Holomorphic Poisson Manifolds and Holomorphic Lie Algebroids

We study holomorphic Poisson manifolds and holomorphic Lie algebroids from the viewpoint of real Poisson geometry. We give a characterization of holomorphic Poisson structures in terms of the Poisson Nijenhuis structures of Magri-Morosi and describe a double complex which computes the holomorphic Poisson cohomology. A holomorphic Lie algebroid structure on a vector bundle $A\to X$ is shown to be equivalent to a matched pair of complex Lie algebroids $(T^{0,1}X,A^{1,0})$, in the sense of Lu. The holomorphic Lie algebroid cohomology of $A$ is isomorphic to the cohomology of the elliptic Lie algebroid $T^{0,1}X\bowtie A^{1,0}$. In the case when $(X,\pi)$ is a holomorphic Poisson manifold and $A=(T^*X)_\pi$, such an elliptic Lie algebroid coincides with the Dirac structure corresponding to the associated generalized complex structure of the holomorphic Poisson manifold.Comment: 29 pages, v2: paper split into two, part 1 of 2, v3: two references added, v4: final version to appear in International Mathematics Research Notice

Integration of Holomorphic Lie Algebroids

We prove that a holomorphic Lie algebroid is integrable if, and only if, its underlying real Lie algebroid is integrable. Thus the integrability criteria of Crainic-Fernandes do also apply in the holomorphic context without any modification. As a consequence we give another proof of the following theorem: a holomorphic Poisson manifold is integrable if, and only if, its real (or imaginary) part is integrable as a real Poisson manifold.Comment: 26 pages, second part of arXiv:0707.4253 which was split into two, v2: example 3.19 and section 3.7 adde

From double Lie groupoids to local Lie 2-groupoids

We apply the bar construction to the nerve of a double Lie groupoid to obtain a local Lie 2-groupoid. As an application, we recover Haefliger's fundamental groupoid from the fundamental double groupoid of a Lie groupoid. In the case of a symplectic double groupoid, we study the induced closed 2-form on the associated local Lie 2-groupoid, which leads us to propose a definition of a symplectic 2-groupoid.Comment: 23 pages, a few minor changes, including a correction to Lemma 6.

From Atiyah Classes to Homotopy Leibniz Algebras

A celebrated theorem of Kapranov states that the Atiyah class of the tangent bundle of a complex manifold $X$ makes $T_X[-1]$ into a Lie algebra object in $D^+(X)$, the bounded below derived category of coherent sheaves on $X$. Furthermore Kapranov proved that, for a K\"ahler manifold $X$, the Dolbeault resolution $\Omega^{\bullet-1}(T_X^{1,0})$ of $T_X[-1]$ is an $L_\infty$ algebra. In this paper, we prove that Kapranov's theorem holds in much wider generality for vector bundles over Lie pairs. Given a Lie pair $(L,A)$, i.e. a Lie algebroid $L$ together with a Lie subalgebroid $A$, we define the Atiyah class $\alpha_E$ of an $A$-module $E$ (relative to $L$) as the obstruction to the existence of an $A$-compatible $L$-connection on $E$. We prove that the Atiyah classes $\alpha_{L/A}$ and $\alpha_E$ respectively make $L/A[-1]$ and $E[-1]$ into a Lie algebra and a Lie algebra module in the bounded below derived category $D^+(\mathcal{A})$, where $\mathcal{A}$ is the abelian category of left $\mathcal{U}(A)$-modules and $\mathcal{U}(A)$ is the universal enveloping algebra of $A$. Moreover, we produce a homotopy Leibniz algebra and a homotopy Leibniz module stemming from the Atiyah classes of $L/A$ and $E$, and inducing the aforesaid Lie structures in $D^+(\mathcal{A})$.Comment: 36 page

Liouville integrability and superintegrability of a generalized Lotka–Volterra system and its Kahan discretization

We prove the Liouville and superintegrability of a generalized Lotka–Volterra system and its Kahan discretization

Properties of field functionals and characterization of local functionals

Functionals (i.e. functions of functions) are widely used in quantum field theory and solid-state physics. In this paper, functionals are given a rigorous mathematical framework and their main properties are described. The choice of the proper space of test functions (smooth functions) and of the relevant concept of differential (Bastiani differential) are discussed. The relation between the multiple derivatives of a functional and the corresponding distributions is described in detail. It is proved that, in a neighborhood of every test function, the support of a smooth functional is uniformly compactly supported and the order of the corresponding distribution is uniformly bounded. Relying on a recent work by Yoann Dabrowski, several spaces of functionals are furnished with a complete and nuclear topology. In view of physical applications, it is shown that most formal manipulations can be given a rigorous meaning. A new concept of local functionals is proposed and two characterizations of them are given: the first one uses the additivity (or Hammerstein) property, the second one is a variant of Peetre's theorem. Finally, the first step of a cohomological approach to quantum field theory is carried out by proving a global Poincar\'e lemma and defining multi-vector fields and graded functionals within our framework.Comment: 32 pages, no figur

Non-classical forms of pemphigus: pemphigus herpetiformis, IgA pemphigus, paraneoplastic pemphigus and IgG/IgA pemphigus

The pemphigus group comprises the autoimmune intraepidermal blistering diseases classically divided into two major types: pemphigus vulgaris and pemphigus foliaceous. Pemphigus herpetiformis, IgA pemphigus, paraneoplastic pemphigus and IgG/IgA pemphigus are rarer forms that present some clinical, histological and immunopathological characteristics that are different from the classical types. These are reviewed in this article. Future research may help definitively to locate the position of these forms in the pemphigus group, especially with regard to pemphigus herpetiformis and the IgG/ IgA pemphigus.Universidade Federal de São Paulo (UNIFESP), Escola Paulista de Medicina (EPM) Dermatology DepartmentUniversidade Federal de São Paulo (UNIFESP), Escola Paulista de Medicina (EPM) Dermatology and Pathology DepartmentsUNIFESP, EPM, Dermatology DepartmentUNIFESP, EPM, Dermatology and Pathology DepartmentsSciEL