In
algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
, the Kawamata–Viehweg vanishing theorem is an extension of the
Kodaira vanishing theorem
In mathematics, the Kodaira vanishing theorem is a basic result of complex manifold theory and complex algebraic geometry, describing general conditions under which sheaf cohomology groups with indices ''q'' > 0 are automatically zero. The implica ...
, on the vanishing of coherent
cohomology group
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewe ...
s, to
logarithmic pairs, proved independently by
Viehweg and
Kawamata in 1982.
The theorem states that if ''L'' is a
big
Big or BIG may refer to:
* Big, of great size or degree
Film and television
* ''Big'' (film), a 1988 fantasy-comedy film starring Tom Hanks
* ''Big!'', a Discovery Channel television show
* ''Richard Hammond's Big'', a television show present ...
nef 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 organisin ...
(for example, an
ample line bundle In mathematics, a distinctive feature of algebraic geometry is that some line bundles on a projective variety can be considered "positive", while others are "negative" (or a mixture of the two). The most important notion of positivity is that of an ...
) on a complex projective manifold with
canonical line bundle In mathematics, the canonical bundle of a non-singular algebraic variety V of dimension n over a field is the line bundle \,\!\Omega^n = \omega, which is the ''n''th exterior power of the cotangent bundle Ω on ''V''.
Over the complex numbers, it ...
''K'',
then the coherent cohomology groups ''H''
''i''(''L''⊗''K'') vanish for all positive ''i''.
References
*
*
Theorems in algebraic geometry
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