Related papers: Special Lagrangian submanifolds in the complex sph…

Let SO(n) act in the standard way on C^n and extend this action in the usual way to C^{n+1}. It is shown that a nonsingular special Lagrangian submanifold L in C^{n+1} that is invariant under this SO(n)-action intersects the fixed line C in…

We study Lagrangian subalgebras of a semisimple Lie algebra with respect to the imaginary part of the Killing form. We show that the variety $\Lagr$ of Lagrangian subalgebras carries a natural Poisson structure $\Pi$. We determine the…

We present a new approach to special lagrangian geometry which works for Bohr - Sommerfeld lagrangian submanifolds of symplectic manifolds with integer symplectic forms. This leads to construction of finite dimensional moduli spaces of SBS…

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…

We extend the "bundle constructions" of calibrated submanifolds, due to Harvey--Lawson in the special Lagrangian case, and to Ionel--Karigiannis--Min-Oo in the cases of exceptional calibrations, by "twisting" the bundles by a special…

Calabi-Yau manifolds are important objects in algebraic geometry and in theoretical physics. A hypothesis of mirror symmetry is that Calabi-Yau manifolds of dimension 3 come in pairs, with the Hodge numbers of one manifold mirroring the…

This proceedings note introduces aspects of the authors' work relating mirror symmetry and integral variations of Hodge structure. The emphasis is on their classification of the integral variations of Hodge structure which can underly…

We study special Lagrangian fibrations of $\mathrm{SU}(3)$-manifolds, not necessarily torsion-free. In the case where the fiber is a unimodular Lie group $G$, we decompose such $\mathrm{SU}(3)$-structures into triples of solder 1-forms,…

We establish the Lagrangian nature of the discrete isospectral and isomonodromic dynamical systems corresponding to the re-factorization transformations of the rational matrix functions on the Riemann sphere. Specifically, in the…

This is the first in a series of five papers math.DG/0211295, math.DG/0302355, math.DG/0302356, math.DG/0303272 studying special Lagrangian submanifolds (SL m-folds) X in (almost) Calabi-Yau m-folds M with singularities x_1,...,x_n locally…

Let $L$ be a special Lagrangian submanifold of a compact, Calabi-Yau manifold $M$ with boundary lying on the symplectic, codimension 2 submanifold $W$. It is shown how deformations of $L$ which keep the boundary of $L$ confined to $W$ can…

Given a Bridgeland stability condition on a 2-Calabi--Yau category, we define a simplicial complex that encodes the Harder--Narasimhan filtrations of spherical objects. For 2-Calabi--Yau categories of type A, we relate this complex to 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 this article we discuss the geometry of moduli spaces of (1) flat bundles over special Lagrangian submanifolds and (2) deformed Hermitian-Yang-Mills bundles over complex submanifolds in Calabi-Yau manifolds. These moduli spaces reflect…

In this paper, which is a natural continuation of our previous paper math.DG/0504557, we describe some special Lagrangians of cohomogeneity one in the resolved conifold. Our main result gives a foliation of the resolved conifold by…

We develop a unifed theory to study geometry of manifolds with different holonomy groups. They are classified by (1) real, complex, quaternion or octonion number they are defined over and (2) being special or not. Specialty is an…

We construct a family of compact almost Calabi--Yau manifolds of complex dimension 3 and therein a corresponding family of compact special Lagrangians with one-point singularities modelled upon that T^2-cone constructed by Harvey--Lawson…

Let $L$ be a closed totally real submanifold of $\mathbb{C}^{n}$, $n\ge 2$, which is not Lagrangian. We observe that small enough tubular neighborhoods of $L$ give exotic examples of weak fillings of $ST^{\ast}L$ endowed with its standard…

We show that two of the Bryant-Salamon G_2-manifolds have a simple topology ; homeomorphic to the complement of some submanifolds of the 7-dimensional sphere. In this connection, we show there exists a complete Ricci-flat (non-flat) metric…

In the first part of this paper we consider compact algebraic manifolds M^2n with an algebraic (n-1)-Torus action. We show that there is a T-invariant meromorphic section $\sigma$ of the canonical bundle of M. Any such $\sigma$ defines a…