Siu's semicontinuity theorem

Last updated

In complex analysis, the Siu semicontinuity theorem implies that the Lelong number of a closed positive current on a complex manifold is semicontinuous. More precisely, the points where the Lelong number is at least some constant form a complex subvariety. This was conjectured by Harvey & King (1972) and proved by Siu  ( 1973 , 1974 ). Demailly (1987) generalized Siu's theorem to more general versions of the Lelong number.

Related Research Articles

The modularity theorem states that elliptic curves over the field of rational numbers are related to modular forms. Andrew Wiles proved the modularity theorem for semistable elliptic curves, which was enough to imply Fermat's Last Theorem. Later, Christophe Breuil, Brian Conrad, Fred Diamond and Richard Taylor extended Wiles' techniques to prove the full modularity theorem in 2001.

In number theory, Iwasawa theory is the study of objects of arithmetic interest over infinite towers of number fields. It began as a Galois module theory of ideal class groups, initiated by Kenkichi Iwasawa (1959), as part of the theory of cyclotomic fields. In the early 1970s, Barry Mazur considered generalizations of Iwasawa theory to abelian varieties. More recently, Ralph Greenberg has proposed an Iwasawa theory for motives.

In topology, a graph manifold is a 3-manifold which is obtained by gluing some circle bundles. They were invented and classified by the German topologist Friedhelm Waldhausen in 1967. This definition allows a very convenient combinatorial description as a graph whose vertices are the fundamental parts and (decorated) edges stand for the description of the gluing, hence the name.

In mathematics, a Barlow surface is one of the complex surfaces introduced by Rebecca Barlow. They are simply connected surfaces of general type with pg = 0. They are homeomorphic but not diffeomorphic to a projective plane blown up in 8 points. The Hodge diamond for the Barlow surfaces is:

In mathematics, the Macdonald identities are some infinite product identities associated to affine root systems, introduced by Ian Macdonald (1972). They include as special cases the Jacobi triple product identity, Watson's quintuple product identity, several identities found by Dyson (1972), and a 10-fold product identity found by Winquist (1969).

In mathematics, surfaces of class VII are non-algebraic complex surfaces studied by (Kodaira 1964, 1968) that have Kodaira dimension −∞ and first Betti number 1. Minimal surfaces of class VII (those with no rational curves with self-intersection −1) are called surfaces of class VII0. Every class VII surface is birational to a unique minimal class VII surface, and can be obtained from this minimal surface by blowing up points a finite number of times.

In mathematics, the Grauert–Riemenschneider vanishing theorem is an extension of the Kodaira vanishing theorem on the vanishing of higher cohomology groups of coherent sheaves on a compact complex manifold, due to Grauert and Riemenschneider (1970).

In mathematics, the Schneider–Lang theorem is a refinement by Lang (1966) of a theorem of Schneider (1949) about the transcendence of values of meromorphic functions. The theorem implies both the Hermite–Lindemann and Gelfond–Schneider theorems, and implies the transcendence of some values of elliptic functions and elliptic modular functions.

In mathematical finite group theory, the trichotomy theorem divides the simple groups of characteristic 2 type and rank at least 3 into three classes. It was proved by Aschbacher for rank 3 and by Gorenstein & Lyons (1983) for rank at least 4. The three classes are groups of GF(2) type, groups of "standard type" for some odd prime, and groups of uniqueness type, where Aschbacher proved that there are no simple groups.

In mathematics, the Walter theorem, proved by John H. Walter, describes the finite groups whose Sylow 2-subgroup is abelian. Bender (1970) used Bender's method to give a simpler proof.

In mathematical group theory, a Demushkin group (also written as Demuškin or Demuskin) is a pro-p groupG having a certain properties relating to duality in group cohomology. More precisely, G must be such that the first cohomology group with coefficients in Fp = Z/pZ has finite rank, the second cohomology group has rank 1, and the cup product induces a non-degenerate pairing

In mathematics, the Poincaré–Lelong equation, studied by Lelong (1964), is the partial differential equation

In mathematics, the Conley–Zehnder theorem, named after Charles C. Conley and Eduard Zehnder, provides a lower bound for the number of fixed points of Hamiltonian diffeomorphisms of standard symplectic tori in terms of the topology of the underlying tori. The lower bound is one plus the cup-length of the torus (thus 2n+1, where 2n is the dimension of the considered torus), and it can be strengthen to the rank of the homology of the torus (which is 22n) provided all the fixed points are non-degenerate, this latter condition being generic in the C1-topology.

In algebraic geometry, the Kempf vanishing theorem, introduced by Kempf (1976), states that the higher cohomology group Hi(G/B,L ) (i > 0) vanishes whenever λ is a dominant weight of B. Here G is a reductive algebraic group over an algebraically closed field, B a Borel subgroup, and L(λ) a line bundle associated to λ. In characteristic 0 this is a special case of the Borel–Weil–Bott theorem, but unlike the Borel–Weil–Bott theorem, the Kempf vanishing theorem still holds in positive characteristic.

In number theory, Mazur's control theorem, introduced by Mazur (1972), describes the behavior in Zp extensions of the Selmer group of an abelian variety over a number field.

In algebraic number theory, the Ferrero–Washington theorem, proved first by Ferrero & Washington (1979) and later by Sinnott (1984), states that Iwasawa's μ-invariant vanishes for cyclotomic Zp-extensions of abelian algebraic number fields.

In algebra, a commutative Noetherian ring A is said to have the approximation property with respect to an ideal I if each finite system of polynomial equations with coefficients in A has a solution in A if and only if it has a solution in the I-adic completion of A. The notion of the approximation property is due to Michael Artin.

Jean-Pierre Demailly French mathematician

Jean-Pierre Demailly is a French mathematician working in complex analysis and differential geometry.

Bernard Shiffman is an American mathematician, specializing in complex geometry and analysis of complex manifolds.

In mathematics, the Cartan–Hadamard conjecture is a fundamental problem in Riemannian geometry and Geometric measure theory which states that the classical isoperimetric inequality may be generalized to spaces of nonpositive sectional curvature, known as Cartan–Hadamard manifolds. The conjecture, which is named after French mathematicians Élie Cartan and Jacques Hadamard, may be traced back to work of André Weil in 1926.

References

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.