A proof of the Borsuk Antipodal Theorem for Fredholm maps

With only the means of elementary analysis and the Smale-Sard lemma, a direct and self-contained proof of the Borsuk Antipodal Theorem for Fredholm maps is given. The discussion extends the arguments of J. C. Alexander and J. A. Yorke used in their analytical proof of the Antipodal Theorem in $\text{R}$ⁿ.     

The Hodge theory of algebraic maps

We give a geometric proof of the Decomposition Theorem of Beilinson, Bernstein, Deligne and Gabber for the direct image of the intersection cohomology complex under a proper map of complex algebraic varieties. The method rests on new Hodge-theoretic results on the cohomology of projective varieties which extend naturally the classical theory and provide new applications.     

Good reductions of Shimura varieties of Hodge type in arbitrary unramified mixed characteristic. Part I

We prove the existence of good smooth integral models of Shimura varieties of Hodge type in arbitrary unramified mixed characteristic (0, p). As a first application we provide a smooth solution (answer) to a conjecture (question) of Langlands for Shimura varieties of Hodge type. As a second application we prove the existence in arbitrary unramified mixed characteristic (0, p) of integral canonical models of projective Shimura varieties of Hodge type with respect to h-hyperspecial subgroups as pro-étale covers of Néron models; this forms progress towards the proof of conjectures of Milne and Reimann. Though the second application was known before in some cases, its proof is new and more of a principle.     

Szegö kernel, regular quantizations and spherical CR-structures

We compute the Szegö kernel of the unit circle bundle of a negative line bundle dual to a regular quantum line bundle over a compact Kähler manifold. As a corollary we provide an infinite family of smoothly bounded strictly pseudoconvex domains on complex manifolds (disk bundles over homogeneous Hodge manifolds) for which the log-terms in the Fefferman expansion of the Szegö kernel vanish and which are not locally CR-equivalent to the sphere. We also give a proof of the fact that, for homogeneous Hodge manifolds, the existence of a locally spherical CR-structure on the unit circle bundle alone implies that the manifold is biholomorphic to a projective space. Our results generalize those obtained by Engli? (Math Z 264(4):901–912, 2010) for Hermitian symmetric spaces of compact type.     

A note on <Emphasis Type="Italic">tt</Emphasis><Superscript>*</Superscript>-bundles over compact nearly Kähler manifolds

First, we generalize a rigidity result for harmonic maps of Gordon (Gordon (1972) Proc AM Math Soc 33: 433–437) to generalized pluriharmonic maps. We give the construction of generalized pluriharmonic maps from metric tt ^*-bundles over nearly Kähler manifolds. An application of the last two results is that any metric tt ^*-bundle over a compact nearly Kähler manifold is trivial (Theorem A). This result we apply to special Kähler manifolds to show that any compact special Kähler manifold is trivial. This is Lu’s theorem (Lu (1999) Math Ann 313: 711–713) for the case of compact special Kähler manifolds. Further we introduce harmonic bundles over nearly Kähler manifolds and study the implications of Theorem A for tt ^*-bundles coming from harmonic bundles over nearly Kähler manifolds.     

Compensated compactness for differential forms in Carnot groups and applications

In this paper we prove a compensated compactness theorem for differential forms of the intrinsic complex of a Carnot group. The proof relies on an Ls-Hodge decomposition for these forms. Because of the lack of homogeneity of the intrinsic exterior differential, Hodge decomposition is proved using the parametrix of a suitable 0-order Laplacian on forms.     

Transformée de Mellin des intégrales-fibres associées aux singularités isolées d'intersection complète quasihomogènes. (Mellin Transform of Fibre Integrals Associated to Isolated Complete Intersection Singularities)

The Mellin transform of the fibre integral is calculated for certain quasihomogeneous isolated complete intersection singularities (above all, unimodal singularities of the list by Giusti and Wall). We show the symmetry property of the Gauss–Manin spectra (Theorem 3.1) and shed light on the lattice structure of the poles of the Mellin transform that are expressed by means of some topological data of the singularities (Theorem 4.3, Theorem 5.2). As an application of these results, we express the Hodge number of the fibre in terms of the Gauss–Manin spectra.     

Two remarks concerning toroidal groups

This paper deals with two quite unrelated properties of toroidal groups. After some preliminary remarks in section 0, we calculate in section 1 the Dolbeault cohomology groups of a toroidal group under an additional assumption which assures that those are at least finite-dimensional. In particular, we obtain a Hodge decomposition for these special toroidal groups. In section 2, we first give a new proof of a theorem of Cousin concerning the sections of topologically trivial line bundles on toroidal groups. As an application, we then show that, in a sense to be made precise, most abelian complex Lie groups of dimension ≥2 do not have any hypersurfaces.     

Topological representations of matroid maps

The Topological Representation Theorem for (oriented) matroids states that every (oriented) matroid arises from the intersection lattice of an arrangement of codimension one homotopy spheres on a homotopy sphere. In this paper, we use a construction of Engström to show that structure-preserving maps between matroids induce topological mappings between their representations; a result previously known only in the oriented case. Specifically, we show that weak maps induce continuous maps and that this process is a functor from the category of matroids with weak maps to the homotopy category of topological spaces. We also give a new and conceptual proof of a result regarding the Whitney numbers of the first kind of a matroid.     

Perfectoid Shimura varieties

This note explains some of the author’s work on understanding the torsion appearing in the cohomology of locally symmetric spaces such as arithmetic hyperbolic 3-manifolds.The key technical tool was a theory of Shimura varieties with infinite level at p: As p-adic analytic spaces, they are perfectoid, and admit a new kind of period map, called the Hodge–Tate period map, towards the flag variety. Moreover, the (semisimple) automorphic vector bundles come via pullback along the Hodge–Tate period map from the flag variety.In the case of the Siegel moduli space, the situation is fully analyzed in [12]. We explain the conjectural picture for a general Shimura variety.     

First and higher order uniform dual ergodic theorems for dynamical systems with infinite measure

We generalize the proof of Karamata’s Theorem by the method of approximation by polynomials to the operator case. As a consequence, we offer a simple proof of uniform dual ergodicity for a very large class of dynamical systems with infinite measure, and we obtain bounds on the convergence rate. In many cases of interest, including the Pomeau-Manneville family of intermittency maps, the estimates obtained through real Tauberian remainder theory are very weak. Building on the techniques of complex Tauberian remainder theory, we develop a method that provides second (and higher) order asymptotics. In the process, we derive a higher order Tauberian theorem for scalar power series which, to our knowledge, has not previously been covered.     

Projektive Einbettung nicht lokal projektiver Räume

It is well known that every locally projective linear space (M,M) with dimM 3, fulfilling the Bundle Theorem (B) can be embedded in a projective space. We give here a new construction for the projective embedding of linear spaces which need not be locally projective. Essentially for this new construction are the assumptions (A) and (C) that for any two bundles there are two points on every line which are incident with a line of each of these bundles. With the Embedding Theorem (7.4) of this note for example a [0,m]-space can be embedded in a projective space.

     

Bundles over Quantum Sphere and Noncommutative Index Theorem

The Noncommutative Index Theorem is used to prove that the Chern numbers of quantum Hopf line bundles over the standard Podle quantum sphere equal the winding numbers of the representations defining these bundles. This result gives an estimate of the positive cone of the algebraic K₀ of the standard quantum sphere.     

Nefness: Generalization to the lc case

This note is devoted to a proof of the b-nefness of the moduli part in the canonical bundle formula for an lc-trivial fibration that is lc and not klt over the generic point of the base. This result is proved in [3, §8] and [4] by using the theory of variation of mixed Hodge structure. Here we present a proof that makes use only of the theory of variation of Hodge structure and follows Ambro's proof of [2, Theorem 0.2].     

The Šilov Boundary for Operator Spaces

Motivated by the recent interest in the examination of unital completely positive maps and their effects in C*-theory, we revisit an older result concerning the existence of the ?ilov ideal. The direct proof of Hamana’s Theorem for the existence of an injective envelope for a unital operator subspace X of some ${\mathcal{B}(H)}$ that we provide implies that the ?ilov ideal is the intersection of C*(X) with any maximal boundary operator subsystem in ${\mathcal{B}(H)}$ . As an immediate consequence we deduce that the ?ilov ideal is the biggest boundary operator subsystem for X in C*(X). The new proof of the existence of the ?ilov ideal that we give does not use the existence of maximal dilations, provided by Drits- chel and McCullough, and so it is independent of the one given by Arveson. As a consequence, the ?ilov ideal can be seen as the set that contains the abnormalities in a C*-cover ${(C, \iota)}$ of X for all the extensions of the identity map ${{\rm id}_{\iota(X)}}$ . The interpretation of our results in terms of ucp maps characterizes the maximal boundary subsystems of X in ${\mathcal{B}(H)}$ as kernels of X-projections that induce completely minimal X-seminorms; equivalently, X-minimal projections with range being an injective envelope, that we view from now on as the ?ilov boundary for X.     

Pure strategy Nash equilibrium points and the Lefschetz fixed point theorem

A pure strategy Nash equilibrium point existence theorem is established for a class ofn-person games with possibly nonacyclic (e.g. disconnected) strategy sets. The principal tool used in the proof is a Lefschetz fixed point theorem for multivalued maps, due to Eilenberg and Montgomery, which extends their better known. Eilenberg-Montgomery fixed point theorem (EMT) [Eilenberg/Montgomery, Theorem 1, p. 215] to nonacyclic spaces. Special cases of the existence theorem are also discussed.     

Differentials of the second kind on mumford curves

We define ap-adic analytic Hodge decomposition for the cohomology of Mumford curves, with values in a local system. When the local system is trivial, we give a new proof of Gerritzen’s theorem, that this decomposition forms a variation of Hodge structure, in a family of Mumford curves.     

The d-very ampleness of adjoint line bundles on quasi-elliptic surfaces

In this paper, we give a numerical criterion of Reider-type for the d-very ampleness of the adjoint line bundles on quasi-elliptic surfaces, and meanwhile we give a new proof of the vanishing theorem on quasi-elliptic surfaces emailed from Langer and show that it is the optimal version.     

Isometric embeddings of real projective spaces into Euclidean spaces

This paper studies isometric embeddings of RPn via non-degenerate symmetric bilinear maps. The main result shows the infimum dimension of target Euclidean spaces among these constructions for RPn is . Next, we construct Veronese maps by induction, which realize the infimum. Finally, we give a simple proof of Rigidity Theorem of Veronese maps.