Related papers: Geometric Poincar\'e Lemma

The Poincare-Hopf Theorem is one of the most used in other areas of science. There are applications of the Poincare-Hopf Theorem in physics, chemistry, biology and even in economics, psychology, etc ... The Poincare-Hopf Theorem connects an…

Let $N$ be a normal subgroup of a finite group $G$ and $V$ be a fixed finite-dimensional $G$-module. The Poincar\'{e} series for the multiplicities of induced modules and restriction modules in the tensor algebra $T(V)=\oplus_{k \geq…

We show that the Poincar\'e lemma we proved elsewhere in the context of crystalline cohomology of higher level behaves well with regard to the Hodge filtration. This allows us to prove the Poincar\'e lemma for transversal crystals of level…

Under Poincar\'e duality, a smooth map of compact oriented manifolds induces a pushforward map in cohomology, called the "Gysin map." It plays an important role in enumerative geometry. Using the equivariant localization formula, the author…

In the first part of this paper we prove some new Poincar\'e inequalities, with explicit constants, for domains of any hypersurface of a Riemannian manifold with sectional curvatures bounded from above. This inequalities involve the first…

We give detailed exposition of modern differential geometry from global coordinate independent point of view as well as local coordinate description suited for actual computations. In introduction, we consider Euclidean spaces and different…

Two of the authors have defined the class $ WDC(M)$ as the class of all subsets of a smooth manifold $M$ that may be expressed in local coordinates as certain sublevel sets of DC (differences of convex) functions. If $M$ is Riemanian and…

We revisit Spakula's uniform K-homology, construct the external product for it and use this to deduce homotopy invariance of uniform K-homology. We define uniform K-theory and on manifolds of bounded geometry we give an interpretation of it…

For a linear subvariety $M$ of a stratum of meromorphic differentials, we investigate its closure in the multi-scale compactification constructed by Bainbridge-Chen-Gendron-Grushevsky-M\"oller. We prove various restrictions on the type of…

We state and prove a generalization of the Poincar\'e-Hopf index theorem for manifolds with boundary. We then apply this result to non-vanishing complex vector fields.

This article presents a Poincare lemma for the Kostant complex, used to compute geometric quantisation, when the polarisation is given by a Lagrangian foliation defined by an integrable system with nondegenerate singularities.

Consider a divisor D with simple normal crossings in a compact K\"ahler manifold X. We show in this article that a K\"ahler metric in an arbitrary class, with constant scalar curvature and cusp singularities along the divisor is unique in…

We study the Fourier transform for compactly supported distributional sections of complex homogeneous vector bundles on symmetric spaces of non-compact type $X = G/K$. We prove a characterisation of their range. In fact, from Delorme's…

We extend the directional Poincar\'e inequality on the torus, introduced by Steinerberger in [Ark. Mat. 54 (2016), pp. 555--569], to the setting of compact Lie groups. We provide necessary and sufficient conditions for the existence of such…

We give a solution to the Poincar\'e Problem, in the formulation of Cerveau and Lins Neto. We obtain a bound on the degree of general leaves of foliations of general type, which is linear in $g$. To achieve this we study the birational…

We prove a spanning result for vector-valued Poincar\'e series on a bounded symmetric domain. We associate a sequence of holomorphic automorphic forms to a submanifold of the domain. When the domain is the unit ball in ${\Bbb{C}}^n$, we…

In earlier papers there were given formulae for the Poincare series of multi-index filtrations on the ring of germs of functions of two variables defined by collections of valuations corresponding to (reducible) plane curve singularities…

Let (N, G), where N is a normal subgroup of G<SL_n(C), be a pair of finite groups and V a finite-dimensional fundamental G-module. We study the G-invariants in the symmetric algebra S(V) by giving explicit formulas of the Poincar\'{e}…

We prove an abstract structure theorem for weighted manifolds supporting a weighted $f$-Poincar\'e inequality and whose ends satisfy a suitable non-integrability condition. We then study how our arguments can be used to obtain full…

Let $ \mathcal D\equiv G/K $ be an irreducible bounded symmetric domain. Using a vector-valued version of Mui\'c's integral non-vanishing criterion for Poincar\'e series on locally compact Hausdorff groups, we study the non-vanishing of…