Positive form explained

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

Λp,p(M)\capΛ2p(M,{R}).

A real (1,1)-form

\omega

is called semi-positive[1] (sometimes just positive[2]), respectively, positive[3] (or positive definite[4]) if any of the following equivalent conditions holds:

-\omega

is the imaginary part of a positive semidefinite (respectively, positive definite) Hermitian form.
  1. For some basis

dz1,...dzn

in the space

Λ1,0M

of (1,0)-forms,

\omega

can be written diagonally, as

\omega=\sqrt{-1}\sumi\alphaidzi\wedged\barzi,

with

\alphai

real and non-negative (respectively, positive).
  1. For any (1,0)-tangent vector

v\inT1,0M

,

-\sqrt{-1}\omega(v,\barv)\geq0

(respectively,

>0

).
  1. For any real tangent vector

v\inTM

,

\omega(v,I(v))\geq0

(respectively,

>0

), where

I:TM\mapstoTM

is the complex structure operator.

Positive line bundles

In algebraic geometry, positive definite (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,

\bar\partial:L\mapstoLΛ0,1(M)

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

\nabla0,1=\bar\partial

.

This connection is called the Chern connection.

The curvature

\Theta

of the Chern connection is always apurely imaginary (1,1)-form. A line bundle L is called positive if

\sqrt{-1}\Theta

is a positive (1,1)-form. (Note that the de Rham cohomology class of

\sqrt{-1}\Theta

is

2\pi

times the first Chern class of L.) The Kodaira embedding theorem claims that a positive line bundle is ample, and conversely, any ample line bundle admits a Hermitian metric with

\sqrt{-1}\Theta

positive.

Positivity for (p, p)-forms

Semi-positive (1,1)-forms on M form a convex cone. When M is a compact complex surface,

dimCM=2

, this cone is self-dual, with respect to the Poincaré pairing :

η,\zeta\mapsto\intMη\wedge\zeta

For (p, p)-forms, where

2\leqp\leqdimCM-2

, there are two different notions of positivity.[5] A form is calledstrongly positive if it is a linear combination of products of semi-positive forms, with positive real coefficients. A real (p, p)-form

η

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

\intMη\wedge\zeta\geq0

.

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

References

Notes and References

  1. Huybrechts (2005)
  2. Demailly (1994)
  3. Huybrechts (2005)
  4. Demailly (1994)
  5. Demailly (1994)