# Kawamata-Viehweg vanishing theorem

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Let be a connected complex projective manifold (cf. Projective scheme). Let denote the canonical bundle of , i.e., the determinant bundle of the cotangent bundle (cf. Tangent bundle) of . A line bundle on (cf. also Vector bundle) is said to be nef if the degree of the restriction of to any effective curve on is non-negative. A line bundle is said to be big if the sections of some positive power of give a birational mapping of into projective space. For a nef line bundle on , bigness is equivalent to , where denotes the first Chern class of . Let be the dimension of the th cohomology group of the sheaf of germs of algebraic or analytic sections of an algebraic line bundle on a projective variety. The Kawamata–Viehweg vanishing theorem states that for a nef and big line bundle on a complex projective manifold , When is a complex compact curve of genus , the bigness of a line bundle is equivalent to the line bundle being ample (cf. also Ample vector bundle), and since , the Kawamata–Viehweg vanishing theorem takes the form if ; or, equivalently, if . For with at least one not-identically-zero section, this vanishing theorem is equivalent to the Roch identification [a15], of the number now (1998) denoted by with , i.e., the one-dimensional Serre duality theorem. In the late 19th century, the numbers intervened in geometric arguments in much the same way as they intervene today, e.g., [a3]. For a very ample line bundle on a two-dimensional complex projective manifold, the Kawamata–Viehweg vanishing theorem was well known as the Picard theorem on the regularity of the adjoint, [a13], Vol. 2; Chap. X111; Sec. IV. This result was based on a description of [a9], Formula I.17, in terms of the double point divisor of a sufficiently general projection of into .

The next large step towards the Kawamata–Viehweg vanishing theorem was due to K. Kodaira [a10]. By means of a curvature technique that S. Bochner [a2] had used to show vanishing of real cohomology groups, Kodaira showed that for an ample line bundle on a compact complex projective manifold, for . Many generalizations of the Kodaira vanishing theorem appeared. Especially notable are results of C.P. Ramanujan [a14], which include the Kawamata–Viehweg vanishing theorem in the two-dimensional case; see also [a12].

The following formulation [a6], [a7], [a4] of the Kawamata–Viehweg vanishing theorem is better adapted to applications. To state it in its simplest form, additive notation is used and is taken to be a line bundle such that , i.e., the -th tensor power of , can be written as a sum of a nef and big line bundle plus an effective divisor (cf. Divisor) , where are positive integers and are smooth irreducible divisors such that any subset of the divisors meet transversely along their intersection. Then, for , where denotes the greatest integer less than or equal to a real number .

For more history and amplifications of these theorems see [a5], [a16]. See [a8] and [a11] for further generalizations of the Kawamata–Viehweg vanishing theorem. The paper [a8] is particularly useful: it contains relative versions of the vanishing theorem with some singularities, for not necessarily Cartier divisors. For applications of the vanishing theorems to classical problems, see [a1].