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Difference between revisions of "Kodaira theorem"

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====References====
 
====References====
<table><TR><TD valign="top">[1]</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">[2]</TD> <TD valign="top"> R.O. Wells jr.,   "Differential analysis on complex manifolds" , Springer (1980)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> D. Mumford,   "Pathologies III" ''Amer. J. Math.'' , '''89''' : 1 (1967) pp. 94–104</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> O. Zariski,   "Algebraic surfaces" , Springer (1971)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> 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</TD></TR></table>
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<table><TR><TD valign="top">[1]</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 {{MR|0066693}} {{ZBL|0053.11701}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> R.O. Wells jr., "Differential analysis on complex manifolds" , Springer (1980) {{MR|0608414}} {{ZBL|0435.32004}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> D. Mumford, "Pathologies III" ''Amer. J. Math.'' , '''89''' : 1 (1967) pp. 94–104 {{MR|0217091}} {{ZBL|0146.42403}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> O. Zariski, "Algebraic surfaces" , Springer (1971) {{MR|0469915}} {{ZBL|0219.14020}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> 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 {{MR|}} {{ZBL|0449.32020}} </TD></TR></table>
  
  
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====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> M. Raynaud,   "Contre-example du "vanishing theorem" en caractéristique <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055640/k05564032.png" />" K.G. Ramanathan (ed.) , ''C.P. Ramanujam, a tribute'' , Springer (1978) pp. 273–278</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> E. Viehweg,   "Vanishing theorems and positivity in algebraic fibre spaces" , ''Proc. Internat. Congress Mathematicians (Berkeley, 1986)'' , '''1''' , Amer. Math. Soc. (1987) pp. 682–688</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> 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</TD></TR></table>
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<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> M. Raynaud, "Contre-example du "vanishing theorem" en caractéristique <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055640/k05564032.png" />" K.G. Ramanathan (ed.) , ''C.P. Ramanujam, a tribute'' , Springer (1978) pp. 273–278</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> E. Viehweg, "Vanishing theorems and positivity in algebraic fibre spaces" , ''Proc. Internat. Congress Mathematicians (Berkeley, 1986)'' , '''1''' , Amer. Math. Soc. (1987) pp. 682–688 {{MR|0934270}} {{ZBL|0685.14013}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> 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 {{MR|0927959}} {{ZBL|0658.14012}} </TD></TR></table>

Revision as of 21:53, 30 March 2012

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

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:

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 MR0066693 Zbl 0053.11701
[2] R.O. Wells jr., "Differential analysis on complex manifolds" , Springer (1980) MR0608414 Zbl 0435.32004
[3] D. Mumford, "Pathologies III" Amer. J. Math. , 89 : 1 (1967) pp. 94–104 MR0217091 Zbl 0146.42403
[4] O. Zariski, "Algebraic surfaces" , Springer (1971) MR0469915 Zbl 0219.14020
[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 Zbl 0449.32020


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 " K.G. Ramanathan (ed.) , C.P. Ramanujam, a tribute , Springer (1978) pp. 273–278
[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 MR0934270 Zbl 0685.14013
[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 MR0927959 Zbl 0658.14012
How to Cite This Entry:
Kodaira theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Kodaira_theorem&oldid=23879
This article was adapted from an original article by I.V. Dolgachev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article