Positive form

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 [math]\displaystyle{ \Lambda^{p,p}(M)\cap \Lambda^{2p}(M,{\mathbb R}). }[/math] A real (1,1)-form [math]\displaystyle{ \omega }[/math] is called semi-positive (sometimes just positive), respectively, positive (or positive definite) if any of the following equivalent conditions holds:

1. [math]\displaystyle{ -\omega }[/math] is the imaginary part of a positive semidefinite (respectively, positive definite) Hermitian form.
2. For some basis [math]\displaystyle{ dz_1, ... dz_n }[/math] in the space [math]\displaystyle{ \Lambda^{1,0}M }[/math] of (1,0)-forms, [math]\displaystyle{ \sqrt{-1}\omega }[/math] can be written diagonally, as [math]\displaystyle{ \sqrt{-1}\omega = \sum_i \alpha_i dz_i\wedge d\bar z_i, }[/math] with [math]\displaystyle{ \alpha_i }[/math] real and non-negative (respectively, positive).
3. For any (1,0)-tangent vector [math]\displaystyle{ v\in T^{1,0}M }[/math], [math]\displaystyle{ -\sqrt{-1}\omega(v, \bar v) \geq 0 }[/math] (respectively, [math]\displaystyle{ \gt 0 }[/math]).
4. For any real tangent vector [math]\displaystyle{ v\in TM }[/math], [math]\displaystyle{ \omega(v, I(v)) \geq 0 }[/math] (respectively, [math]\displaystyle{ \gt 0 }[/math]), where [math]\displaystyle{ I:\; TM\mapsto TM }[/math] 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,

[math]\displaystyle{ \bar\partial:\; L\mapsto L\otimes \Lambda^{0,1}(M) }[/math]

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

[math]\displaystyle{ \nabla^{0,1}=\bar\partial }[/math].

This connection is called the Chern connection.

The curvature [math]\displaystyle{ \Theta }[/math] of the Chern connection is always a purely imaginary (1,1)-form. A line bundle L is called positive if [math]\displaystyle{ \sqrt{-1}\Theta }[/math] is a positive (1,1)-form. (Note that the de Rham cohomology class of [math]\displaystyle{ \sqrt{-1}\Theta }[/math] is [math]\displaystyle{ 2\pi }[/math] 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 [math]\displaystyle{ \sqrt{-1}\Theta }[/math] positive.

Positivity for (p, p)-forms

Semi-positive (1,1)-forms on M form a convex cone. When M is a compact complex surface, [math]\displaystyle{ dim_{\mathbb C}M=2 }[/math], this cone is self-dual, with respect to the Poincaré pairing :[math]\displaystyle{ \eta, \zeta \mapsto \int_M \eta\wedge\zeta }[/math]

For (p, p)-forms, where [math]\displaystyle{ 2\leq p \leq dim_{\mathbb C}M-2 }[/math], there are two different notions of positivity. A form is called strongly positive if it is a linear combination of products of semi-positive forms, with positive real coefficients. A real (p, p)-form [math]\displaystyle{ \eta }[/math] 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 [math]\displaystyle{ \int_M \eta\wedge\zeta\geq 0 }[/math].

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
• Griffiths, Phillip (3 January 2020). "Positivity and Vanishing Theorems". https://hdl.handle.net/20.500.12111/7881. 
• 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).
• Complex Geometry: An Introduction, Springer, 2005, ISBN 3-540-21290-6 
• Hodge Theory and Complex Algebraic Geometry (2 vols.), Cambridge University Press, 2007, doi:10.1017/CBO9780511615344, ISBN 978-0-521-71801-1