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Let $X$ be a connected complex projective manifold (cf. [[Projective scheme|Projective scheme]]). Let $K _ { X }$ denote the canonical bundle of $X$, i.e., the determinant bundle of the cotangent bundle (cf. [[Tangent bundle|Tangent bundle]]) of $X$. A line bundle $L$ on $X$ (cf. also [[Vector bundle|Vector bundle]]) is said to be nef if the degree of the restriction of $L$ to any effective curve on $X$ is non-negative. A line bundle is said to be big if the sections of some positive power of $L$ give a [[Birational mapping|birational mapping]] of $X$ into projective space. For a nef line bundle $L$ on $X$, bigness is equivalent to $c _ { 1 } ( L ) ^ { \operatorname { dim } X } &gt; 0$, where $ c  _ { 1 } (  L  )$ denotes the first [[Chern class|Chern class]] of $L$. Let $h ^ { i } ( E )$ be the dimension of the $i$th cohomology group of the sheaf of germs of algebraic or analytic sections of an algebraic line bundle $E$ on a projective variety. The Kawamata–Viehweg vanishing theorem states that for a nef and big line bundle on a complex projective manifold $X$,

\begin{equation*} h ^ { i } ( K _ { X } \bigotimes L ) = 0 , \quad i &gt; 0. \end{equation*}

When $X$ is a complex compact curve of genus $g$, the bigness of a line bundle $L$ is equivalent to the line bundle being ample (cf. also [[Ample vector bundle|Ample vector bundle]]), and since $\operatorname { deg } K _ { X } = 2 g - 2$, the Kawamata–Viehweg vanishing theorem takes the form $h ^ { 1 } ( L ) = 0$ if $\operatorname { deg } L &gt; 2 g - 2$; or, equivalently, $h ^ { 0 } ( K_{ X} \otimes L ^ { * } ) = 0$ if $\operatorname { deg } L &gt; 2 g - 2$. For $L$ with at least one not-identically-zero section, this vanishing theorem is equivalent to the Roch identification [[#References|[a15]]], of the number now (1998) denoted by $h ^ { 1 } ( L )$ with $h ^ { 0 } ( K _ { X } \otimes L ^ { * } )$, i.e., the one-dimensional Serre duality theorem. In the late 19th century, the numbers $h ^ { i } ( L )$ intervened in geometric arguments in much the same way as they intervene today, e.g., [[#References|[a3]]]. For a very ample line bundle $L$ 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, [[#References|[a13]]], Vol. 2; Chap. X111; Sec. IV. This result was based on a description of $K _ { X }$ [[#References|[a9]]], Formula I.17, in terms of the double point divisor of a sufficiently general projection of $X$ into $P^3$.

The next large step towards the Kawamata–Viehweg vanishing theorem was due to K. Kodaira [[#References|[a10]]]. By means of a curvature technique that S. Bochner [[#References|[a2]]] had used to show vanishing of real cohomology groups, Kodaira showed that for an ample line bundle $L$ on a compact complex projective manifold, $h ^ { i } ( K _ { X } \otimes L ) = 0$ for $i &gt; 0$. Many generalizations of the Kodaira vanishing theorem appeared. Especially notable are results of C.P. Ramanujan [[#References|[a14]]], which include the Kawamata–Viehweg vanishing theorem in the two-dimensional case; see also [[#References|[a12]]].

The following formulation [[#References|[a6]]], [[#References|[a7]]], [[#References|[a4]]] of the Kawamata–Viehweg vanishing theorem is better adapted to applications. To state it in its simplest form, additive notation is used and $L$ is taken to be a line bundle such that $N L$, i.e., the $N$-th tensor power of $L$, can be written as a sum $E + D$ of a nef and big line bundle $E$ plus an effective divisor (cf. [[Divisor|Divisor]]) $D = \sum _ { k = 1 } ^ { r } a _ { k } D _ { k }$, where $a_k$ are positive integers and $D _ { k }$ are smooth irreducible divisors such that any subset of the divisors meet transversely along their intersection. Then, for $i &gt; 0$,

\begin{equation*} h ^ { i } \left( K _ { X } + j L - \sum _ { k = 1 } ^ { r } \left[ \frac { j a _ { k } } { N } \right] D _ { k } \right) = 0, \end{equation*}

where $[ q ]$ denotes the greatest integer less than or equal to a real number $q$.

For more history and amplifications of these theorems see [[#References|[a5]]], [[#References|[a16]]]. See [[#References|[a8]]] and [[#References|[a11]]] for further generalizations of the Kawamata–Viehweg vanishing theorem. The paper [[#References|[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 [[#References|[a1]]].

====References====
<table><tr><td valign="top">[a1]</td> <td valign="top"> M. Beltrametti, A.J. Sommese, "The adjunction theory of complex projective varieties" , ''Experim. Math.'' , '''16''' , W. de Gruyter (1995)</td></tr><tr><td valign="top">[a2]</td> <td valign="top"> S. Bochner, "Curvature and Betti numbers I–II" ''Ann. of Math.'' , '''49/50''' (1948/9) pp. 379–390; 77–93</td></tr><tr><td valign="top">[a3]</td> <td valign="top"> G. Castelnuovo, F. Enriques, "Sur quelques résultat nouveaux dans la théorie des surfaces algébriques" E. Picard (ed.) G. Simart (ed.) , ''Théorie des Fonctions Algébriques'' , '''I–II'''</td></tr><tr><td valign="top">[a4]</td> <td valign="top"> E. Viehweg, "Vanishing theorems" ''J. Reine Angew. Math.'' , '''335''' (1982) pp. 1–8</td></tr><tr><td valign="top">[a5]</td> <td valign="top"> H. Esnault, E. Viehweg, "Lectures on vanishing theorems" , ''DMV-Sem.'' , '''20''' , Birkhäuser (1992)</td></tr><tr><td valign="top">[a6]</td> <td valign="top"> Y. Kawamata, "On the cohomology of $Q$-divisors" ''Proc. Japan Acad. Ser. A'' , '''56''' (1980) pp. 34–35</td></tr><tr><td valign="top">[a7]</td> <td valign="top"> Y. Kawamata, "A generalization of Kodaira–Ramanujam's vanishing theorem" ''Math. Ann.'' , '''261''' (1982) pp. 43–46</td></tr><tr><td valign="top">[a8]</td> <td valign="top"> Y. Kawamata, K. Matsuda, K. Matsuki, "Introduction to the minimal model problem" , ''Algebraic Geometry, Sendai 1985'' , ''Adv. Stud. Pure Math.'' , '''10''' (1987) pp. 283–360 {{MR|0946243}} {{ZBL|0672.14006}} </td></tr><tr><td valign="top">[a9]</td> <td valign="top"> S.L. Kleiman, "The enumerative theory of singularities" P. Holme (ed.) , ''Real and Complex Singularities, Oslo 1976'' , Sijthoff&amp;Noordhoff (1977) pp. 297–396</td></tr><tr><td valign="top">[a10]</td> <td valign="top"> K. Kodaira, "On a differential-geometric method in the theory of analytic stacks" ''Proc. Nat. Acad. Sci. USA'' , '''39''' (1953) pp. 1268–1273</td></tr><tr><td valign="top">[a11]</td> <td valign="top"> J. Kollár, "Higher direct images of dualizing sheaves I–II" ''Ann. of Math.'' , '''123/4''' (1986) pp. 11–42; 171–202</td></tr><tr><td valign="top">[a12]</td> <td valign="top"> Y. Miyaoka, "On the Mumford–Ramanujam vanishing theorem on a surface" , ''Journees de Geometrie Algebrique, Angers/France 1979'' (1980) pp. 239–247</td></tr><tr><td valign="top">[a13]</td> <td valign="top"> Picard, G. Simart, "Théorie des fonctions algébriques I–II" , Chelsea, reprint (1971)</td></tr><tr><td valign="top">[a14]</td> <td valign="top"> C.P. Ramanujam, "Remarks on the Kodaira vanishing theorem" ''J. Indian Math. Soc.'' , '''36''' (1972) pp. 41–51 (See also the Supplement: J. Indian Math. Soc. 38 (1974), 121-124)</td></tr><tr><td valign="top">[a15]</td> <td valign="top"> G. Roch, "Über die Anzahl der willkürlichen Constanten in algebraischen Funktionen" ''J. de Crelle'' , '''44''' (1864) pp. 207–218</td></tr><tr><td valign="top">[a16]</td> <td valign="top"> B. Shiffman, A.J. Sommese, "Vanishing theorems on complex manifolds" , ''Progr. Math.'' , '''56''' , Birkhäuser (1985)</td></tr></table>