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

Λ ^{p,p} (M)\cap Λ ^{2p} (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

dz _{1,} ... dz _{n} in the space Λ ^{1,0} M of (1,0)-forms, \omega can be written diagonally, as \omega = \sqrt{ -1 } \sum _{i} \alpha _{i} dz _{i\wedge} d\bar z _{i,} with \alpha _{i} real and non-negative (respectively, positive).

1. For any (1,0)-tangent vector

v\in T ^{1,0} M , - \sqrt{ -1 }\omega(v, \bar v) \geq 0 (respectively, > 0 ).

1. For any real tangent vector

v\in TM , \omega(v, I(v)) \geq 0 (respectively, > 0 ), where I: TM\mapsto TM 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\mapsto L ⊗ Λ ^{0,1} (M)

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

\nabla ^{0,1} = \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,

dim _{C} M =2 , this cone is self-dual, with respect to the Poincaré pairing :η , \zeta \mapsto \int _{M} η \wedge\zeta

For (p, p)-forms, where

2 \leq p \leq dim _{C} M -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 \int _{M} η \wedge\zeta\geq 0 .

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.
• Web site: Positivity and Vanishing Theorems . 20.500.12111/7881 . 3 January 2020 . Griffiths . Phillip .
• J.-P. Demailly, 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).

Notes and References

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