Parametrization of a singular Lagrangian variety

Regularizing a Singular Special Lagrangian Variety

Suppose M1 and M2 are two special Lagrangian submanifolds with boundary of R , n ≥ 3, that intersect transversally at one point p. The set M1 ∪M2 is a singular special Lagrangian variety with an isolated singularity at the point of intersection. Suppose further that the tangent planes at the intersection satisfy an angle criterion (which always holds in dimension n = 3). Then, M1 ∪ M2 is regula...

Regularising a Singular Special Lagrangian Variety

Suppose M1 and M2 are two special Lagrangian submanifolds of R 2n with boundary that intersect transversally at one point p. The set M1 ∪M2 is a singular special Lagrangian variety with an isolated singularity at the point of intersection. Suppose further that the tangent planes at the intersection satisfy an angle condition (which always holds in dimension n = 3). Then, M1 ∪ M2 is regularisabl...

Global nilpotent variety is Lagrangian

Let (M,ω) be a smooth symplectic algebraic variety. A (possibly singular) algebraic subvariety Y ⊂ M is said to be isotropic, resp. Lagrangian, if the tangent space, TyY , at any regular point y ∈ Y is an isotropic, resp. Lagrangian, vector subspace in the symplectic vector space TyM (we always assume Y to be reduced, but not necessarily irreducuble). The following characterisation of isotropic...

On the Variety of Lagrangian Subalgebras, Ii

On the variety of Lagrangian subalgebras

We study Lagrangian subalgebras of a semisimple Lie algebra with respect to the imaginary part of the Killing form. We show that the variety L of Lagrangian subalgebras carries a natural Poisson structure Π. We determine the irreducible components of L, and we show that each irreducible component is a smooth fiber bundle over a generalized flag variety, and that the fiber is the product of the ...