Related papers: Lectures on Calabi-Yau and special Lagrangian geom…
Let $M_1$ and $M_2$ be special Lagrangian submanifolds of a compact Calabi-Yau manifold $X$ that intersect transversely at a single point. We can then think of $M_1\cup M_2$ as a singular special Lagrangian submanifold of $X$ with a single…
Type II string theory compactified on a Calabi-Yau manifold, with a singularity modeled by a hypersurface in an orbifold, is considered. In the limit of vanishing string coupling, one expects a non gravitational theory concentrated at the…
This is the third in a series of papers constructing explicit examples of special Lagrangian submanifolds in C^m. The previous paper in the series, math.DG/0008155, defined the idea of evolution data, which includes an (m-1)-submanifold P…
In 1996, Strominger, Yau and Zaslow made a conjecture about the geometric relationship between two mirror Calabi-Yau manifolds. Roughly put, if X and Y are a mirror pair of such manifolds, then X should possess a special Lagrangian torus…
A categorical formalism is introduced for studying various features of the symplectic geometry of Lefschetz fibrations and the algebraic geometry of Tyurin degenerations. This approach is informed by homological mirror symmetry, derived…
With a bird's-eye view, we survey the landscape of Calabi-Yau threefolds, compact and non-compact, smooth and singular. Emphasis will be placed on the algorithms and databases which have been established over the years, and how they have…
The moduli dependence of $(2,2)$ superstring compactifications based on Calabi--Yau hypersurfaces in weighted projective space has so far only been investigated for Fermat-type polynomial constraints. These correspond to Landau-Ginzburg…
Lecture notes from 1993 Park City lectures and 1994 Trento lectures. The focus of these lectures is on giving a mathematical description of the A-model and B-model correlation functions on a Calabi--Yau manifold, and a precise mathematical…
M-theory suggests the study of 11-dimensional space-times compactified on some 7-manifolds. From its intimate relation to superstrings, one possible class of such 7-manifolds are those that have Calabi-Yau threefolds as boundary. In this…
We study the geometry of M5-branes wrapping a 2-cycle which is Special Lagrangian with respect to a specific complex structure in a Calabi-Yau two-fold. Using methods recently applied to the three-fold case, we are again able find a…
M-theory compactification leads one to consider 7-manifolds obtained by rolling Calabi-Yau threefolds in the web of Calabi-Yau moduli spaces. The resulting 7-space in general has singularities governed by the extremal transition undergone.…
The moduli space of multiply-connected Calabi-Yau threefolds is shown to contain codimension-one loci on which the corresponding variety develops a certain type of hyperquotient singularity. These have local descriptions as discrete…
We call a projective Calabi-Yau (CY) 3-fold almost generic if it has only isolated nodes as singularities and the homology classes of all of the exceptional curves in an analytic small resolution are non-trivial but torsion. Such a…
We study the relation between discrete gauge symmetries in F-theory compactifications and torsion homology on the associated Calabi-Yau manifold. Focusing on the simplest example of a $\mathbb Z_2$ symmetry, we show that there are two…
This paper considers the natural geometric structure on the moduli space of deformations of a compact special Lagrangian submanifold $L^n$ of a Calabi-Yau manifold. From the work of McLean this is a smooth manifold with a natural $L^2$…
We introduce some new algebraic structures arising naturally in the geometry of Calabi-Yau manifolds and mirror symmetry. We give a universal construction of Calabi-Yau algebras in terms of a noncommutative symplectic DG algebra resolution.…
We construct some examples of special Lagrangian submanifolds and Lagrangian self-similar solutions in almost Calabi-Yau cones over toric Sasaki manifolds. For example, for any integer g>0, we can construct a real 6 dimensional Calabi-Yau…
Our main results are: (1) The complex a Lagrangian points of a non-complex Lagrangian $2n$-dimensional submanifold $F:M\ra N$, immersed with parallel mean curvature and with equal Kaehler angles into a Kaehler-Einstein manifold $(N,J,g)$ of…
An exact Calabi-Yau structure, originally introduced by Keller, is a special kind of smooth Calabi-Yau structure in the sense of Kontsevich-Vlassopoulos. For a Weinstein manifold $M$, the existence of an exact Calabi-Yau structure on the…
We present a novel way to classify Calabi-Yau threefolds by systematically studying their infinite volume limits. Each such limit is at infinite distance in Kahler moduli space and can be classified by an associated limiting mixed Hodge…