Positive form

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

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

[edit] (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

$\Lambda^{p,p}(M)\cap \Lambda^{2p}(M,{\Bbb R}).$

A real (1,1)-form $ω$ is called positive if any of the following equivalent conditions hold

$\sqrt{-1}\omega$ is an imaginary part of a positive (not necessarily positive definite) Hermitian form.
For some basis $d z 1,... d z n$ in the space $Λ 1,0 M$ of (1,0)-forms, $\sqrt{-1}\omega$ can be written diagonally, as $\sqrt{-1}\omega = \sum_i \alpha_i dz_i\wedge d\bar z_i,$ with $α i$ real and non-negative.
For any (1,0)-tangent vector $v\in T^{1,0}M$ , $-\sqrt{-1}\omega(v, \bar v) \geq 0$
For any real tangent vector $v\in TM$ , $\omega(v, I(v)) \geq 0$ , where $I:\; TM\mapsto TM$ is the complex structure operator.

[edit] 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,

$\bar\partial:\; L\mapsto L\otimes \Lambda^{0,1}(M)$

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 $Θ$ of a Chern connection is always a purely imaginary (1,1)-form. A line bundle L is called positive if

$\sqrt{-1}\Theta$

is a positive (1,1)-form. The Kodaira vanishing theorem claims that a positive line bundle is ample, and conversely, any ample line bundle admits a Hermitian metric with $\sqrt{-1}\Theta$ positive.

[edit] Positivity for (p, p)-forms

Positive (1,1)-forms on M form a convex cone. When M is a compact complex surface, $dim_{\Bbb C}M=2$ , this cone is self-dual, with respect to the Poincaré pairing

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For (p, p)-forms, where $2\leq p \leq dim_{\Bbb C}M-2$ , 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 $η$ 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 $\int_M \eta\wedge\zeta\geq 0$ .