“Positive form”的意思、由来-开放百科全书

词条

Positive form

释义

(1,1)-forms
Positive line bundles
Positivity for (p, p)-forms
References

In complex geometry, the term positive form refers to several classes of real differential forms of Hodge type (p, p).

(1,1)-forms

Real (p,p)-forms on a complex manifold M are forms which are of type (p,p) and real, that is, lie in the intersection : A real (1,1)-form is called positive if any of the following equivalent conditions hold

is an imaginary part of a positive (not necessarily positive definite) Hermitian form.
For some basis in the space of (1,0)-forms, can be written diagonally, as with real and non-negative.
For any (1,0)-tangent vector ,
For any real tangent vector , , where is the complex structure operator.

Positive line bundles

In algebraic geometry, positive (1,1)-forms arise as curvature forms of ample line bundles (also known as positive line bundles). Let L be a holomorphic Hermitian line bundle on a complex manifold,

its complex structure operator. Then L is equipped with a unique connection preserving the Hermitian structure and satisfying

.

This connection is called the Chern connection.

The curvature of a Chern connection is always a

purely imaginary (1,1)-form. A line bundle L is called positive if

is a positive definite (1,1)-form. The Kodaira embedding theorem claims that a positive line bundle is ample, and conversely, any ample line bundle admits a Hermitian metric with positive.

Positivity for (p, p)-forms

Positive (1,1)-forms on M form a convex cone. When M is a compact complex surface, , this cone is self-dual, with respect to the Poincaré pairing :

For (p, p)-forms, where , there are two different notions of positivity. A form is called

strongly positive if it is a linear combination of products of positive forms, with positive real coefficients. A real (p, p)-form on an n-dimensional complex manifold M is called weakly positive if for all strongly positive (n-p, n-p)-forms ζ with compact support, we have .

Weakly positive and strongly positive forms form convex cones. On compact manifolds these cones are dual with respect to the Poincaré pairing.

References

P. Griffiths and J. Harris (1978), Principles of Algebraic Geometry, Wiley. {{isbn|0-471-32792-1}}
J.-P. Demailly, [https://arxiv.org/abs/alg-geom/9410022 L² vanishing theorems for positive line bundles and adjunction theory, Lecture Notes of a CIME course on "Transcendental Methods of Algebraic Geometry" (Cetraro, Italy, July 1994)].

3 : Complex manifolds|Algebraic geometry|Differential forms

随便看

开放百科全书收录14589846条英语、德语、日语等多语种百科知识，基本涵盖了大多数领域的百科知识，是一部内容自由、开放的电子版国际百科全书。