Related papers: Construct $b$-symplectic toric manifolds from tori…

Zariski decomposition plays an important role in the theory of algebraic surfaces due to many applications. For irreducible symplectic manifolds Boucksom provided a characterization of his divisorial Zariski decomposition in terms of the…

Previously work of the author with Meier and Starkston showed that every closed symplectic manifold $(X,\omega)$ with a rational symplectic form admits a trisection compatible with the symplectic topology. In this paper, we describe the…

We show that compact toric cosymplectic manifolds are mapping tori of equivariant symplectomorphisms of toric symplectic manifolds.

We extend work of Davis and Januszkiewicz by considering {\it omnioriented} toric manifolds, whose canonical codimension-2 submanifolds are independently oriented. We show that each omniorientation induces a canonical stably complex…

We use the techniques of integration of Poisson manifolds into symplectic Lie groupoids to build symplectic resolutions (= desingularizations) of the closure of a symplectic leaf. More generally, we show how Lie groupoids can be used to…

Recently Kontsevich solved the classification problem for deformation quantizations of all Poisson structures on a manifold. In this paper we study those Poisson structures for which the explicit methods of Fedosov can be applied, namely…

In this paper, we study Hamiltonian R-actions on symplectic orbifolds [M/S], where R and S are tori. We prove an injectivity theorem and generalize Tolman-Weitsman's proof of the GKM theorem in this setting. The main example is the…

We introduce an equivariant Pontrjagin-Thom construction which identifies equivariant cohomotopy classes with certain fixed point bordism classes. This provides a concrete geometric model for equivariant cohomotopy which works for any…

We prove a normal form theorem for Poisson structures around Poisson transversals (also called cosymplectic submanifolds), which simultaneously generalizes Weinstein's symplectic neighborhood theorem from symplectic geometry and Weinstein's…

Let $M$ be a compact, connected symplectic manifold with a Hamiltonian action of a compact $n$-dimensional torus $T$. Suppose that $M$ is equipped with an anti-symplectic involution $\sigma$ compatible with the $T$-action. The real locus of…

Toric geometry provides a bridge between the theory of polytopes and algebraic geometry: one can associate to each lattice polytope a polarized toric variety. In this thesis we explore this correspondence to classify smooth lattice…

Let M be a compact, connected symplectic 2n-dimensional manifold on which an(n-2)-dimensional torus T acts effectively and Hamiltonianly. Under the assumption that there is an effective complementary 2-torus acting on M with symplectic…

Log-symplectic structures are Poisson structures that are determined by a symplectic form with logarithmic singularities. We construct moduli spaces of curves with values in a log-symplectic manifold. Among the applications, we classify…

Given a strictly unbounded toric symplectic 4-manifold, we explicitly construct complete toric scalar-flat K\"ahler metrics on the complement of a toric divisor. These symplectic 4-manifolds correspond to a specific class of non-compact…

We construct, for each convex polytope, possibly nonrational and nonsimple, a family of compact spaces that are stratified by quasifolds, i.e. each of these spaces is a collection of quasifolds glued together in an suitable way. A quasifold…

In [FOOO12], K. Fukaya, Y. Oh, H. Ohta, and K. Ono (FOOO) obtained the monotone symplectic manifold $S^2\times S^2$ by resolving the singularity of a toric degeneration of a Hirzebruch surface. They identified a continuum of toric fibers in…

We show that the symplectic contraction map of Hilgert-Manon-Martens -- a symplectic version of Popov's horospherical contraction -- is simply the quotient of a Hamiltonian manifold $M$ by a "stratified null foliation" that is determined by…

Symplectic and complex toric quasifolds are a generalization of toric manifolds and orbifolds to the nonrational case. In this paper, we reframe these notions from the viewpoint of algebraic geometry.

Consider an effective Hamiltonian torus action $T\times M \to M$ on a topologically twisted,generalized complex manifold $M$ of dimension $2n$. We prove that the $rank(T) \leq n-2$ and that the topological twisting survives Hamiltonian…

We translate the equivariant decomposition theorem (in the case of a proper morphism of toric varieties) in to the language of combinatorially defined ``shifted minimal complexes''.