arXiv research

Study blow-ups in generalized complex geometry using holomorphic ideals.

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 splitting theorem. Our approach turns out to be essentially canonical, and as a…

We study the transverse Poisson structure to adjoint orbits in a complex semi-simple Lie algebra. The problem is first reduced to the case of nilpotent orbits. We prove then that in suitably chosen quasi-homogeneous coordinates the quasi-degree of the transverse Poisson structure is -2. In the particular case of {\emph…

Constructs integrable systems for Lie-Poisson structures at nilpotent elements.

Characterizes Hilbert schemes and their geometric properties.

The paper starts with an interpretation of the complete lift of a Poisson structure from a manifold M to its tangent bundle TM by means of the Schouten- Nijenhuis bracket of covariant symmetric tensor fields defined by the co- tangent Lie algebroid of M. Then, we discuss Poisson structures of TM which have a graded res…

We prove an equivariant version of the local splitting theorem for tame Poisson structures and Poisson actions of compact Lie groups. As a consequence, we obtain an equivariant linearization result for Poisson structures whose transverse structure has semisimple linear part of compact type.

Abstract: Geometrically describes Poisson cohomology groups around symplectic leaves.

The paper normalizes Poisson saturation of coregular submanifolds.

We consider Lagrangian-like submanifolds in certain even-dimensional 'symplectic-like' Poisson manifolds. We show, under suitable transversality hypotheses, that the pair consisting of the ambient Poisson manifold and the submanifold has unobstructed deformations and that the deformations automatically preserve the Lag…

Study of symplectic and Poisson reduction, proposing Poisson implosion.

First, we review the notion of a Poisson structure on a noncommutative algebra due to Block-Getzler and Xu and introduce a notion of a Hamiltonian vector field on a noncommutative Poisson algebra. Then we describe a Poisson structure on a noncommutative algebra associated with a transversely symplectic foliation and co…

Unimodularity criteria for Poisson structures on foliated manifolds are established.

Let $M^{2n}$ be a Poisson manifold with Poisson bivector field $Π$. We say that $M$ is b-Poisson if the map $Π^n:M\toΛ^{2n}(TM)$ intersects the zero section transversally on a codimension one submanifold $Z\subset M$. This paper will be a systematic investigation of such Poisson manifolds. In particular, we will study …

Characterizes blowups of Dirac structures on manifolds.

The study introduces a new equivalence for Poisson modules on complex projective varieties.

On an orientable manifold M, we consider a regular even dimensional foliation F which is globally defined by a set of k-independent 1-forms. We give necessary and sufficient conditions for the existence of a regular Poisson structure on M whose Characteristic foliation is precisely F. Moreover, introducing a special cl…

We give a notion of entropy for general gemetric structures, which generalizes well-known notions of topological entropy of vector fields and geometric entropy of foliations, and which can also be applied to singular objects, e.g. singular foliations, singular distributions, and Poisson structures. We show some basic p…

We study a number of local and global classification problems in generalized complex geometry. In the first topic, we characterize the local structure of generalized complex manifolds by proving that a generalized complex structure near a complex point arises from a holomorphic Poisson structure. In the proof we use a …

Given a manifold M with an action of a quadratic Lie algebra d, such that all stabilizer algebras are co-isotropic in d, we show that the product M\times d becomes a Courant algebroid over M. If the bilinear form on d is split, the choice of transverse Lagrangian subspaces g_1, g_2 of d defines a bivector field on M, w…

The notion of homogeneous tensors is discussed. We show that there is a one-to-one correspondence between multivector fields on a manifold $M$, homogeneous with respect to a vector field $Δ$ on $M$, and first-order polydifferential operators on a closed submanifold $N$ of codimension 1 such that $Δ$ is transversal to $…

Classifies complex Dirac structures with invariants and local structure.

We give a local normal form for Dirac structures. As a consequence, we show that the dimensions of the pre-symplectic leaves of a Dirac manifold have the same parity. We also show that, given a point $m$ of a Dirac manifold $M$, there is a well-defined transverse Poisson structure to the pre-symplectic leaf $P$ through…

We answer the natural question: when are a regular Poisson structure along with a complex structure transverse to its symplectic leaves induced by generalized complex structure? The leafwise symplectic form and transverse complex structure determine an obstruction class in a certain cohomology, which vanishes if and on…

We study local normal forms for completely integrable systems on Poisson manifolds in the presence of additional symmetries. The symmetries that we consider are encoded in actions of compact Lie groups. The existence of Weinstein's splitting theorem for the integrable system is also studied giving some examples in whic…

This paper studies regular Poisson manifolds of compact types, revealing their geometric and algebraic properties.

Study of higher-order Dirac structures in field theory.

New concept of coisotropic structures for differentiable stacks defined.

Simplified method for $L_\infty$ algebra of Dirac structures.

Study harmonic measures and rigidity in Seifert 3-manifolds using $S^1$-connections.

The blow-down map is studied in Lie algebroid cohomology.

We show that every Lie algebroid $A$ over a manifold $P$ has a natural representation on the line bundle $Q_A = \wedge^{top}A \otimes \wedge^{top} T^*P$. The line bundle $Q_A$ may be viewed as the Lie algebroid analog of the orientation bundle in topology, and sections of $Q_A$ may be viewed as transverse measures to $…

In this paper we analyze the obstructions to the existence of global action-angle variables for regular non-commutative integrable systems (NCI systems) on Poisson manifolds. In contrast with local action-angle variables, which exist as soon as the fibers of the momentum map of such an integrable system are compact, gl…

We prove that the classical $W$-algebra associated to a nilpotent orbit in a simple Lie-algebra can be constructed by preforming bihamiltonian, Drinfeld-Sokolov or Dirac reductions. We conclude that the classical $W$-algebra depends only on the nilpotent orbit but not on the choice of a good grading or an isotropic sub…

Paper characterizes generic transversality, improving on Mather's result.

Extends symplectic flow results to foliations.

New algebraic structures for Lie 2-algebroids and their connections.

Log-symplectic structures on fibrations are explored using symplectic techniques.

Paper generalizes Schwarz lemma for harmonic maps between Riemannian manifolds.

On the level of Lie algebras, the contraction procedure is a method to create a new Lie algebra from a given Lie algebra by rescaling generators and letting the scaling parameter tend to zero. One of the most well-known examples is the contraction from su(2) to e(2), the Lie algebra of upper-triangular matrices with ze…

Two new transversality theorems for linearly perturbed mappings are proven.

The paper proves generic transversality and regularity for minimal submanifolds and area-minimizing currents.

This thesis studies geometric structures using Lie algebroids to simplify singular behavior.

New theorem allows transverse links to be braided with rational book structure.

Study transverse Dolbeault cohomology for almost complex structures.

Formula calculates residues for maps near holomorphic distributions.

Paper develops theory of transverse generalized complex structures and proves a key lemma.

Maps to manifolds transverse to certain distributions satisfy an $h$-principle.