arXiv research

Floer theory constructs filtrations on quantum cohomology for symplectic manifolds.

New method recovers hyperkähler metrics from twistor models.

Proves stable degeneration preserves symplectic forms and confirms Kaledin's conjecture.

The paper generalizes convex and star-shaped concepts to symplectic spaces and studies variational problems.

In this note we discuss the problem of resolving conically singular cscK varieties to construct smooth cscK manifolds, showing a glueing result for (some) crepant resolutions of cscK varieties with discrete automorphism groups.

In this note we introduce the notion of a smooth structure on a conical pseudomanifold $M$ in terms of $C^\infty$-rings of smooth functions on $M$. For a finitely generated smooth structure $C^\infty (M)$ we introduce the notion of the Nash tangent bundle, the Zariski tangent bundle, the tangent bundle of $M$, and the …

Constructs scalar-flat Kähler metrics on toric symplectic manifolds.

New conical metrics found on toric varieties with convex cones.

The study connects conic connections and torsion-free principal connections on G-structures.

Study of symplectic Monge-Ampère equations using moment maps and contact structures.

Study on how incomplete smooth metrics degenerate from asymptotically conical Ricci-flat Kähler metrics.

Study intersection cohomology and Lagrangian fibrations in symplectic varieties.

This note gives a simple formula for the unique asymptotically conical Calabi-Yau metrics on the canonical bundle of a flag variety known to exist by the work of R. Goto and others. This is done by generalizing the well known Calabi Ansatz to general Kähler classes. We give some examples of explicit families, in partic…

We generalize Fujiki relation of Beauville-Bogomolov quadratic form on a projective symplectic variety. As an application, we study a fibre space structure of a projective symplectic variety.

Proves non-hyperbolicity of symplectic varieties with specific properties.

In this paper is to extend the Cheeger-Colding Theory to the class of conic Kahler-Einstein metrics. This extension provides a technical tool for [LTW] in which we prove a version of the Yau-Tian-Donaldson conjecture for Fano varieties with certain singularity.

Krein's formula for conic Laplacians on compact Riemann surfaces

Non-exact Poisson structures found on toric varieties.

The study shows boundedness of certain fibered varieties in algebraic geometry.

Quiver varieties' geometry at infinity studied using Nakajima metric.

We show there is a class of symplectic Lie algebra representations over any field of characteristic not 2 or 3 that have many of the exceptional algebraic and geometric properties of both symmetric three forms in two dimensions and alternating three forms in six dimensions. All nonzero orbits are coisotropic and the co…

We establish connections between contact isometry groups of certain contact manifolds and compactly supported symplectomorphism groups of their symplectizations. We apply these results to investigate the space of symplectic embeddings of balls with a single conical singularity at the origin. Using similar ideas, we als…

In this paper two Poisson structures on the moduli space of hyperbolic surfaces with conical points are compared: the Weil-Petersson one and the ηcoming from the representation variety. We show that they are multiple of each other, if the angles do not exceed 2π. Moreover, we exhibit an explicit formula for ηin terms o…

Study of symplectic manifolds degenerating into singular spaces.

The paper derives formulas for symplectic volume forms on surface representation varieties.

By results of the author there exists a projective (holomorphic) symplectic desingularization of the moduli space of rank-two torsion-free sheaves on a genus-two Jacobian with $c_1=0$ and $c_2=2$. This desingularization has a natural map to the self-product of the Jacobian. We show that the fiber over $(0,0)$ is a 6-di…

Estimates ends of Ricci shrinkers, focusing on smooth and singular cases.

The paper studies symplectic structures on character varieties of Sasakian threefolds.

A mapping bends Teichmüller spaces into character varieties, preserving symplectic structure.

Minimal action of mapping class group on character variety.

These notes give an introduction to Geometric Invariant Theory and symplectic reduction, with lots of pictures and simple examples. We describe their applications to moduli of bundles and varieties, and their infinite dimensional analogues in gauge theory and the theory of special metrics on algebraic varieties. Donald…

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

Cluster varieties are geometric objects that have recently found applications in several areas of mathematics and mathematical physics. This thesis studies the geometry of a large class of cluster varieties associated to compact oriented surfaces with boundary. The main original contribution of this thesis is to develo…

There are some similarities between cohomology of SU(2)-representation varieties of the fundamental group of some link complements and the Khovanov homology of the links. We start here a program to explain a possible source of these similarities. We introduce a symplectic manifold ${\mathscr M}$ with an action of the b…

We describe a natural $q$-deformation of Fock and Goncharov's canonical basis for the algebra of regular functions on a cluster variety associated to a quiver of type $A$. We then describe an extension of this construction involving a cluster variety called the symplectic double.

Hypertoric varieties are hyperkähler analogues of toric varieties, and are constructed as abelian hyperkähler quotients of a quaternionic affine space. Just as symplectic toric orbifolds are determined by labelled polytopes, orbifold hypertoric varieties are intimately related to the combinatorics of hyperplane arrange…

For simple and simply-connected complex algebraic group G, we conjecture the existence of a functor eta_G from the category of 2-bordisms to the category of holomorphic symplectic varieties with Hamiltonian action, such that gluing of boundaries corresponds to the holomorphic symplectic quotient with respect to the dia…

In the first part of the paper, we build a foundation for further work on Hamiltonian actions on symplectic orbifolds. Most importantly we prove the orbifold versions of the abelian connectedness and convexity theorems. In the second half, we prove that compact symplectic orbifolds with completely integrable torus acti…

Study of measured laminations on surfaces using Newton polytopes and Poisson brackets.

Uniqueness proven for a specific type of complex manifold's solitons.

Study of Kähler-Ricci flow on toric Fano varieties with preserved symplectic condition.

Smooth resolutions found for quotient of R^2 by infinite discrete groups.

Anosov representations of surface groups are complex manifolds.

There is a well-known correspondence between the symplectic variety of representations of the fundamental group of a punctured Riemann surface into a compact Lie group G, with fixed conjugacy classes at the punctures, and a complex variety of holomorphic bundles on the unpunctured surface with a parabolic structure at …

Study of Hilbert schemes and Coulomb branches of hypertoric varieties.

The paper maps two types of hyperkähler manifolds and identifies their symplectic structures.

The paper uses Seshadri constants to construct symplectic ellipsoid embeddings.

Given a six-dimensional symplectic manifold $(M, B)$, a nondegenerate, co-closed four-form $C$ introduces a dual symplectic structure $\widetilde{B} = *C$ independent of $B$ via the Hodge duality $*$. We show that the doubling of symplectic structures due to the Hodge duality results in two independent classes of nonc…