Metric SYZ conjecture proved using non-archimedean geometry.
problem Proving the metric SYZ conjecture in large generality.
method Using non-archimedean geometry to support the conjecture.
result Metric SYZ conjecture can be proved in large generality.
Proves SYZ conjecture for certain toric Fano hypersurfaces.
problem Proving the metric SYZ conjecture for specific Calabi-Yau hypersurfaces.
method Solving a variational problem related to the real Monge-Ampère equation on polytopes.
result Minimizer of the variational problem interpreted as a global solution to the real Monge-Ampère equation.
In this thesis, we study a class of special Lagrangian submanifolds of toric Calabi-Yau manifolds and construct their mirrors using some techniques developed in the SYZ programme. We present a justification on the conjecture on the mirror construction of D- branes in Aganagic-Vafa [2]. We apply the techniques employed …
The paper proves a precise SYZ conjecture for toric Calabi-Yau manifolds with singular fibers.
problem Proving the SYZ conjecture for Calabi-Yau manifolds with singular fibers.
method Using family Floer mirror construction and Gross Lagrangian fibration.
result The dual singular fibration is compatible with the family Floer mirror construction.
Researchers resolve a SYZ conjecture for A_n singularities using quantum-corrected T-duality.
problem Resolving a mathematically precise SYZ conjecture for A_n singularities.
method Building a quantum-corrected T-duality between two singular torus fibrations.
result Constructing a parameter-dependent SYZ mirror fibration partner with matching singular loci and integral affine structure.
In this note, we study the SYZ mirror construction for a toric Calabi-Yau manifold using instanton corrections coming from Woodward's quasimap Floer theory instead of Fukaya-Oh-Ohta-Ono's Lagrangian Floer theory. We show that the resulting SYZ mirror coincides with the one written down via physical means (as expected).
Proves a conjecture for Calabi-Yau manifolds.
problem Maximal degeneration of Calabi-Yau manifolds.
method Valuative independence condition for section ring.
result Metric SYZ conjecture proven.
Survey on SYZ mirror symmetry using Morse theory.
problem Understanding SYZ mirror symmetry with quantum corrections.
method Using Witten-Morse theory and Fukaya's combinatorial structures.
result Explicit relation between geometric and combinatorial structures.
Proves stability in Weyl polytopes using optimal transport.
problem Stability of Weyl polytopes under optimal transport.
method Optimal transport stability for reflexive Weyl polytopes.
result Weak metric SYZ conjecture holds for Delzant reflexive Weyl polytopes.
The paper proves a version of the SYZ conjecture for hyperkahler manifolds.
problem Proving the SYZ conjecture for hyperkahler manifolds with specific conditions.
method Introduced a Teichmuller space to parametrize pairs of hyperkahler manifolds up to isotopy, used a version of the global Torelli theorem to deduce the deformation invariance of semiampleness.
result Proved the SYZ conjecture for hyperkahler manifolds with specific conditions.
Non-archimedean SYZ fibration constructed for Calabi-Yau hypersurfaces.
problem Analyzing Calabi-Yau hypersurfaces using non-archimedean geometry.
method Yamamoto's tropical contractions and Li's Fermat degeneration, with toric plurisubharmonic metrics.
result Constant potential along fibers of retraction under discrete symmetry assumption.
Homological mirror symmetry for toric Fano surfaces using Morse homotopy.
problem Establishing homological mirror symmetry for toric Fano surfaces.
method Applying SYZ construction and using Morse homotopy of the moment polytope.
result Homological mirror symmetry achieved for toric Fano surfaces.
We survey recent developments in the study of SYZ mirror symmetry for compact toric and toric Calabi-Yau varieties, with a special emphasis on works of the author and his collaborators.
Mathematically proves SYZ conjecture for conifold transition.
problem Tackles the SYZ conjecture for conifold transition.
method Uses family Floer context and non-archimedean setting.
result Explicitly writes singular T-duality fibers and confirms missing points in mirror cluster variety.
The paper explores non-Kähler SYZ mirrors for solvmanifolds, proving cohomological properties and constructing new mirror pairs.
problem Understanding non-Kähler SYZ mirrors for solvmanifolds and their cohomological properties.
method Investigates geometric and cohomological properties, proving relationships and constructing mirror pairs.
result Proves the Fourier-Mukai transform exchanges type-A and type-B cycles, and provides criteria for non-Kähler SYZ mirror pairs.
Study explores unstable 3-forms on Calabi-Yau 3-folds.
problem Understanding degenerations of Calabi-Yau 3-folds via 3-forms.
method Investigates geometries of 3-forms on symplectic 6-manifolds.
result Unstable 3-forms reveal rich geometric properties related to SYZ conjecture.
Constructs Calabi-Yau metrics on 3-folds with properties similar to Taub-NUT and Ooguri-Vafa.
problem Creating metrics on Calabi-Yau 3-folds with specific properties.
method Constructs families of Calabi-Yau metrics on $\C^3$ and Calabi-Yau 3-folds with properties similar to Taub-NUT and Ooguri-Vafa.
result Demonstrates construction of Calabi-Yau metrics with properties similar to Taub-NUT and Ooguri-Vafa.
The paper solves Monge-Ampère equations on reflexive polytopes, linking solvability to SYZ conjecture and tropical geometry.
problem Solvability of Monge-Ampère equations on reflexive polytopes.
method Analyzes reflexive polytopes with height functions, proving conditions for Monge-Ampère solvability and linking to SYZ conjecture.
result Conditions for Monge-Ampère solvability are necessary and sufficient, and solvability implies the SYZ conjecture for Calabi-Yau hypersurfaces.
In this paper, we study the geometry of the SYZ transform on a semi-flat Lagrangian torus fibration. Our starting point is an investigation on the relation between Lagrangian surgery of a pair of straight lines in a symplectic 2-torus and extension of holomorphic vector bundles over the mirror elliptic curve, via the S…
Survey on metric SYZ conjecture and non-archimedean geometry.
problem Existence of special Lagrangian fibrations on Calabi-Yau manifolds.
method Pluripotential theory and non-archimedean geometry.
result Subtleties and open questions in the conjectural picture.
Mathematical framework for brane quantization using SYZ mirror symmetry.
problem Developing a mathematical framework for brane quantization.
method Applying SYZ mirror symmetry to construct and analyze branes.
result Established a mathematical definition of endomorphism algebras and their isomorphisms.
We study SYZ mirror symmetry in the context of non-Kaehler Calabi-Yau manifolds. In particular, we study the six-dimensional Type II supersymmetric SU(3) systems with Ramond-Ramond fluxes, and generalize them to higher dimensions. We show that Fourier-Mukai transform provides the mirror map between these Type IIA and…
Constructs mirror pairs for solvmanifolds using Lie groups.
problem Finding mirror pairs for non-Kaehler solvmanifolds.
method Left-invariant affine structures on Lie groups.
result Explicitly finds SYZ mirror symmetric partners for all known compact 6D solvmanifolds.
Paper explains scattering diagrams' role in mirror symmetry.
problem Reconstruction problem in mirror symmetry.
method Introduction of scattering diagrams and their role in SYZ and HMS conjectures.
result Scattering diagrams help in understanding mirror symmetry.
Identifies special Lagrangian submanifolds in non-Kähler Calabi-Yau manifolds.
problem Determining special Lagrangian submanifolds in non-Kähler Calabi-Yau manifolds.
method Algebraic equations, invariant distributions, deformation theory, SYZ mirror symmetry.
result Existence of topologically distinct SLags and non-Kähler SYZ mirrors.
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 a series of papers of the author and his collaborators Daniel Pomerleano and Kazushi Ueda where we show how Strominger-Yau-Zaslow (SYZ) transforms can be …
In this paper we give a construction of Lagrangian torus fibration for Calabi-Yau hypersurface in toric variety via the method of gradient flow. Using our construction of Lagrangian torus fibration, we are able to prove the symplectic topological version of SYZ mirror conjecture for generic Calabi-Yau hypersurface in t…
The SYZ Conjecture explains Mirror Symmetry between mirror Calabi-Yau 3-folds M,M' in terms of special Lagrangian fibrations f : M --> B and f' : M' --> B over the same base B, whose fibres are dual 3-tori, except for singular fibres. One of the main problems in proving the SYZ Conjecture (or even in finding the right …
This paper proves a bijection between complex and symplectic categories of tori.
problem Ambiguities in transition functions cause difficulties in constructing a functor.
method SYZ construction and transform to solve ambiguities and prove bijection.
result Proves the existence of a bijection between complex and symplectic categories of tori.
The mirror of a projective toric manifold XΣ is given by a Landau-Ginzburg model (Y,W). We introduce a class of Lagrangian submanifolds in (Y,W) and show that, under the SYZ mirror transformation, they can be transformed to torus-invariant hermitian metrics on holomorphic line bundles over XΣ. Through this ge…
Constructs hyperbolic affine spheres and Calabi-Yau metrics.
problem Constructing non-isometric Calabi-Yau metrics.
method Non-Abelian Hodge theory for parabolic Higgs bundles.
result Answers a question about Calabi-Yau metrics.
Researchers match complex affine structures in mirror constructions.
problem Matching complex affine structures in SYZ fibrations of Del Pezzo surfaces.
method Floer-theoretical gluing method to construct mirrors using immersed Lagrangians.
result The constructed mirror agrees with Carl-Pomperla-Siebert's mirror.
Constructing brane quantization for An-resolutions using SYZ mirror symmetry.
problem Quantizing branes on singular fibers using SYZ mirror symmetry.
method Constructing coisotropic A-branes and their mirrors via fiberwise geometric quantization.
result Establishing a mirror isomorphism between endomorphism algebras.
Constructs special Lagrangian submanifolds in Calabi-Yau 3-folds.
problem Constructing special Lagrangian submanifolds in collapsing Calabi-Yau 3-folds.
method Constructs special Lagrangian submanifolds in collapsing Calabi-Yau 3-folds fibered by K3 surfaces.
result Special Lagrangian submanifolds shrink to 1-dimensional graphs in the base as the 3-folds collapse.
The paper explores complex geometries of 3-forms on symplectic 6-manifolds.
problem Understanding the geometry of 3-forms on symplectic 6-manifolds.
method Investigation of geometries associated with 3-forms of various orbital types.
result Rich geometric structures attached to unstable 3-forms from Calabi-Yau degeneration.
The abstract proves a conjecture about geometric structures in Calabi-Yau orbifolds.
problem Proving a conjecture about geometric structures in Calabi-Yau orbifolds.
method Using the Koopman--von Neumann formulation of Landau--Ginzburg theory and a Lagrangian torus fibration.
result The base of the SYZ fibration is a Monge--Ampère domain (the open simplex) for all Berglund--Hübsch--Krawitz mirror pairs.
The Kobayashi pseudometric on a complex manifold is the maximal pseudometric such that any holomorphic map from the Poincaré disk to the manifold is distance-decreasing. Kobayashi has conjectured that this pseudometric vanishes on Calabi-Yau manifolds. Using ergodicity of complex structures, we prove this conjecture fo…
We study the moduli spaces of flat SL(r)- and PGL(r)-connections, or equivalently, Higgs bundles, on an algebraic curve. These spaces are noncompact Calabi-Yau orbifolds; we show that they can be regarded as mirror partners in two different senses. First, they satisfy the requirements laid down by Strominger-Yau-Zaslow…
The paper explores Higgs bundles and their moduli spaces on Riemann surfaces.
problem Developing the moduli theory of Higgs bundles and understanding their geometric properties.
method Establishing non-abelian Hodge correspondences and studying Hitchin fibration.
result Computing the Poincaré polynomial of rank 2 moduli space and verifying topological mirror symmetry.
Extends noncommutative deformations of holomorphic line bundles on complex tori and their mirror partners.
problem Noncommutative deformations of holomorphic line bundles on complex tori.
method Real nonformal deformation quantization and SYZ construction.
result Extended construction of noncommutative deformations of holomorphic line bundles.
New category theory for complex projective plane sections.
problem Defining multi-valued Morse homotopy for complex projective plane.
method Introducing multi-valued Morse homotopy category and showing equivalence to DG category of holomorphic vector bundles.
result Multi-valued Morse homotopy category is equivalent to DG category of holomorphic vector bundles.
This article surveys the development of the SYZ conjecture since it was proposed by Strominger, Yau and Zaslow in their famous 1996 paper, and discusses how it has been leading us to a thorough understanding of the geometry underlying mirror symmetry.
In this survey paper, we briefly review various aspects of the SYZ approach to mirror symmetry for non-Calabi-Yau varieties, focusing in particular on Lagrangian fibrations and wall-crossing phenomena in Floer homology. Various examples are presented, some of them new.
Study Monge-Ampère equations on Calabi-Yau hypersurfaces, proving unique solutions and implications for special Lagrangian fibrations.
problem Existence of special Lagrangian fibrations in Calabi-Yau hypersurfaces.
method Non-Archimedean and tropical Monge-Ampère equations on Berkovich and skeleton spaces, proving uniqueness and deriving solutions.
result Unique solutions to tropical and non-Archimedean Monge-Ampère equations, leading to existence of special Lagrangian fibrations.
In this article we construct Lagrangian torus fibrations for general quintic \cy hypersurfaces near the large complex limit and their mirror manifolds using gradient flow method. Then we prove the Strominger-Yau-Zaslow mirror conjecture for this class of \cy manifolds in symplectic category.
We introduce special Lagrangian submanifolds in C^m and in (almost) Calabi-Yau manifolds, and survey recent results on singularities of special Lagrangian submanifolds, and their application to the SYZ Conjecture. The paper is aimed at graduate students in Geometry, String Theorists, and others wishing to learn the sub…
We study hyperkahler metrics and hyperholomorphic connections of Hitchin's moduli spaces after Gaiotto, Moore and Neitzke. Their construction via the twistor technique produces intricate wall crossing behaviors. For certain four dimensional Hitchin's moduli spaces local models and degeneration to local models near sing…
Exposes how Hessian manifold duality aids in solving optimal transport problems.
problem Solving Monge-Ampère equations and understanding mirror symmetry.
method Explains duality theory for Hessian manifolds and its application to optimal transport.
result Provides a natural setting for optimal transport and solves Monge-Ampère equations.