Related papers: The symplectic structure of a toric conic transfor…

We study the half-form Kaehler quantization of a smooth symplectic toric manifold $(X,\omega)$, such that $[\omega/2\pi]-c_{1}(X)/2 \in H^{2}(X,{\mathbb{Z}})$ and is nonnegative. We define the half-form corrected quantization of…

Given a root system $\Phi$ of type $A_n$, $B_n$, $C_n$, or $D_n$ in Euclidean space $E$, let $W$ be the associated Weyl group. For a point $p \in E$ not orthogonal to any of the roots in $\Phi$, we consider the $W$-permutohedron $P_W$,…

An origami manifold is a manifold equipped with a closed 2-form which is symplectic except on a hypersurface where it is like the pullback of a symplectic form by a folding map and its kernel fibrates with oriented circle fibers over a…

Let $M$ be a compact K\"ahler manifold equipped with a pre-quantum line bundle $L$. In [9], using $T$-symmetry, we constructed a polarization $\mathcal{P}_{\mathrm{mix}}$ on $M$, which generalizes real polarizations on toric manifolds. In…

As in the case of irreducible holomorphic symplectic manifolds, the period domain $Compl$ of compact complex tori of even dimension $2n$ contains twistor lines. These are special $2$-spheres parametrizing complex tori whose complex…

Let G be a torus of dimension n > 1 and M a compact Hamiltonian G-manifold with $M^G$ finite. A circle, $S^1$, in G is generic if $M^G = M^{S^1}$. For such a circle the moment map associated with its action on M is a perfect Morse function.…

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…

Let $M^n$, $n \in \{4,5,6\}$, be a compact, simply connected $n$-manifold which admits some Riemannian metric with non-negative curvature and an isometry group of maximal possible rank. Then any smooth, effective action on $M^n$ by a torus…

This paper studies Hamiltonian circle actions, i.e. circle subgroups of the group Ham(M,\om) of Hamiltonian symplectomorphisms of a closed symplectic manifold (M,\om). Our main tool is the Seidel representation of \pi_1(\Ham(M,\om)) in the…

Given an oriented $2$-manifold $M$, a locally constant sheaf of lattices $\Lambda$ over $M$, and a pointed morphism $q : \textsf B^2\Lambda \rightarrow \textsf B^4\mathbf C^{\times}$, we define an $\mathbb E_M$-category…

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…

We give a classification of toric anti-self-dual conformal structures on compact 4-orbifolds with positive Euler characteristic. Our proof is twistor theoretic: the interaction between the complex torus orbits in the twistor space and the…

We survey recent results about the Torelli question for holomorphic-symplectic varieties. Following are the main topics. A Hodge theoretic Torelli theorem. A study of the subgroup W, of the isometry group of the weight 2 Hodge structure,…

We show that for a Hamiltonian action of a compact torus $G$ on a compact, connected symplectic manifold $M$, the $G$-equivariant cohomology is determined by the residual $S^1$ action on the submanifolds of $M$ fixed by codimension-1 tori.…

We introduce the notion of a multi-fan. It is a generalization of that of a fan in the theory of toric variety in algebraic geometry. Roughly speaking a toric variety is an algebraic variety with an action of algebraic torus of the same…

A basic problem in the classification theory of compact complex manifolds is to give simple characterizations of complex tori. It is well known that a compact K\"ahler manifold $X$ homotopically equivalent to a a complex torus is…

We establish a local model for the moduli space of holomorphic symplectic structures with logarithmic poles, near the locus of structures whose polar divisor is normal crossings. In contrast to the case without poles, the moduli space is…

In 2002 Polterovich has notably established that on closed aspherical symplectic manifolds, Hamiltonian diffeomorphisms of finite order, which we call Hamiltonian torsion, must in fact be trivial. In this paper we prove the first…

This paper is concerned with the rational symplectic field theory in the Floer case. For this observe that in the general geometric setup for symplectic field theory the contact manifolds can be replaced by mapping tori of symplectic…

We prove that a compact toric locally conformally K\"ahler manifold which is not K\"ahler admits a toric Vaisman structure, a fact which was conjectured in \cite{mmp}. This is the final step leading to the classification of compact toric…