Kodaira theorem
Kodaira's vanishing theorem
A theorem on the vanishing of the cohomology groups ,
, where
is the sheaf of holomorphic sections of the negative vector bundle
of rank
on a compact complex manifold
. An equivalent statement of Kodaira's vanishing theorem is that
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for any positive vector bundle of rank 1 (here denotes the canonical line bundle on
). In terms of divisors (cf. Divisor) Kodaira's vanishing theorem is stated as the equation
for
and any divisor
such that for some
,
is a hyperplane section in some projective imbedding of
.
The theorem was proved by transcendental methods by K. Kodaira [1] (see also [2]) as a generalization to arbitrary dimension of the classical theorem on the regularity of an adjoint system on an algebraic surface. There exists an example of a normal algebraic surface over a field of positive characteristic for which Kodaira's vanishing theorem is false [4].
Kodaira's theorem also holds for holomorphic vector bundles of arbitrary rank that are negative in the sense of J. Nakano. The following result is also a generalization of Kodaira's theorem:
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where is a weakly-positive vector bundle of rank
on the compact complex manifold
, and
is the sheaf of holomorphic forms (cf. Holomorphic form) of degree
with values in
. For weakly-negative vector bundles
, vanishing takes place when
. Analogues of these theorems have been obtained for weakly-complete manifolds
, that is, manifolds admitting a smooth pluriharmonic function
such that the set
is relatively compact in
for all
, and for compact complex spaces
having
algebraically-independent meromorphic functions [5].
References
[1] | K. Kodaira, "On a differential geometric method in the theory of analytic stacks" Proc. Nat. Acad. Sci. USA , 39 (1953) pp. 1268–1273 |
[2] | R.O. Wells jr., "Differential analysis on complex manifolds" , Springer (1980) |
[3] | D. Mumford, "Pathologies III" Amer. J. Math. , 89 : 1 (1967) pp. 94–104 |
[4] | O. Zariski, "Algebraic surfaces" , Springer (1971) |
[5] | A.L. Onishchik, "Pseudoconvexity in the theory of complex spaces" J. Soviet Math. , 14 : 4 (1980) pp. 1363–1407 Itogi Nauk. Algebra Topol. Geom. , 15 (1977) pp. 93–171 |
Comments
Counterexamples to Kodaira's theorem for non-singular varieties over a field of positive characteristic were given by M. Raynaud [a1]. There exists a much stronger version of Kodaira's theorem, due to E. Viehweg and Y. Kawamata [a2].
Recently, many generalizations of Kodaira vanishing have been found, see [a3].
References
[a1] | M. Raynaud, "Contre-example du "vanishing theorem" en caractéristique ![]() |
[a2] | E. Viehweg, "Vanishing theorems and positivity in algebraic fibre spaces" , Proc. Internat. Congress Mathematicians (Berkeley, 1986) , 1 , Amer. Math. Soc. (1987) pp. 682–688 |
[a3] | J. Kollar, "Vanishing theorems for cohomology groups" S.J. Bloch (ed.) , Algebraic geometry (Bowdoin, 1985) , Proc. Symp. Pure Math. , 46 : 2 , Amer. Math. Soc. (1987) pp. 233–243 |
Kodaira theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Kodaira_theorem&oldid=23879