Explicit Decomposition Theorem for special Schubert varieties

We give a short and self-contained proof of the Decomposition Theorem for the non-small resolution of a Special Schubert variety. We also provide an explicit description of the perverse cohomology sheaves. As a by-product of our approach, we obtain a simple proof of the Relative Hard Lefschetz Theorem.Comment: This is an extensively revised version of my previous paper, taking care of the referee's comment

Non-Associative Geometry of Quantum Tori

We describe how to obtain the imprimitivity bimodules of the noncommutative torus from a "principal bundle" construction, where the total space is a quasi-associative deformation of a 3-dimensional Heisenberg manifold

Some remarks on K-lattices and the Adelic Heisenberg Group for CM curves

We define an adelic version of a CM elliptic curve $E$ which is equipped with an action of the profinite completion of the endomorphism ring of $E$. The adelic elliptic curve so obtained is provided with a natural embedding into the adelic Heisenberg group. We embed into the adelic Heisenberg group the set of commensurability classes of arithmetic $1$-dimensional $\mathbb{K}$-lattices (here and subsequently, $\mathbb{K}$ denotes a quadratic imaginary number field) and define theta functions on it. We also embed the groupoid of commensurability modulo dilations into the union of adelic Heisenberg groups relative to a set of representatives of elliptic curves with $R$-multiplication ($R$ is the ring of algebraic integers of $\mathbb{K}$). We thus get adelic theta functions on the set of $1$-dimensional $\mathbb{K}$-lattices and on the groupoid of commensurability modulo dilations. Adelic theta functions turn out to be acted by the adelic Heisenberg group and behave nicely under complex automorphisms (Theorems 6.12 and 6.14).Comment: 25 pages, no figures. Extensively revised version according to the comments of the reviewer

On the topology of a resolution of isolated singularities

Let $Y$ be a complex projective variety of dimension $n$ with isolated singularities, $\pi:X\to Y$ a resolution of singularities, $G:=\pi^{-1}{\rm{Sing}}(Y)$ the exceptional locus. From Decomposition Theorem one knows that the map $H^{k-1}(G)\to H^k(Y,Y\backslash {\rm{Sing}}(Y))$ vanishes for $k>n$. Assuming this vanishing, we give a short proof of Decomposition Theorem for $\pi$. A consequence is a short proof of the Decomposition Theorem for $\pi$ in all cases where one can prove the vanishing directly. This happens when either $Y$ is a normal surface, or when $\pi$ is the blowing-up of $Y$ along ${\rm{Sing}}(Y)$ with smooth and connected fibres, or when $\pi$ admits a natural Gysin morphism. We prove that this last condition is equivalent to say that the map $H^{k-1}(G)\to H^k(Y,Y\backslash {\rm{Sing}}(Y))$ vanishes for any $k$, and that the pull-back $\pi^*_k:H^k(Y)\to H^k(X)$ is injective. This provides a relationship between Decomposition Theorem and Bivariant Theory.Comment: 18 page

N\'eron-Severi group of a general hypersurface

In this paper we extend the well known theorem of Angelo Lopez concerning the Picard group of the general space projective surface containing a given smooth projective curve, to the intermediate N\'eron-Severi group of a general hypersurface in any smooth projective variety.Comment: 14 pages, to appear on Communications in Contemporary Mathematic