Complex geometry

In mathematics, complex geometry is the study of complex manifolds and functions of several complex variables. Application of transcendental methods to algebraic geometry falls in this category, together with more geometric aspects of complex analysis.

Throughout this article, "analytic" is often dropped for simplicity; for instance, subvarieties or hypersurfaces refer to analytic ones. Following the convention in Wikipedia, varieties are assumed to be irreducible.

Definitions

An analytic subset of a complex-analytic manifold M is locally the zero-locus of some family of holomorphic functions on M. It is called an analytic subvariety if it is irreducible in the Zariski topology.

Line bundles and divisors

Throughout this section, X denotes a complex manifold. Accordance with the definitions of the paragraph "line bundles and divisors" in "projective varieties", let the regular functions on X be denoted ${\mathcal {O}}$ and its invertible subsheaf $\mathcal{O}^*$ .　And let　 $\mathcal{M}_X$ be the sheaf on X associated with $U \mapsto$ the total ring of fractions of $\Gamma(U, \mathcal{O}_X)$ , where $U_{i}$ are the open affine charts. Then a global section of $\mathcal{M}_X^*/\mathcal{O}_X^*$ (* means multiplicative group) is called a Cartier divisor on X.

Let $\operatorname {Pic} (X)$ be the set of all isomorphism classes of line bundles on X. It is called the Picard group of X and is naturally isomorphic to $H^1(X, \mathcal{O}^*)$ . Taking the short exact sequence of

0 \to \mathbb{Z} \to \mathcal{O} \to \mathcal{O}^* \to 0

where the second map is $f \mapsto \exp (2\pi i f)$ yields a homomorphism of groups:

\operatorname{Pic}(X) \to H^2(X, \mathbb{Z}).

The image of a line bundle ${\mathcal {L}}$ under this map is denoted by $c_1(\mathcal{L})$ and is called the first Chern class of ${\mathcal {L}}$ .

A divisor D on X is a formal sum of hypersurfaces (subvariety of codimension one):

D = \sum a_i V_i, \quad a_i \in \mathbb{Z}

that is locally a finite sum.^[1] The set of all divisors on X is denoted by $\operatorname{Div}(X)$ . It can be canonically identified with $H^0(X, \mathcal{M}^*/\mathcal{O}^*)$ . Taking the long exact sequence of the quotient $\mathcal{M}^*/\mathcal{O}^*$ , one obtains a homomorphism:

\operatorname{Div}(X) \to \operatorname{Pic}(X).

A line bundle is said to be positive if its first Chern class is represented by a closed positive real $(1,1)$ -form. Equivalently, a line bundle is positive if it admits a hermitian structure such that the induced connection has Griffiths-positive curvature. A complex manifold admitting a positive line bundle is kähler.

The Kodaira embedding theorem states that a line bundle on a compact kähler manifold is positive if and only if it is ample.

Complex vector bundles

Let X be a differentiable manifold. The basic invariant of a complex vector bundle $\pi: E \to X$ is the Chern class of the bundle. By definition, it is a sequence $c_1, c_2, \dots$ such that $c_i(E)$ is an element of $H^{2i}(X, \mathbb{Z})$ and that satisfies the following axioms:^[2]

$c_i(f^*(E)) = f^*(c_i(E))$ for any differentiable map $f: Z \to X$ .
$c(E \oplus F) = c(E) \cup c(F)$ where F is another bundle and $c = 1 + c_1 + c_2 + \dots.$
$c_i(E) = 0$ for $i > \operatorname{rk}E$ .
$-c_1(E_1)$ generates $H^2(\mathbb{C}\mathbf{P}^1, \mathbb{Z})$ where $E_{1}$ is the canonical line bundle over $\mathbb{C}\mathbf{P}^1$ .

If L is a line bundle, then the Chern character of L is given by

\operatorname{ch}(L) = e^{c_1(L)}

More generally, if E is a vector bundle of rank r, then we have the formal factorization: $\sum c_i(E)t^i = \prod_1^r (1+ \eta_i t)$ and then we set

\operatorname{ch}(E) = \sum e^{\eta_i}

Methods from harmonic analysis

Some deep results in complex geometry are obtained with the aid of harmonic analysis.

Vanishing theorem

There are several versions of vanishing theorems in complex geometry for both compact and non-compact complex manifolds. They are however all based on the Bochner method.

References

↑ This last condition is automatic for a noetherian scheme or a compact complex manifold.
↑ Kobayashi–Nomizu & 1996 Ch XII

Huybrechts, Daniel (2005). Complex Geometry: An Introduction. Springer. ISBN 3-540-21290-6.
Griffiths, Phillip; Harris, Joseph (1994), Principles of algebraic geometry, Wiley Classics Library, New York: John Wiley & Sons, ISBN 978-0-471-05059-9, MR 1288523
Hörmander, Lars (1990) [1966], An Introduction to Complex Analysis in Several Variables, North–Holland Mathematical Library, 7 (3rd (Revised) ed.), Amsterdam–London–New York–Tokyo: North-Holland, ISBN 0-444-88446-7, MR 1045639, Zbl 0685.32001
S. Kobayashi, K. Nomizu. Foundations of Differential Geometry (Wiley Classics Library) Volume 1, 2.
E. H. Neville (1922) Prolegomena to Analytical Geometry in Anisotropic Euclidean Space of Three Dimensions, Cambridge University Press.

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