arXiv research

A holomorphic Lagrangian fibration on a holomorphically symplectic manifold is a holomorphic map with Lagrangian fibers. It is known that a given compact manifold admits only finitely many holomorphic symplectic structures, up to deformation. We prove that a given compact manifold with $b_2 \geq 7$ admits only finitely…

The paper proves compactness for holomorphic curves with boundary on nearby Lagrangians.

Modified proof constructs holomorphic quilts on closed surfaces.

Proves uniqueness of holomorphic quilts on surfaces.

The paper studies deformations of Lagrangian fibrations on symplectic manifolds.

Study of symplectic manifolds degenerating into singular spaces.

Classifies actions on complex space forms with Lagrangian orbits.

For a given embedded Lagrangian in the complement of a complex hypersurface we show existence of a holomorphic disc in the complement having boundary on that Lagrangian.

Let X be a compact complex manifold, $D_c^b(X)$ be the bounded derived category of constructible sheaves on $X$, and $Fuk(T^*X)$ be the Fukaya category of $T^*X$. A Lagrangian brane in $Fuk(T^*X)$ is holomorphic if the underlying Lagrangian submanifold is complex analytic in $T^*X_{\mathbb{C}}$, the holomorphic cotange…

A hypercomplex manifold is a manifold equipped with a triple of complex structures satisfying the quaternionic relations. A holomorphic Lagrangian variety on a hypercomplex manifold with trivial canonical bundle is a holomorphic subvariety which is calibrated by a form associated with the holomorphic volume form; this …

Paper studies Lagrangian submanifolds and their homological monodromy.

Holomorphic disks on compact Lagrangian surfaces are shown to exist.

New symplectic forms derived from Lagrangian fibrations on symplectic manifolds.

The paper generalizes hyperkahler metrics near Lagrangian submanifolds.

Estimates holomorphic sections on Bohr-Sommerfeld Lagrangian submanifolds.

Computes colored HOMFLYPT invariants using holomorphic curves.

The paper explores the Thomas-Yau conjecture using holomorphic curves and Floer theory.

Holomorphic symplectic structure on Lagrangian moduli space.

Let X be a compact hyperkähler manifold containing a complex torus L as a Lagrangian subvariety. Beauville posed the question whether X admits a Lagrangian fibration with fibre L. We show that this is indeed the case if X is not projective. If X is projective we find an almost holomorphic Lagrangian fibration with fibr…

Study shows special Kähler geometry on base of holomorphic Lagrangian fibrations implies projective space.

We describe a family of calibrations arising naturally on a hyperkähler manifold $M$. These calibrations calibrate the holomorphic Lagrangian, holomorphic isotropic and holomorphic coisotropic subvarieties. When $M$ is an HKT (hyperkaehler with torsion) manifold with holonomy $SL(n, {\Bbb H})$, we construct another fam…

Constructs Lagrangian correspondences for Higgs bundles and holomorphic connections.

Following earlier work of Loftin-McIntosh, we study minimal Lagrangian immersions of the universal cover of a closed surface (of genus at least 2) into CH2, with prescribed data of a conformal structure plus a holomorphic cubic differential. We show existence and non-uniqueness of such minimal Lagrangian immersions. We…

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…

We prove that the length of the boundary of a $J$-holomorphic curve with Lagrangian boundary conditions is dominated by a constant times its area. The constant depends on the symplectic form, the almost complex structure, the Lagrangian boundary conditions and the genus. A similar result holds for the length of the rea…

In this article we study complex properties of minimal Lagrangian submanifolds in Kaehler ambient spaces, and how they depend on the ambient curvature. In particular, we prove that, in the negative curvature case, minimal Lagrangians do not admit fillings by holomorphic discs. The proof relies on a mix of holomorphic c…

Let $M$ be a holomorphic symplectic Kähler manifold equipped with a Lagrangian fibration $π$ with compact fibers. The base of this manifold is equipped with a special Kähler structure, that is, a Kähler structure $(I, g, ω)$ and a symplectic flat connection $\nabla$ such that the metric $g$ is locally the Hessian of a …

This paper solves the generalized Kähler problem by linking it to symplectic geometry.

Constructs special Lagrangian submanifolds in Calabi-Yau 3-folds.

Study of conformal limits in Nakajima quiver varieties.

We survey Lagrangian fibrations of holomorphic symplectic varieties, both compact and non-compact, whose fibres are Jacobians and Prym varieties.

The abstract proves a Moser-like theorem for C-symplectic structures and applies it to complex manifolds.

A well-known conjecture of Caratheodory states that the number of umbilic points on a closed convex surface in ${\mathbb E}^3$ must be greater than one. In this paper we prove this for $C^{3+α}$-smooth surfaces. The Conjecture is first reformulated in terms of complex points on a Lagrangian surface in $TS^2$, viewed as…

We use holomorphic disks to describe the formation of singularities in the mean curvature flow of monotone Lagrangian submanifolds in $\mathbb C^{n}$.

The purpose of this note is to establish the following theorem: Let N be a Kahler manifold, L be a compact oriented immersed minimal Lagrangian submanifold in N and V be a holomorphic vector field in a neighbourhood of L in N. Let div(V) be the (complex) divergence of V. Then the integral of div(V) over L is 0. Vice ve…

The mirror of a projective toric manifold $X_Σ$ is given by a Landau-Ginzburg model $(Y,W)$. We introduce a class of Lagrangian submanifolds in $(Y,W)$ and show that, under the SYZ mirror transformation, they can be transformed to torus-invariant hermitian metrics on holomorphic line bundles over $X_Σ$. Through this ge…

A Lagrangian submanifold in an almost Calabi-Yau manifold is called positive if the real part of the holomorphic volume form restricted to it is positive. An exact isotopy class of positive Lagrangian submanifolds admits a natural Riemannian metric. We compute the Riemann curvature of this metric and show all sectional…

Constructs special Lagrangian 3-spheres in non-Kähler compact threefolds.

Constructs families of monotone Lagrangians in Brieskorn-Pham hypersurfaces.

In this paper we prove that there does not exists any closed Pre-Lagrangian submanifolds in any closed contact manifolds by using the $J-$holomorphic curves and Gromov's nonlinear Fredholm alternative.

We construct smooth families of compact special Lagrangian submanifolds embedded in some toric hyper-K\"ahler manifolds, which never become holomorphic Lagrangian submanifolds via any hyper-Kähler rotations. These families converge to special Lagrangian immersions with self-intersection points in the sense of current…

We study complex Lagrangian submanifolds of a compact hyper-Kähler manifold and prove two results: (a) that an involution of a hyper-Kähler manifold which is antiholomorphic with respect to one complex structure and which acts non-trivially on the corresponding symplectic form always has a fixed point locus which is co…

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…

A 2-category categorifies complex Lagrangians in hyperkähler manifolds.

We define relative Floer theoretic invariants arising from 'quilted pseudo-holomorphic surfaces': Collections of pseudoholomorphic maps to various target spaces with 'seam conditions' in Lagrangian correspondences. As application we construct a morphism on quantum homology associated to any monotone Lagrangian correspo…

Classifies surfaces in hyperbolic space with constant Gaussian curvature.

Let $\OO$ be an orbit of the group of Hamiltonian symplectomorphisms acting on the space of Lagrangian submanifolds of a symplectic manifold $(X,ω).$ We define a functional $\CC:\OO \to \R$ for each differential form $β$ of middle degree satisfying $β\wedge ω= 0$ and an exactness condition. If the exactness condition d…

The generic fiber of a Lagrangian fibration on an irreducible holomorphic symplectic manifold is an abelian variety. Associate a polarization type to such Lagrangian fibrations coming from polarizations on a generic fiber. We prove that this polarization type is constant in families of Lagrangian fibrations. Further, w…