We obtain a correct generalization of Shokurov’s nonvanishing theorem for log canonical pairs. It implies the base point free theorem for log canonical pairs. We also prove the rationality theorem for log canonical pairs. As a corollary, we obtain the cone theorem for log canonical pairs. We do not need Ambro’s theory of quasi-log varieties.

For a given set of points S, a Euclidean distance field is defined by associating with every point p of Euclidean space Ed a value that is equal to the Euclidean distance from p to S. Such distance fields have numerous computational applications, but are expensive to compute and may not be sufficiently smooth for some applications. Instead, popular implicit modeling techniques rely on various a...

The idea in this paper is to show how a generalized “log” version of the Kodaira vanishing theorem can be employed to improve the results of Theorem 1 (which was also proved by Kodaira vanishing) when we have more knowledge about the equations for Y . The idea is to find a hypersurface F ⊂ P which has high multiplicity along Y , is “log canonical” near Y , and has relatively small degree, then ...

It is demonstrated that grouping together features which satisfy a geometric relationship can be used both for (automatic) detection and estimation of vanishing points and lines. We describe the geometry of three commonly occurring types of geometric grouping and present eecient grouping algorithms which exploit these ge-ometries. The three types of grouping are : (1) a family of equally spaced...

The work described here began as an attempt to understand the structure of the cohomology groups associated to a subgroup ? of nite index of SL 2 (Z) which are not congruence subgroups. One knows that to the space of cusp forms of weight w > 2 on ? (whose dimension we denote by d) one can 15] attach a motive M, which is pure of weight w?1 and of rank 2d, deened over some number eld; it is a dir...

— It is shown here that, for any finitely generated lattice r in certain semi simple groups over local fields of positive characteristics, H (r. Ad) is nonvanishing; this is in sharp contrast with the situation in characteristic zero. Let Kbe a local field (i. e. a non-discrete locally compact field), and let G be a connected semi simple algebraic group defined over K. Let G = G (K), and let r ...