L’objectif dans ce travail est de présenter une généralisation pour l’obstruction d’Euler locale d’une fonction holomorphe singulière à l’origine dans le cas d’une application holomorphe $f:(V,0)\to ({ℂ}^k,0)$, où $(V,0)$ est un germe de variété analytique complexe, équidimensionnel de dimension $n\ge k$. Le résultat principal (Théorème 6.1) exprime l’obstruction d’Euler locale, définie pour un $k$-repère par Brasselet, Seade, Suwa, en fonction de l’obstruction d’Euler relative à $f$.

Classical results of Weil, Néron and Tate are generalized to local heights of subvarieties with respect to hermitian pseudo-divisors. The local heights are well-defined if the intersection of supports is empty. In the archimedean case, the metrics are hermitian and the local heights are defined by a refined version of the $*$-product of Gillet-Soulé developped on compact varieties without assuming regularity. In the non-archimedean case, the local heights are intersection numbers using methods from...

In this paper we review the moduli theory of polarized CY manifolds. We briefly sketched Kodaira-Spencer-Kuranishi local deformation theory developed by the author and G. Tian. We also construct the Teichmüller space of polarized CY manifolds following the ideas of I. R. Shafarevich and I. I. Piatetski-Shapiro. We review the fundamental result of E. Viehweg about the existence of the course moduli space of polarized CY manifolds as a quasi-projective variety. Recently S. Donaldson computed the moment...

Let $A$ be a closed semialgebraic subset of euclidean space of codimension at least one, and containing the origin $O$ as a non-isolated point. We prove that, for every real $s\ge 1$, there exists an algebraic set $V$ which approximates $A$ to order $s$ at $O$. The special case $s=1$ generalizes the result of the authors that every semialgebraic cone of codimension at least one is the tangent cone of an algebraic set.

The local Picard group at a generic point of the one-dimensional Baily-Borel boundary of a Hermitean symmetric quotient of type $\mathrm{O}(2,n)$ is computed. The main ingredient is a local version of Borcherds’ automorphic products. The local obstructions for a Heegner divisor to be principal are given by certain theta series with harmonic coefficients. Sometimes they generate Borcherds’ space of global obstructions. In these particular cases we obtain a simple proof of a result due to the first author: Suppose...

We show that every n-dimensional smooth algebraic variety X can be covered by Zariski open subsets $U_i$ which are isomorphic to closed smooth hypersurfaces in ${ℂ}^{n+1}$. As an application we show that forevery (pure) n-1-dimensional ℂ-uniruled variety $X\subset {ℂ}^m$ there is a generically-finite (even quasi-finite) polynomial mapping $f:{ℂ}^n\to {ℂ}^m$ such that $X\subset S_f$. This gives (together with [3]) a full characterization of irreducible components of the set $S_f$ for generically-finite polynomial mappings $f:{ℂ}^n\to {ℂ}^m$.

We propose a new method for defining a notion of support for objects in any compactly generated triangulated category admitting small coproducts. This approach is based on a construction of local cohomology functors on triangulated categories, with respect to a central ring of operators. Special cases are, for example, the theory for commutative noetherian rings due to Foxby and Neeman, the theory of Avramov and Buchweitz for complete intersection local rings, and varieties for representations of...

Let $I$ be an ideal of a commutative Noetherian ring $R$. It is shown that the $R$-modules $H_I^j\left(M\right)$ are $I$-cofinite for all finitely generated $R$-modules $M$ and all $j\in {\mathbb{N}}_0$ if and only if the $R$-modules ${\mathrm{Ext}}_R^i(N,H_I^j\left(M\right))$ and ${\mathrm{Tor}}_i^R(N,H_I^j\left(M\right))$ are $I$-cofinite for all finitely generated $R$-modules $M$, $N$ and all integers $i,j\in {\mathbb{N}}_0$.

Using a recently introduced correspondence of Emerton-Kisin we give a description of Lyubeznik’s local cohomology invariants in terms of local étale cohomology with $𝐙/p𝐙$ coefficients.

Lines on hypersurfaces with isolated singularities are classified. New normal forms of simple singularities with respect to lines are obtained. Several invariants are introduced.