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We give a version in characteristic $p>0$ of Mumfords theorem characterizing a smooth complex germ of surface $(X,x)$ by the triviality of the topological fundamental group of $U=Xsetminus {x}$. This note relies on discussions the authors had during the Christmas break 2009/10 in Ivry. They have been written down by Hel`ene in the night when Eckart died, as a despaired sign of love.
In this paper we generalize the definitions of singularities of pairs and multiplier ideal sheaves to pairs on arbitrary normal varieties, without any assumption on the variety being Q-Gorenstein or the pair being log Q-Gorenstein. The main features
We show various properties of smooth projective D-affine varieties. In particular, any smooth projective D-affine variety is algebraically simply connected and its image under a fibration is D-affine. In characteristic zero such D-affine varieties ar
We classify all $n$-dimensional reduced Cohen-Macaulay modular quotient variety $mathbb{A}_mathbb{F}^n/C_{2p}$ and study their singularities, where $p$ is a prime number and $C_{2p}$ denotes the cyclic group of order $2p$. In particular, we present a
By the fundamental work of Griffiths one knows that, under suitable assumption, homological and algebraic equivalence do not coincide for a general hypersurface section of a smooth projective variety $Y$. In the present paper we prove the same result in case $Y$ has isolated singularities.
We prove that the kernel bundle of the evaluation morphism of global sections, namely the syzygy bundle, of a sufficiently ample line bundle on a smooth projective variety is slope stable with respect to any polarization. This settles a conjecture of Ein-Lazarsfeld-Mustopa.