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We briefly review our results on the Lie theory underlying vector bundles over Lie groupoids and Lie algebroids, pointing out the role of Poisson geometry in extending these results to double Lie algebroids and LA-groupoids.
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Let $M$ be a smooth closed orientable manifold and $mathcal{P}(M)$ the space of Poisson structures on $M$. We construct a Poisson bracket on $mathcal{P}(M)$ depending on a choice of volume form. The Hamiltonian flow of the bracket acts on $mathcal{P}(M)$ by volume-preserving diffeomorphism of $M$. We then define an invariant of a Poisson structure that describes fixed points of the flow equation and compute it for regular Poisson 3-manifolds, where it detects unimodularity. For unimodular Poisson structures we define a further, related Poisson bracket and show that for symplectic structures the associated invariant counting fixed points of the flow equation is given in terms of the $d d^Lambda$ and $d+ d^Lambda$ symplectic cohomology groups defined by Tseng and Yau.
In this paper, we study the analytic continuation to complex time of the Hamiltonian flow of certain $Gtimes T$-invariant functions on the cotangent bundle of a compact connected Lie group $G$ with maximal torus $T$. Namely, we will take the Hamiltonian flows of one $Gtimes G$-invariant function, $h$, and one $Gtimes T$-invariant function, $f$. Acting with these complex time Hamiltonian flows on $Gtimes G$-invariant Kahler structures gives new $Gtimes T$-invariant, but not $Gtimes G$-invariant, Kahler structures on $T^*G$. We study the Hilbert spaces ${mathcal H}_{tau,sigma}$ corresponding to the quantization of $T^*G$ with respect to these non-invariant Kahler structures. On the other hand, by taking the vertical Schrodinger polarization as a starting point, the above $Gtimes T$-invariant Hamiltonian flows also generate families of mixed polarizations $mathcal{P}_{0,sigma}, sigma in {mathbb C}, {rm Im}(sigma) >0$. Each of these mixed polarizations is globally given by a direct sum of an integrable real distribution and of a complex distribution that defines a Kahler structure on the leaves of a foliation of $T^*G$. The geometric quantization of $T^*G$ with respect to these mixed polarizations gives rise to unitary partial coherent state transforms, corresponding to KSH maps as defined in [KMN1,KMN2].
VB-groupoids are vector bundles in the category of Lie groupoids. They encompass several classical objects, including Lie group representations and 2-vector spaces. Moreover, they provide geometric pictures for 2-term representations up to homotopy of Lie groupoids. We attach to every VB-groupoid a cochain complex controlling its deformations and discuss its fundamental features, such as Morita invariance and a van Est theorem. Several examples and applications are given.
In this paper, we study pre-Lie analogues of Poisson-Nijenhuis structures and introduce ON-structures on bimodules over pre-Lie algebras. We show that an ON-structure gives rise to a hierarchy of pairwise compatible O-operators. We study solutions of the strong Maurer-Cartan equation on the twilled pre-Lie algebra associated to an O-operator, which gives rise to a pair of ON-structures which are naturally in duality. We show that KVN-structures and HN-structures on a pre-Lie algebra g are corresponding to ON-structures on the bimodule $(mathfrak g^*;mathrm{ad}^*,-R^*)$, and $KVOmega$-structures are corresponding to solutions of the strong Maurer-Cartan equation on a twilled pre-Lie algebra associated to an $s$-matrix.
We introduce several families of filtrations on the space of vector bundles over a smooth projective variety. These filtrations are defined using the large k asymptotics of the kernel of the Dolbeault Dirac operator on a bundle twisted by the kth power of an ample line bundle. The filtrations measure the failure of the bundle to admit a holomorphic structure. We study compatibility under the Chern isomorphism of these filtrations with the Hodge filtration on cohomology.
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