arXiv research

Non-archimedean SYZ fibration constructed for Calabi-Yau hypersurfaces.

The paper proves a precise SYZ conjecture for toric Calabi-Yau manifolds with singular fibers.

Mathematically proves SYZ conjecture for conifold transition.

Researchers resolve a SYZ conjecture for A_n singularities using quantum-corrected T-duality.

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…

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…

Survey on metric SYZ conjecture and non-archimedean geometry.

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 …

Researchers match complex affine structures in mirror constructions.

Homological mirror symmetry for toric Fano surfaces using Morse homotopy.

The abstract proves a conjecture about geometric structures in Calabi-Yau orbifolds.

We construct a family of Calabi-Yau metrics on $\C^3$ with properties analogous to the Taub-NUT metric on $\C^2$, and construct a family of Calabi-Yau 3-fold metric models on the positive and negative vertices of SYZ fibrations with properties analogous to the Ooguri-Vafa metric.

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.

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.

Study Monge-Ampère equations on Calabi-Yau hypersurfaces, proving unique solutions and implications for special Lagrangian fibrations.

The paper explores Higgs bundles and their moduli spaces on Riemann surfaces.

For the sake of hyperk{ä}hler SYZ conjecture, finding holomorphic Lagrangian fibrations becomes an important issue. Toric hyperk{ä}hler manifolds are real dimension $4n$ non-compact hyperk{ä}hler manifolds which are quaternion analog of toric varieties. The $n$ dimensional residue circle action on it admitting a hyperk…

Parabolic automorphisms on hyperkahler manifolds act ergodically on fibers.

Identifies special Lagrangian submanifolds in non-Kähler Calabi-Yau manifolds.

The paper explores non-Kähler SYZ mirrors for solvmanifolds, proving cohomological properties and constructing new mirror pairs.

We produce special Lagrangian $T^n$-fibrations on the generic regions of some Calabi-Yau hypersurfaces in the Fermat family $X_s=\{ Z_0\ldots Z_{n+1}+ e^{-s} ( Z_0^{n+2}+ \ldots Z_{n+1}^{n+2} ) =0 \}\subset \mathbb{CP}^{n+1}$ near the large complex structure limit $s\to +\infty$.

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…

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…

The study shows boundedness of certain fibered varieties in algebraic geometry.

This paper focuses on a topological version on the Strominger-Yau-Zaslow mirror symmetry conjecture. Roughly put, the SYZ conjecture suggests that mirror pairs of Calabi-Yau manifolds are related by the existence of dual special Lagrangian torus fibrations. We explore this conjecture without reference to the special La…

Proves non-hyperbolicity of symplectic varieties with specific properties.

Mathematical framework for brane quantization using SYZ mirror symmetry.

By the SYZ construction, a mirror pair $(X,\check{X})$ of a complex torus $X$ and a mirror partner $\check{X}$ of the complex torus $X$ is described as the special Lagrangian torus fibrations $X \rightarrow B$ and $\check{X} \rightarrow B$ on the same base space $B$. Then, by the SYZ transform, we can construct a simpl…

This is a survey of the author's series of three papers math.DG/0111324, math.DG/0111326, math.DG/0204343 using analysis to investigate special Lagrangian 3-folds (SL 3-folds) in C^3 invariant under the U(1)-action (z_1,z_2,z_3) --> (gz_1,g^{-1}z_2,z_3) for unit complex numbers g, and their sequel math.DG/0011179 on sp…

Extends noncommutative deformations of holomorphic line bundles on complex tori and their mirror partners.

Proves SYZ mirror symmetry for del Pezzo and rational elliptic surfaces.

This survey was written for the Current Developments in Mathematics conference, 2012, and is an updating of my article "The Strominger-Yau-Zaslow conjecture: From torus fibrations to degenerations," in the Seattle 2005 proceedings. We trace progress and thinking about the SYZ conjecture since its introduction in 1996. …

Metric SYZ conjecture proved using non-archimedean geometry.

Proves SYZ conjecture for certain toric Fano hypersurfaces.

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 …

Proves stability condition for Lagrangian sections in toric weak Fano manifolds.

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.

Proves stability in Weyl polytopes using optimal transport.

The paper proves a version of the SYZ conjecture for hyperkahler manifolds.

This paper gives a leisurely introduction to Calabi-Yau manifolds and special Lagrangian submanifolds from the differential geometric point of view, followed by a survey of recent results on singularities of special Lagrangian submanifolds, and their application to the SYZ Conjecture. It is aimed at graduate students i…

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.

Study explores unstable 3-forms on Calabi-Yau 3-folds.

The paper solves Monge-Ampère equations on reflexive polytopes, linking solvability to SYZ conjecture and tropical geometry.

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.

The paper proves mirror symmetry for del Pezzo surfaces and computes related structures.

Paper explains scattering diagrams' role in mirror symmetry.