Smoothable varieties with torsion free canonical sheaf

It will be shown that a smoothable Cohen-Macaulay variety over a field of characteristic zero is smooth if and only if its canonical sheaf is torsion free.partially supported by NSF     

Deformations of canonical singularities

We prove that small deformations of canonical singularities are canonical.

     

Log canonical foliation singularities on surfaces

We give a classification of the dual graphs of the exceptional divisors on the minimal resolutions of log canonical foliation singularities on surfaces. As an application, we show the set of foliated minimal log discrepancies for foliated surface triples satisfies the ascending chain condition and a Grauert–Riemenschneider–type vanishing theorem for foliated surfaces with special log canonical foliation singularities.     

Extended canonical algebras and Fuchsian singularities

The authors introduce a new class of finite dimensional algebras called extended canonical, and determine the shape of their derived categories. Extended canonical algebras arise from a canonical algebra ?? by onepoint extension or coextension by an indecomposable projective module. Our main results concern the case of negative Euler characteristic of the corresponding weighted projective line ${\mathbb{X}}$ ; more specifically we establish, for a base field of arbitrary characteristic, a link to the Fuchsian singularity R of ${\mathbb{X}}$ which for the base field of complex numbers is isomorphic to an algebra of automorphic forms. By means of a recent result of Orlov we show that the triangulated category of the graded singularities of R (in the sense of Buchweitz and Orlov) admits a tilting object whose endomorphism ring is the corresponding extended canonical algebra. Of particular interest are those cases where the attached Coxeter transformation has spectral radius one. A K-theoretic analysis then shows that this happens exactly for 38 cases including Arnold??s 14 exceptional unimodal singularities. The paper is related to recent independent work by Kajiura, Saito and Takahashi.     

Del Pezzo foliations with log canonical singularities

In this paper, we classify del Pezzo foliations of rank at least 3 on projective manifolds and with log canonical singularities in the sense of McQuillan.     

Hypersurface non-rational singularities which look canonical from their Newton boundaries

We study the Newton boundaries of hypersurface singularities from the viewpoint of the theory of filtered blowing-ups. We can construct counter examples to Reid's conjecture in the -dimensional cases which is related to the problem about the existence of good embedding for the criterion about the rationality (or the non-rationality). For 3-dimensional case, our examples contains a simple K3 singularity which does not belong to the famous list consisting of 95 types Received March 5, 1999; in final form October 28, 1999 / Published online February 5, 2001     

Rational, Log Canonical, Du Bois Singularities II: Kodaira Vanishing and Small Deformations

Kollár's conjecture, that log canonical singularities are Du Bois, is proved in the case of Cohen–Macaulay 3-folds. This in turn is used to derive Kodaira vanishing for this class of varieties. Finally it is proved that small deformations of Du Bois singularities are again Du Bois.     

The Nash sheaf of a complete resolution

Every three-dimensional complex algebraic variety with isolated singular point has a resolution factoring through the Nash blowup and the blowup of the maximal ideal over which the second Fitting ideal sheaf is locally principal. In such resolutions one can construct Hsiang–Pati coordinates and thus obtain generators for the Nash sheaf that are the differentials of monomial functions. Received: 9 April 2001 / Revised version: 26 July 2001     

Algebraic surfaces with log canonical singularities and the fundamental groups of their smooth parts

Let be a log surface with at worst log canonical singularities and reduced boundary such that is nef and big. We shall prove that either has finite fundamental group or is affine-ruled. Moreover, and the structure of are determined in some sense when .

     

A canonical form of a system of microdifferential equations with non-involutory characteristics and branching of singularities

We study a system of microdifferential (=pseudodifferential) equations. We assume that the characteristic varietyV of takes the formV=V ₁V₂; hereV ₁andV ₂are regular involutory submanifolds and intersect normally, andV ₁V₂is non-involutory and 1-codimensional both inV ₁and inV ₂. We also assume that has regular singularities alongV. Then we give a canonical form of in the complex domain. This enables us to investigate the branching of supports of microfunction solutions of when is hyperbolic.     

The etale sheaf cohomology of a class of singular surfaces

Summary In this paper the techniques of étale sheaf cohomology are used to estimate trigonometric sums on certain singular algebraic surfaces. This involves the computation of Betti numbers, made complicated by the presence of singularities. This leads to the introduction of some new (non-singular) surfaces. Their Betti numbers are then computed and related to the original problem which is thereby solved.

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Riassunto In questo lavoro si utilizza la coomologia étale per valutare le somme trigonometriche per certe superfici algebriche singolari. Per il calcolo dei numeri di Betti necessari, reso difficile per la presenza delle singolarità, si utilizzano delle superfici ausiliari lisce i cui numeri di Betti consentono di risolvere il problema iniziale.