Kodaira vanishing theorem

Last updated April 27, 2024

In mathematics, the Kodaira vanishing theorem is a basic result of complex manifold theory and complex algebraic geometry, describing general conditions under which sheaf cohomology groups with indices q > 0 are automatically zero. The implications for the group with index q = 0 is usually that its dimension — the number of independent global sections — coincides with a holomorphic Euler characteristic that can be computed using the Hirzebruch–Riemann–Roch theorem.

The complex analytic case

The statement of Kunihiko Kodaira's result is that if M is a compact Kähler manifold of complex dimension n, L any holomorphic line bundle on M that is positive, and K_M is the canonical line bundle, then

H^{q}(M,K_{M}\otimes L)=0

for q > 0. Here $K_{M}\otimes L$ stands for the tensor product of line bundles. By means of Serre duality, one also obtains the vanishing of $H^{q}(M,L^{\otimes -1})$ for q < n. There is a generalisation, the Kodaira–Nakano vanishing theorem, in which $K_{M}\otimes L\cong \Omega ^{n}(L)$ , where Ωⁿ(L) denotes the sheaf of holomorphic (n,0)-forms on M with values on L, is replaced by Ω^r(L), the sheaf of holomorphic (r,0)-forms with values on L. Then the cohomology group H^q(M, Ω^r(L)) vanishes whenever q + r > n.

The algebraic case

The Kodaira vanishing theorem can be formulated within the language of algebraic geometry without any reference to transcendental methods such as Kähler metrics. Positivity of the line bundle L translates into the corresponding invertible sheaf being ample (i.e., some tensor power gives a projective embedding). The algebraic Kodaira–Akizuki–Nakano vanishing theorem is the following statement:

If k is a field of characteristic zero, X is a smooth and projective k-scheme of dimension d, and L is an ample invertible sheaf on X, then

H^{q}(X,L\otimes \Omega _{X/k}^{p})=0{\text{ for }}p+q>d,{\text{ and}}

H^{q}(X,L^{\otimes -1}\otimes \Omega _{X/k}^{p})=0{\text{ for }}p+q<d,

where the Ω^p denote the sheaves of relative (algebraic) differential forms (see Kähler differential).

Raynaud (1978) showed that this result does not always hold over fields of characteristic p > 0, and in particular fails for Raynaud surfaces. Later Sommese (1986) give a counterexample for singular varieties with non-log canonical singularities,^[1] and also,Lauritzen & Rao (1997) gave elementary counterexamples inspired by proper homogeneous spaces with non-reduced stabilizers.

Until 1987 the only known proof in characteristic zero was however based on the complex analytic proof and the GAGA comparison theorems. However, in 1987 Pierre Deligne and Luc Illusie gave a purely algebraic proof of the vanishing theorem in ( Deligne & Illusie 1987 ). Their proof is based on showing that the Hodge–de Rham spectral sequence for algebraic de Rham cohomology degenerates in degree 1. This is shown by lifting a corresponding more specific result from characteristic p > 0 — the positive-characteristic result does not hold without limitations but can be lifted to provide the full result.

Consequences and applications

Historically, the Kodaira embedding theorem was derived with the help of the vanishing theorem. With application of Serre duality, the vanishing of various sheaf cohomology groups (usually related to the canonical line bundle) of curves and surfaces help with the classification of complex manifolds, e.g. Enriques–Kodaira classification.

Note

↑ ( Fujino 2009 , Proposition 2.64)

Related Research Articles

In mathematics, complex geometry is the study of geometric structures and constructions arising out of, or described by, the complex numbers. In particular, complex geometry is concerned with the study of spaces such as complex manifolds and complex algebraic varieties, functions of several complex variables, and holomorphic constructions such as holomorphic vector bundles and coherent sheaves. Application of transcendental methods to algebraic geometry falls in this category, together with more geometric aspects of complex analysis.

In mathematics, the Hodge conjecture is a major unsolved problem in algebraic geometry and complex geometry that relates the algebraic topology of a non-singular complex algebraic variety to its subvarieties.

The Riemann–Roch theorem is an important theorem in mathematics, specifically in complex analysis and algebraic geometry, for the computation of the dimension of the space of meromorphic functions with prescribed zeros and allowed poles. It relates the complex analysis of a connected compact Riemann surface with the surface's purely topological genus g, in a way that can be carried over into purely algebraic settings.

In algebraic geometry, a projective variety over an algebraically closed field k is a subset of some projective n-space $over k that is the zero-locus of some finite family of homogeneous polynomials of n + 1 variables with coefficients in k, that generate a prime ideal, the defining ideal of the variety. Equivalently, an algebraic variety is projective if it can be embedded as a Zariski closed subvariety of .$

In algebraic geometry, a branch of mathematics, Serre duality is a duality for the coherent sheaf cohomology of algebraic varieties, proved by Jean-Pierre Serre. The basic version applies to vector bundles on a smooth projective variety, but Alexander Grothendieck found wide generalizations, for example to singular varieties. On an n-dimensional variety, the theorem says that a cohomology group $is the dual space of another one, . Serre duality is the analog for coherent sheaf cohomology of Poincaré duality in topology, with the canonical line bundle replacing the orientation sheaf.$

In mathematics and especially differential geometry, a Kähler manifold is a manifold with three mutually compatible structures: a complex structure, a Riemannian structure, and a symplectic structure. The concept was first studied by Jan Arnoldus Schouten and David van Dantzig in 1930, and then introduced by Erich Kähler in 1933. The terminology has been fixed by André Weil. Kähler geometry refers to the study of Kähler manifolds, their geometry and topology, as well as the study of structures and constructions that can be performed on Kähler manifolds, such as the existence of special connections like Hermitian Yang–Mills connections, or special metrics such as Kähler–Einstein metrics.

In mathematics, Hodge theory, named after W. V. D. Hodge, is a method for studying the cohomology groups of a smooth manifold M using partial differential equations. The key observation is that, given a Riemannian metric on M, every cohomology class has a canonical representative, a differential form that vanishes under the Laplacian operator of the metric. Such forms are called harmonic.

In mathematics, Kähler differentials provide an adaptation of differential forms to arbitrary commutative rings or schemes. The notion was introduced by Erich Kähler in the 1930s. It was adopted as standard in commutative algebra and algebraic geometry somewhat later, once the need was felt to adapt methods from calculus and geometry over the complex numbers to contexts where such methods are not available.

In algebraic geometry and the theory of complex manifolds, a logarithmic differential form is a differential form with poles of a certain kind. The concept was introduced by Pierre Deligne. In short, logarithmic differentials have the mildest possible singularities needed in order to give information about an open submanifold.

In mathematics, the Hirzebruch–Riemann–Roch theorem, named after Friedrich Hirzebruch, Bernhard Riemann, and Gustav Roch, is Hirzebruch's 1954 result generalizing the classical Riemann–Roch theorem on Riemann surfaces to all complex algebraic varieties of higher dimensions. The result paved the way for the Grothendieck–Hirzebruch–Riemann–Roch theorem proved about three years later.

In mathematics, the Kodaira embedding theorem characterises non-singular projective varieties, over the complex numbers, amongst compact Kähler manifolds. In effect it says precisely which complex manifolds are defined by homogeneous polynomials.

In mathematics, specifically in algebraic geometry and algebraic topology, the Lefschetz hyperplane theorem is a precise statement of certain relations between the shape of an algebraic variety and the shape of its subvarieties. More precisely, the theorem says that for a variety X embedded in projective space and a hyperplane section Y, the homology, cohomology, and homotopy groups of X determine those of Y. A result of this kind was first stated by Solomon Lefschetz for homology groups of complex algebraic varieties. Similar results have since been found for homotopy groups, in positive characteristic, and in other homology and cohomology theories.

In mathematics, a holomorphic vector bundle is a complex vector bundle over a complex manifold $X$ such that the total space $E$ is a complex manifold and the projection map $π : E \to X$ is holomorphic. Fundamental examples are the holomorphic tangent bundle of a complex manifold, and its dual, the holomorphic cotangent bundle. A holomorphic line bundle is a rank one holomorphic vector bundle.

In mathematics, a Hodge structure, named after W. V. D. Hodge, is an algebraic structure at the level of linear algebra, similar to the one that Hodge theory gives to the cohomology groups of a smooth and compact Kähler manifold. Hodge structures have been generalized for all complex varieties in the form of mixed Hodge structures, defined by Pierre Deligne (1970). A variation of Hodge structure is a family of Hodge structures parameterized by a manifold, first studied by Phillip Griffiths (1968). All these concepts were further generalized to mixed Hodge modules over complex varieties by Morihiko Saito (1989).

In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaf cohomology is a technique for producing functions with specified properties. Many geometric questions can be formulated as questions about the existence of sections of line bundles or of more general coherent sheaves; such sections can be viewed as generalized functions. Cohomology provides computable tools for producing sections, or explaining why they do not exist. It also provides invariants to distinguish one algebraic variety from another.

This is a glossary of algebraic geometry.

In mathematics, the Hodge–de Rham spectral sequence is an alternative term sometimes used to describe the Frölicher spectral sequence. This spectral sequence describes the precise relationship between the Dolbeault cohomology and the de Rham cohomology of a general complex manifold. On a compact Kähler manifold, the sequence degenerates, thereby leading to the Hodge decomposition of the de Rham cohomology.

In mathematics, specifically in the study of vector bundles over complex Kähler manifolds, the Nakano vanishing theorem, sometimes called the Akizuki–Nakano vanishing theorem, generalizes the Kodaira vanishing theorem. Given a compact complex manifold M with a holomorphic line bundle F over M, the Nakano vanishing theorem provides a condition on when the cohomology groups $equal zero. Here, denotes the sheaf of holomorphic (p,0)-forms taking values on F . The theorem states that, if the first Chern class of F is negative,$

In algebraic geometry and differential geometry, the nonabelian Hodge correspondence or Corlette–Simpson correspondence is a correspondence between Higgs bundles and representations of the fundamental group of a smooth, projective complex algebraic variety, or a compact Kähler manifold.

In algebraic geometry, Le Potier's vanishing theorem is an extension of the Kodaira vanishing theorem, on vector bundles. The theorem states the following

Le Potier (1975): Let X be a n-dimensional compact complex manifold and E a holomorphic vector bundle of rank r over X, here $is Dolbeault cohomology group, where denotes the sheaf of holomorphic p -forms on X . If E is an ample, then$
from Dolbeault theorem,
By Serre duality, the statements are equivalent to the assertions:

References

Deligne, Pierre; Illusie, Luc (1987), "Relèvements modulo p² et décomposition du complexe de de Rham", Inventiones Mathematicae, 89 (2): 247–270, Bibcode:1987InMat..89..247D, doi:10.1007/BF01389078, S2CID 119635574
Esnault, Hélène; Viehweg, Eckart (1992), Lectures on vanishing theorems (PDF), DMV Seminar, vol. 20, Birkhäuser Verlag, ISBN 978-3-7643-2822-1, MR 1193913
Phillip Griffiths and Joseph Harris, Principles of Algebraic Geometry
Kodaira, Kunihiko (1953), "On a differential-geometric method in the theory of analytic stacks", Proc. Natl. Acad. Sci. USA, 39 (12): 1268–1273, Bibcode:1953PNAS...39.1268K, doi: 10.1073/pnas.39.12.1268 , PMC 1063947 , PMID 16589409
Lauritzen, Niels; Rao, Prabhakar (1997), "Elementary counterexamples to Kodaira vanishing in prime characteristic", Proc. Indian Acad. Sci. Math. Sci., 107, Springer Verlag: 21–25, arXiv: alg-geom/9604012 , doi:10.1007/BF02840470, S2CID 16736679
Raynaud, Michel (1978), "Contre-exemple au vanishing theorem en caractéristique p>0", C. P. Ramanujam---a tribute, Tata Inst. Fund. Res. Studies in Math., vol. 8, Berlin, New York: Springer-Verlag, pp. 273–278, MR 0541027
Fujino, Osamu (2009). "Introduction to the log minimal model program for log canonical pairs". arXiv: 0907.1506 [math.AG].
Sommese, Andrew John (1986). "On the adjunction theoretic structure of projective varieties". Complex Analysis and Algebraic Geometry. Lecture Notes in Mathematics. Vol. 1194. pp. 175–213. doi:10.1007/BFb0077004. ISBN 978-3-540-16490-6.

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.

[1] ( Fujino 2009 , Proposition 2.64)

[1]