# Kawamata-Viehweg vanishing theorem

Let $X$ be a connected complex projective manifold (cf. Projective scheme). Let $K _ { X }$ denote the canonical bundle of $X$, i.e., the determinant bundle of the cotangent bundle (cf. Tangent bundle) of $X$. A line bundle $L$ on $X$ (cf. also 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 of $X$ into projective space. For a nef line bundle $L$ on $X$, bigness is equivalent to $c _ { 1 } ( L ) ^ { \operatorname { dim } X } > 0$, where $ c _ { 1 } ( L )$ denotes the first 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 > 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), and since $\operatorname { deg } K _ { X } = 2 g - 2$, the Kawamata–Viehweg vanishing theorem takes the form $h ^ { 1 } ( L ) = 0$ if $\operatorname { deg } L > 2 g - 2$; or, equivalently, $h ^ { 0 } ( K_{ X} \otimes L ^ { * } ) = 0$ if $\operatorname { deg } L > 2 g - 2$. For $L$ 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 $h ^ { 1 } ( L )$ 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., [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, [a13], Vol. 2; Chap. X111; Sec. IV. This result was based on a description of $K _ { X }$ [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 [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 $L$ on a compact complex projective manifold, $h ^ { i } ( K _ { X } \otimes L ) = 0$ for $i > 0$. 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 $L$ 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) $D = \sum _ { k = 1 } ^ { r } a _ { k } D _ { k }$, 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 $i > 0$,

\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 [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].

#### References

[a1] | M. Beltrametti, A.J. Sommese, "The adjunction theory of complex projective varieties" , Experim. Math. , 16 , W. de Gruyter (1995) |

[a2] | S. Bochner, "Curvature and Betti numbers I–II" Ann. of Math. , 49/50 (1948/9) pp. 379–390; 77–93 |

[a3] | G. Castelnuovo, F. Enriques, "Sur quelques résultats nouveaux dans la théorie des surfaces algébriques" E. Picard (ed.) G. Simart (ed.) , Théorie des Fonctions Algébriques , I–II |

[a4] | E. Viehweg, "Vanishing theorems" J. Reine Angew. Math. , 335 (1982) pp. 1–8 |

[a5] | H. Esnault, E. Viehweg, "Lectures on vanishing theorems" , DMV-Sem. , 20 , Birkhäuser (1992) |

[a6] | Y. Kawamata, "On the cohomology of $Q$-divisors" Proc. Japan Acad. Ser. A , 56 (1980) pp. 34–35 |

[a7] | Y. Kawamata, "A generalization of Kodaira–Ramanujam's vanishing theorem" Math. Ann. , 261 (1982) pp. 43–46 |

[a8] | 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 MR0946243 Zbl 0672.14006 |

[a9] | S.L. Kleiman, "The enumerative theory of singularities" P. Holme (ed.) , Real and Complex Singularities, Oslo 1976 , Sijthoff&Noordhoff (1977) pp. 297–396 |

[a10] | K. Kodaira, "On a differential-geometric method in the theory of analytic stacks" Proc. Nat. Acad. Sci. USA , 39 (1953) pp. 1268–1273 |

[a11] | J. Kollár, "Higher direct images of dualizing sheaves I–II" Ann. of Math. , 123/4 (1986) pp. 11–42; 171–202 |

[a12] | Y. Miyaoka, "On the Mumford–Ramanujam vanishing theorem on a surface" , Journées de Géométrie Algébrique, Angers/France 1979 (1980) pp. 239–247 |

[a13] | Picard, G. Simart, "Théorie des fonctions algébriques I–II" , Chelsea, reprint (1971) |

[a14] | 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) |

[a15] | G. Roch, "Über die Anzahl der willkürlichen Constanten in algebraischen Funktionen" J. de Crelle , 44 (1864) pp. 207–218 |

[a16] | B. Shiffman, A.J. Sommese, "Vanishing theorems on complex manifolds" , Progr. Math. , 56 , Birkhäuser (1985) |

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Kawamata-Viehweg vanishing theorem.

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