Regularization of currents and entropy

Let T be a positive closed (p,p)-current on a compact Kähler manifold X. We prove the existence of smooth positive closed (p,p)-forms and such that weakly. Moreover, where cX>0 is a constant independent of T. We also extend this result to positive pluriharmonic currents. Then we study the wedge product of positive closed (1,1)-currents having continuous potential with positive pluriharmonic currents. As an application, we give an estimate for the topological entropy of meromorphic maps on compact Kähler manifolds.     

Existence results for a class of evolution equations of mixed type

We give an existence result for the evolution equation in the space where is a Banach space and is a non-invertible operator (the equation may be partially elliptic and partially parabolic, both forward and backward) and we study the “Cauchy-Dirichlet” problem associated to this equation (indeed also for the inclusion . We also investigate continuous and compact embeddings of and regularity in time of the solution. At the end we give some examples of different .     

On the Clark Ocone formula for the abstract Wiener space

Fix an abstract Wiener space where is a separable Hilbert space densely embedded into a Banach space . A pathwise construction of the Itô integral as a continuous square integrable martingale is given, where the integrands are -valued processes and the integrator is a -valued Brownian motion. We use this approach to the vector integral to prove that each Malliavin differentiable functional ? defined on the space of continuous -valued functions on [0,1], endowed with the Wiener measure, can be decomposed into the sum of the expected value of ? and the Itô integral of the conditional expectation of the Malliavin derivative of ? with respect to the Brownian filtration. The Malliavin derivative of ? is an -valued stochastic process. In a second application, it is shown that the iterated Itô integral, defined as a process on , is a continuous square integrable martingale.     

Calderón-Zygmund operators on Herz type Hardy spaces

We consider the boundedness of Calderón-Zygmund operators from to , where is the Hardy space associated with the Herz space and is the local version of . We show Calderón's commutator is bounded from to .     

Bounded Kähler class rigidity of actions on Hermitian symmetric spaces

We prove that the conjugacy class of a Zariski dense representation , q>p?1, of a finitely generated group Γ is completely determined by the pull-back via π of a bounded cohomology class in defined in terms of the Kähler form on the associated symmetric space. Under the assumption that is finite dimensional, we show that, up to equivalence, there is only a finite number of such representations for fixed q>p?1; moreover, under the hypothesis that injects into , we estimate the total number of such representations (for all q>p?1) to be bounded above by .     

Refined limiting imbeddings for Sobolev spaces of vector-valued functions

We prove a refined limiting imbedding theorem of the Brézis-Wainger type in the first critical case, i.e. , for Sobolev spaces and Bessel potential spaces of functions with values in a general Banach space E. In particular, the space E may lack the UMD property.     

Cohomological constraint on deformations of compact Kähler manifolds

We prove that for every compact Kähler manifold X the cup product

     

On the twistor space of pseudo-spheres

We give a proof that the sphere S⁶ does not admit an integrable orthogonal complex structure using simple differential geometric methods. This appears as a corollary of a general analogous result concerning pseudo-spheres.We study the twistor space of a pseudo-Riemannian manifold in both the holomorphic and pseudo-Riemannian directions. In particular, we construct the twistor space of a pseudo-sphere as a known pseudo-Kähler symmetric space. This leads to the explicit, unexpected computation of the exterior derivative of the Kähler form on the base manifold.     

On duality between étale groupoids and Hopf algebroids

For any étale Lie groupoid G over a smooth manifold M, the groupoid convolution algebra of smooth functions with compact support on G has a natural coalgebra structure over the commutative algebra which makes it into a Hopf algebroid. Conversely, for any Hopf algebroid A over we construct the associated spectral étale Lie groupoid over M such that is naturally isomorphic to G. Both these constructions are functorial, and is fully faithful left adjoint to . We give explicit conditions under which a Hopf algebroid is isomorphic to the Hopf algebroid of an étale Lie groupoid G.     

Geodesic currents and Teichmüller space

Consider a hyperbolic surface X of infinite area. The Liouville map assigns to any quasiconformal deformation of X a measure on the space of geodesics of the universal covering X? of X. We show that the Liouville map is a homeomorphism from the Teichmüller space onto its image, and that the image is closed and unbounded. The set of asymptotic rays to consists of all bounded measured laminations on X. Hence, the set of projective bounded measured laminations is a natural boundary for . The action of the quasiconformal mapping class group on continuously extends to this boundary for .     

Hecke curves and Hitchin discriminant

Let C be a smooth projective curve of genus g?4 over the complex numbers and be the moduli space of stable vector bundles of rank r with a fixed determinant of degree d. In the projectivized cotangent space at a general point E of , there exists a distinguished hypersurface SE consisting of cotangent vectors with singular spectral curves. In the projectivized tangent space at E, there exists a distinguished subvariety CE consisting of vectors tangent to Hecke curves in through E. Our main result establishes that the hypersurface SE and the variety CE are dual to each other. As an application of this duality relation, we prove that any surjective morphism , where C^′ is another curve of genus g, is biregular. This confirms, for , the general expectation that a Fano variety of Picard number 1, excepting the projective space, has no non-trivial self-morphism and that morphisms between Fano varieties of Picard number 1 are rare. The duality relation also gives simple proofs of the non-abelian Torelli theorem and the result of Kouvidakis-Pantev on the automorphisms of .     

ParaHermitian and paraquaternionic manifolds

A set of canonical paraHermitian connections on an almost paraHermitian manifold is defined. ParaHermitian version of the Apostolov-Gauduchon generalization of the Goldberg-Sachs theorem in General Relativity is given. It is proved that the Nijenhuis tensor of a Nearly paraKähler manifolds is parallel with respect to the canonical connection. Salamon's twistor construction on quaternionic manifold is adapted to the paraquaternionic case. A hyper-paracomplex structure is constructed on Kodaira-Thurston (properly elliptic) surfaces as well as on the Inoe surfaces modeled on . A locally conformally flat hyper-paraKähler (hypersymplectic) structure with parallel Lee form on Kodaira-Thurston surfaces is obtained. Anti-self-dual non-Weyl flat neutral metric on Inoe surfaces modeled on is presented. An example of anti-self-dual neutral metric which is not locally conformally hyper-paraKähler is constructed.     

Ideals of operators and the metric approximation property

We prove that a Banach space X has the metric approximation property if and only if , the space of all finite rank operators, is an ideal in , the space of all bounded operators, for every Banach space Y. Moreover, X has the shrinking metric approximation property if and only if is an ideal in for every Banach space Y.Similar results are obtained for u-ideals and the corresponding unconditional metric approximation properties.