May 4, 2004
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July 12, 2024
We consider a toric degeneration $\mathcal{X}$ of Calabi--Yau complete intersections of Batyrev--Borisov in the Gross--Siebert program. For the toric degeneration $\mathcal{X}$, we study the real Monge--Amp\`{e}re equation corresponding to the non-archimedean Monge--Amp\`{e}re equation that yields the non-archimedean Calabi--Yau metric. Our main theorem describes the real Monge--Amp\`{e}re equation in terms of tropical geometry and proves the metric SYZ conjecture for the tor...
January 30, 2023
We prove the metric version of the SYZ conjecture for a class of Calabi-Yau hypersurfaces inside toric Fano manifolds, by solving a variational problem whose minimizer may be interpreted as a global solution of the real Monge-Amp\`ere equation on certain polytopes. This does not rely on discrete symmetry.
August 13, 2001
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 in Geometry, String Theorists, and others wishing to learn the subject, and is designed to be fairly self-contained. It is based on lecture courses given at No...
November 26, 2010
An AH (affine hypersurface) structure is a pair comprising a projective equivalence class of torsion-free connections and a conformal structure satisfying a compatibility condition which is automatic in two dimensions. They generalize Weyl structures, and a pair of AH structures is induced on a co-oriented non-degenerate immersed hypersurface in flat affine space. The author has defined for AH structures Einstein equations, which specialize on the one hand to the usual Einste...
February 3, 2017
We come up with infinite-dimensional prequantum line bundles and moment map interpretations of three different sets of equations - the generalised Monge-Amp`ere equation, the almost Hitchin system, and the Calabi-Yang-Mills equations. These are all perturbations of already existing equations. Our construction for the generalised Monge-Amp`ere equation is conditioned on a conjecture from algebraic geometry. In addition, we prove that for small values of the perturbation parame...
May 31, 2010
In the neighborhood of a regular point, generalized Kahler geometry admits a description in terms of a single real function, the generalized Kahler potential. We study the local conditions for a generalized Kahler manifold to be a generalized Calabi-Yau manifold and we derive a non-linear PDE that the generalized Kahler potential has to satisfy for this to be true. This non-linear PDE can be understood as a generalization of the complex Monge-Ampere equation and its solutions...
March 5, 2015
The classes of Monge-Amp\`ere systems, decomposable and bi-decomposable Monge-Amp\`ere systems, including equations for improper affine spheres and hypersurfaces of constant Gauss-Kronecker curvature are introduced. They are studied by the clear geometric setting of Lagrangian contact structures, based on the existence of Lagrangian pairs in contact structures. We show that the Lagrangian pair is uniquely determined by such a bi-decomposable system up to the order, if the num...
October 12, 2015
Let M be a real analytic Riemannian manifold. An adapted complex structure on $TM$ is a complex structure on a neighborhood of the zero section such that the leaves of the Riemann foliation are complex submanifolds. This structure is called entire if it may be extended to the whole of $TM$. We prove here that the only real analytic Zoll metric on the $n$-sphere with an entire adapted complex structure on $TM$ is the round sphere. Using similar ideas, we answer a special case ...
May 31, 2013
A C^2 function on C^n is called (n-1)-plurisubharmonic in the sense of Harvey-Lawson if the sum of any n-1 eigenvalues of its complex Hessian is nonnegative. We show that the associated Monge-Ampere equation can be solved on any compact Kahler manifold. As a consequence we prove the existence of solutions to an equation of Fu-Wang-Wu, giving Calabi-Yau theorems for balanced, Gauduchon and strongly Gauduchon metrics on compact Kahler manifolds.
March 21, 2003
This is the last in a series of five papers math.DG/0211294, math.DG/0211295, math.DG/0302355, math.DG/0302356 studying compact special Lagrangian submanifolds (SL m-folds) X in (almost) Calabi-Yau m-folds M with singularities x_1,...,x_n locally modelled on special Lagrangian cones C_1,...,C_n in C^m with isolated singularities at 0. Readers are advised to begin with this paper. We survey the major results of the previous four papers, giving brief explanations of the proof...