On the compactification of hyperconcave ends

We find a class of manifolds whose ‘pseudoconcave holes’ can be filled in, even in dimension two. To cite this article: G. Marinescu, T.-C. Dinh, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

On the compactification of concave ends

Let X be a complex manifold which admits a proper strictly plurisubharmonic function ρ : X →]a, b[, where −∞ ≤ a < b ≤ ∞. It is well-known that the Hausdorffness of H ¹(X) is necessary for the existence of a Stein completion of X. We show that this condition is also sufficient.     

A Nadel vanishing theorem via injectivity theorems

The purpose of this paper is to establish Nadel type vanishing theorems with multiplier ideal sheaves of singular metrics admitting an analytic Zariski decomposition (such as, metrics with minimal singularities and Siu’s metrics). For this purpose, we generalize Kollár’s injectivity theorem to an injectivity theorem for line bundles equipped with singular metrics, by making use of the theory of harmonic integrals. Moreover we give asymptotic cohomology vanishing theorems for high tensor powers of line bundles.     

On compactification of schemes

For a Serre fibration with a fibre of theK(π, n)'s product type, obstructions to the section problem in each degree are defined by means of the Hirsch complex of fibration. This allows us to give the homotopy classification of sections (maps) as well as other applications. In particular, forG-bundles, these obstructions are related to theA _∞-module structure on the homology of the fibre and, consequently, some results in the fixed point theory are obtained.     

On compactification of metric spaces

Iff:X →X* is a homeomorphism of a metric separable spaceX into a compact metric spaceX* such thatf(X)=X*, then the pair (f,X*) is called a metric compactification ofX. An absoluteG _δ-space (F _σ-space)X is said to be of the first kind, if there exists a metric compactification (f,X*) ofX such that , whereG _i are sets open inX* and dim[Fr(G _i)]<dimX. (Fr(G _i) being the boundary ofG _i and dimX — the dimension ofX). An absoluteG _δ-space (F _σ-space), which is not of the first kind, is said to be of the second kind. In the present paper spaces which are both absoluteG _δ andF _σ-spaces of the second kind are constructed for any positive finite dimension, a problem related to one of A. Lelek in [11] is solved, and a sufficient condition onX is given under which dim [X* −f(X)]≧k, for any metric compactification (f,X*) ofX, wherek≦dimX is a given number. This research has been sponsored by the U.S. Navy through the Office of Naval Research under contract No. 62558-3315.     

On the Natanzon-Turaev compactification of the Hurwitz space

Natanzon and Turaev have constructed by topological methods a compactification of the Hurwitz space, that is, the space of simple branched covers of the two-sphere. Here we show that this compactification is homeomorphic to a compactification mentioned by Diaz and Edidin (in 1996) that was constructed by algebraic methods. Using this we are able to show by example that the Natanzon-Turaev compactification can be singular, that is, not a manifold.

     

Minimax theorems for infinite graphs with the ends as ideal points

For a set of pairwise disjoint sets of ends of an infinite graph, we define the concepts of -paths and of -separators, and we determine the maximum number of pairwise disjoint -path.     

On the theorems of Drake and Lenz

A generalization and an improvement of the results of Drake and Lenz on the constructions of projective Hjelmslev planes are obtained. Using this, some new series of invariant pairs including 4 new pairs (t, 2),t 1000, for projective Hjelmslev planes are obtained.     

On the theorems of Levine and Zund

Summary Some results of Levine and Zund are extended and applied to Maxwell's theory. It is deduced that a null Maxwell field in a conformally flat space-time is necessarily expansion free and twistfree as well as shearfree. An analogous result is given for null Maxwell fields which are type N in the Petrov classification. Earl Grey Memorial Fellow. Entrata in Redazione il 25 febbraio 1972.     

On Ambarzumyan-type theorems

Generalizations of the classical Ambarzumyan theorem are provided for wide classes of self-adjoint differential operators with arbitrary self-adjoint boundary conditions: scalar Sturm–Liouville operators, higher-order differential operators, matrix Sturm–Liouville operators and operators on spatial networks.     

关于极小极大原理

通过减弱条件,获得了一个新的结果。由此,我们给出一些新的极小极大定理和极小极大不等式。