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''Kodaira's vanishing theorem''

A theorem on the vanishing of the cohomology groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055640/k0556401.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055640/k0556402.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055640/k0556403.png" /> is the sheaf of holomorphic sections of the [[Negative vector bundle|negative vector bundle]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055640/k0556404.png" /> of rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055640/k0556405.png" /> on a compact [[Complex manifold|complex manifold]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055640/k0556406.png" />. An equivalent statement of Kodaira's vanishing theorem is that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055640/k0556407.png" /></td> </tr></table>

for any [[Positive vector bundle|positive vector bundle]] of rank 1 (here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055640/k0556408.png" /> denotes the canonical line bundle on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055640/k0556409.png" />). In terms of divisors (cf. [[Divisor|Divisor]]) Kodaira's vanishing theorem is stated as the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055640/k05564010.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055640/k05564011.png" /> and any divisor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055640/k05564012.png" /> such that for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055640/k05564013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055640/k05564014.png" /> is a hyperplane section in some projective imbedding of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055640/k05564015.png" />.

The theorem was proved by transcendental methods by K. Kodaira [[#References|[1]]] (see also [[#References|[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 [[#References|[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:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055640/k05564016.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055640/k05564017.png" /> is a weakly-positive vector bundle of rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055640/k05564018.png" /> on the compact complex manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055640/k05564019.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055640/k05564020.png" /> is the sheaf of holomorphic forms (cf. [[Holomorphic form|Holomorphic form]]) of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055640/k05564021.png" /> with values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055640/k05564022.png" />. For weakly-negative vector bundles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055640/k05564023.png" />, vanishing takes place when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055640/k05564024.png" />. Analogues of these theorems have been obtained for weakly-complete manifolds <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055640/k05564025.png" />, that is, manifolds admitting a smooth [[Pluriharmonic function|pluriharmonic function]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055640/k05564026.png" /> such that the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055640/k05564027.png" /> is relatively compact in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055640/k05564028.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055640/k05564029.png" />, and for compact complex spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055640/k05564030.png" /> having <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055640/k05564031.png" /> algebraically-independent meromorphic functions [[#References|[5]]].

====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 {{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>



====Comments====
Counterexamples to Kodaira's theorem for non-singular varieties over a field of positive characteristic were given by M. Raynaud [[#References|[a1]]]. There exists a much stronger version of Kodaira's theorem, due to E. Viehweg and Y. Kawamata [[#References|[a2]]].

Recently, many generalizations of Kodaira vanishing have been found, see [[#References|[a3]]].

====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 {{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>