Kawamata–Viehweg vanishing theorem
Encyclopedia
In algebraic geometry
, the Kawamata–Viehweg vanishing theorem is an extension of the Kodaira vanishing theorem, on the vanishing of coherent cohomology groups, to logarithmic pair
s, proved independently by and .
The theorem states that if L is a big nef line bundle
(for example, an ample line bundle
) on a complex projective manifold with canonical line bundle
K,
then the coherent cohomology groups Hi(L⊗K) vanish for all positive i.
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
, the Kawamata–Viehweg vanishing theorem is an extension of the Kodaira vanishing theorem, on the vanishing of coherent cohomology groups, to logarithmic pair
Logarithmic pair
In algebraic geometry, a logarithmic pair consists of a variety, together with a divisor along which one allows mild logarithmic singularities. They were studied by .-Definition:...
s, proved independently by and .
The theorem states that if L is a big nef line bundle
Line bundle
In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. For example a curve in the plane having a tangent line at each point determines a varying line: the tangent bundle is a way of organising these...
(for example, an ample line bundle
Ample line bundle
In algebraic geometry, a very ample line bundle is one with enough global sections to set up an embedding of its base variety or manifold M into projective space. An ample line bundle is one such that some positive power is very ample...
) on a complex projective manifold with canonical line bundle
Canonical bundle
In mathematics, the canonical bundle of a non-singular algebraic variety V of dimension n is the line bundle\,\!\Omega^n = \omegawhich is the nth exterior power of the cotangent bundle Ω on V. Over the complex numbers, it is the determinant bundle of holomorphic n-forms on V.This is the dualising...
K,
then the coherent cohomology groups Hi(L⊗K) vanish for all positive i.