algebraic geometry Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...

, 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 implicat ...

, 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 viewed ...

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), ''Big'' (film), a 1988 fantasy-comedy film starring Tom Hanks * ''Big'', a 2023 Taiwanese children's film starring Van Fan and Chie Tanaka * ''Big!'', a ...

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 organis ...

(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 ...

) 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 nth exterior power of the cotangent bundle \Omega on V. Over the complex numbers, it is ...

''K'', then the coherent cohomology groups ''H''^''i''(''L''⊗''K'') vanish for all positive ''i''.

References

* * Theorems in algebraic geometry {{abstract-algebra-stub