Multiplier ideal sheaves in complex and algebraic geometry

The application of the method of multiplier ideal sheaves to effective problems in algebraic geometry is briefly discussed. Then its application to the deformational invari-ance of plurigenera for general compact algebraic manifolds is presented and discussed. Finally its application to the conjecture of the finite generation of the canonical ring is explored, and the use of complex algebraic geometry in complex Neumann estimates is discussed.     

Coherent algebraic sheaves in real algebraic geometry

Summary In the following paper we give examples of vector bundlesF→X defined on a regular, compact, connected affine varietyX such thatF has no algebraic structure.

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Riassunto Nel presente lavoro si danno esempi di fibrati vettorialiF→X, definiti su varietà affini, regolari, compatte, connesseX che non ammettono struttura algebrica.

     

Multiplier ideal sheaves and integral invariants on toric Fano manifolds

We extend Nadel’s results on some conditions for the multiplier ideal sheaves to satisfy which are described in terms of an obstruction defined by the first author. Applying our extension we can determine the multiplier ideal subvarieties on toric del Pezzo surfaces which do not admit Kähler–Einstein metrics. We also show that one can define multiplier ideal sheaves for Kähler–Ricci solitons and extend the result of Nadel using the holomorphic invariant defined by Tian and Zhu.     

Nadel-type multiplier ideal sheaves on complex spaces with singularities

In this article, we introduce multiplier ideal sheaves on complex spaces with singularities(not necessarily normal) via Ohsawa’s extension measure, as a special case of which, it turns out to be the socalled Mather-Jacobian multiplier ideals in the algebro-geometric setting. Inspired by Nadel’s coherence and Guan-Zhou’s strong openness properties of the multiplier ideal sheaves, we discuss similar properties for the generalized multiplier ideal sheaves. As applications, we obtain a reasonable ge...     

Coherent sheaves and algebraic vector bundles

Summary LetK be a non algebraically closed field andV an affine subvariety ofK ⁿ. It is know that theoremsA, B are false in this case. The purpose of this paper is to study some properties of coherent sheaves and of algebraic vector bundles onV in the spirit of the famous article F.A.C. of J. P. Serre.

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Riassunto SiaK un corpo non algebricamente chiuso eV una varietà affine diK ⁿ. è noto che i teoremiA, B non valgono in questo caso. Scopo di questo lavoro è studiare alcune proprietà dei fasci coerenti edei fibrati vettoriali algebrici suV nello spirito del famoso articolo diJ. P. Serre: F.A.C.

     

A complex Frobenius theorem, multiplier ideal sheaves and Hermitian-Einstein metrics on stable bundles

A complex Frobenius theorem is proved for subsheaves of a holomorphic vector bundle satisfying a finite generation condition and a differential inclusion relation. A notion of `multiplier ideal sheaf' for a sequence of Hermitian metrics is defined. The complex Frobenius theorem is applied to the multiplier ideal sheaf of a sequence of metrics along Donaldson's heat flow to give a construction of the destabilizing subsheaf appearing in the Donaldson-Uhlenbeck-Yau theorem, in the case of algebraic surfaces.

     

An analogue of the Novikov Conjecture in complex algebraic geometry

We introduce an analogue of the Novikov Conjecture on higher signatures in the context of the algebraic geometry of (nonsingular) complex projective varieties. This conjecture asserts that certain ``higher Todd genera' are birational invariants. This implies birational invariance of certain extra combinations of Chern classes (beyond just the classical Todd genus) in the case of varieties with large fundamental group (in the topological sense). We prove the conjecture under the assumption of the ``strong Novikov Conjecture' for the fundamental group, which is known to be correct for many groups of geometric interest. We also show that, in a certain sense, our conjecture is best possible.

     

Inner metric geometry of complex algebraic surfaces with isolated singularities

We produce examples of complex algebraic surfaces with isolated singularities such that these singularities are not metrically conical, i.e., the germs of the surfaces near singular points are not bi‐Lipschitz equivalent, with respect to the inner metric, to cones. The technique used to prove the nonexistence of the metric conical structure is related to a development of metric homology. The class of the examples is rather large and includes some Kleinian singularities. © 2008 Wiley Periodicals, Inc.     

Gromov-Witten invariants in algebraic geometry

Gromov-Witten invariants for arbitrary non-singular projective varieties and arbitrary genus are constructed using the techniques from [K. Behrend, B. Fantechi. The Intrinsic Normal Cone.] Oblatum 26-II-1996 & 27-VI-1996     

Deformation quantization in algebraic geometry

We study deformation quantizations of the structure sheaf O_X of a smooth algebraic variety X in characteristic 0. Our main result is that when X is D-affine, any formal Poisson structure on X determines a deformation quantization of O_X (canonically, up to gauge equivalence). This is an algebro-geometric analogue of Kontsevich's celebrated result.     

Logarithmic geometry and algebraic stacks

We construct algebraic moduli stacks of log structures and give stack-theoretic interpretations of K. Kato's notions of log flat, log smooth, and log étale morphisms. In the last section we describe the local structure of these moduli stacks in terms of toric stacks.     

Spherical subcategories in algebraic geometry

We study objects in triangulated categories which have a two‐dimensional graded endomorphism algebra. Given such an object, we show that there is a unique maximal triangulated subcategory, in which the object is spherical. This general result is then applied to examples from algebraic geometry.     

Baxter's equations and algebraic geometry

Functional Analysis and Its Applications -