth. Then, as explained for example in [3,13], every complex…”

Section: 2mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Exponentially Twisted Cyclic Homology

Shklyarov

2016

Int Math Res Notices

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“…In this case, the mirror is given by a Landau-Ginzburg (LG) model (X, W ) -a pair consisting of a noncompact Kähler manifoldX and a holomorphic function W :X → C (called the superpotential). More generally, one can consider manifolds with an effective anti-canonical divisor [10] or even general type manifolds [57,55,47], for which mirrors are again given by LG models. In this setting, the HMS conjecture has been verified for P 2 and P 1 × P 1 [68, 67], toric del Date: December 22, 2016. Pezzo surfaces [77,78], weighted projective planes and Hirzebruch surfaces [13], del Pezzo surfaces [12], projective spaces [34] and more general projective toric varieties [3,4,35,36, 37], 1 higher genus Riemann surfaces [71,32] and Fano hypersurfaces in projective spaces [73].…”

Section: Introductionmentioning

confidence: 99%

Homological Mirror Symmetry for Local Calabi-Yau Manifolds via SYZ

Chan

2017

Taiwanese J. Math.

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This is a write-up of the author's talk in the conference Algebraic Geometry in East Asia 2016 held at the University of Tokyo in January 2016. We give a survey on the series of papers [16,24, 23,22] where the author and his collaborators Daniel Pomerleano and Kazushi Ueda show how Strominger-Yau-Zaslow (SYZ) transforms can be applied to understand the geometry of Kontsevich's homological mirror symmetry (HMS) conjecture for certain local Calabi-Yau manifolds.

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“…Evidence for mirror symmetry to apply to varieties of positive Kodaira dimension has been given in [Sei08], [KKOY09], [Ef09], [GKR12], [AAK12]. Gross, Katzarkov and the author suggest in [GKR12] that the mirror dual of a curve Z of genus g ≥ 2 is a union of 3g − 3 projective lines that meet in 2g − 2 points such that exactly three components meet in each point.…”

Section: Introductionmentioning

confidence: 99%