Mirror Symmetry and Fano Manifolds

@inproceedings{Coates2012MirrorSA,
  title={Mirror Symmetry and Fano Manifolds},
  author={Tom Coates and Alessio Corti and Sergey Galkin and Vasily Golyshev and Alexander M. Kasprzyk},
  year={2012}
}
We consider mirror symmetry for Fano manifolds, and describe how one can recover the classification of 3-dimensional Fano manifolds from the study of their mirrors. We sketch a program to classify 4-dimensional Fano manifolds using these ideas. 
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References

SHOWING 1-10 OF 43 REFERENCES
Mirror symmetry, mirror map and applications to Calabi-Yau hypersurfaces
Mirror Symmetry, Picard-Fuchs equations and instanton corrected Yukawa couplings are discussed within the framework of toric geometry. It allows to establish mirror symmetry of Calabi-Yau spaces forExpand
The versal deformation of an isolated toric Gorenstein singularity
Given a lattice polytope Q ⊆ ℝn, we define an affine scheme that reflects the possibilities of splitting Q into a Minkowski sum. Denoting by Y the toric Gorenstein singularity induced by Q, weExpand
Hodge theoretic aspects of mirror symmetry
We discuss the Hodge theory of algebraic non-commutative spaces and analyze how this theory interacts with the Calabi-Yau condition and with mirror symmetry. We develop an abstract theory ofExpand
Surveys in geometry and number theory : reports on contemporary Russian mathematics
Preface 1. Affine embeddings of homogeneous spaces I. V. Arzhantsev 2. Formal groups over local fields: a constructive approach M. V. Bondarko 3. Classification problems and mirror duality V. V.Expand
Complete classification of reflexive polyhedra in four dimensions
Four dimensional reflexive polyhedra encode the data for smooth Calabi-Yau threefolds that are hypersurfaces in toric varieties, and have important applications both in perturbative and inExpand
Gravitational Quantum Cohomology
We discuss how the theory of quantum cohomology may be generalized to "gravitational quantum cohomology" by studying topological σ models coupled to two-dimensional gravity. We first consider σExpand
QUANTUM COHOMOLOGY OF A
The rank 4 degeneracy locus of a general skew-symmetric 7 × 7-matrix with Γ(OP6(1))-coefficients defines a non-complete intersection Calabi-Yau 3-fold X3 with h1,1 = 1. We recall some results ofExpand
Algebraic D-modules
Presented here are recent developments in the algebraic theory of D-modules. The book contains an exposition of the basic notions and operations of D-modules, of special features of coherent,Expand
Classification of Reflexive Polyhedra in Three Dimensions
We present the last missing details of our algorithm for the classification of reflexive polyhedra in arbitrary dimensions. We also present the results of an application of this algorithm to the caseExpand
Upper Bounds for Mutations of Potentials
In this note we provide a new, algebraic proof of the excessive Laurent phe- nomenon for mutations of potentials (in the sense of (Galkin S., Usnich A., Preprint IPMU 10-0100, 2010)) by introducingExpand
...
1
2
3
4
5
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