Mirror Symmetry and Fano Manifolds

  title={Mirror Symmetry and Fano Manifolds},
  author={Tom Coates and Alessio Corti and Sergey Galkin and Vasily Golyshev and Alexander M. Kasprzyk},
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. 
Quantum Periods for Certain Four-Dimensional Fano Manifolds
A list of known four-dimensional Fano manifolds is collected and their quantum periods are computed to derive certain complete intersections in projective bundles. Expand
Four-dimensional Fano toric complete intersections
We find at least 527 new four-dimensional Fano manifolds, each of which is a complete intersection in a smooth toric Fano manifold.
We review mirror symmetry for the quantum cohomology D-module of a compact weak-Fano toric manifold. We also discuss the relationship to the GKZ system, the Stanley-Reisner ring, the Mellin-BarnesExpand
Lagrangian torus fibration models of Fano threefolds
Inspired by the work of Gross on topological Mirror Symmetry we construct candidate Lagrangian torus fibration models for the 105 families of smooth Fano threefolds. We prove, in the case the secondExpand
Some examples of non-smoothable Gorenstein Fano toric threefolds
We present a combinatorial criterion on reflexive polytopes of dimension 3 which gives a local-to-global obstruction for the smoothability of the corresponding Fano toric threefolds. As a result, weExpand
Laurent inversion
We describe a practical and effective method for reconstructing the deformation class of a Fano manifold X from a Laurent polynomial f that corresponds to X under Mirror Symmetry. We exploreExpand
On Mirror Symmetry for Fano varieties and for singularities
In this thesis we discuss some aspects of Mirror Symmetry for Fano varieties and toric singularities. We formulate a conjecture that relates the quantum cohomology of orbifold del Pezzo surfaces to aExpand
Quantum reconstruction for Fano bundles on projective space
Abstract We present a reconstruction theorem for Fano vector bundles on projective space which recovers the small quantum cohomology for the projectivization of the bundle from a small number ofExpand
An Example of Mirror Symmetry for Fano Threefolds
In this note we illustrate the Fanosearch programme of Coates, Corti, Galkin, Golyshev, and Kasprzyk in the example of the anticanonical cone over the smooth del Pezzo surface of degree 6.
Degenerations, transitions and quantum cohomology
Given a singular variety I discuss the relations between quantum cohomology of its resolution and smoothing. In particular, I explain how toric degenerations helps with computing Gromov--WittenExpand


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
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