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 :$\Lambda^{p,p}(M)\cap \Lambda^{2p}(M,{\Bbb R}).$ A real (1,1)-form $\omega$ is called positive if any of the following equivalent conditions hold

1. $-\omega$ is an imaginary part of a positive (not necessarily positive definite) Hermitian form.
2. For some basis $dz_1, ... dz_n$ in the space $\Lambda^{1,0}M$ 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 $\alpha_i$ real and non-negative.
3. For any (1,0)-tangent vector $v\in T^{1,0}M$, $-\sqrt{-1}\omega(v, \bar v) \geq 0$
4. For any real tangent vector $v\in TM$, $\omega(v, I(v)) \geq 0$, where $I:\; TM\mapsto TM$ 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,

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

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 $\Theta$ 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 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 $\sqrt{-1}\Theta$ positive.

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 :$\eta, \zeta \mapsto \int_M \eta\wedge\zeta$

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 $\eta$ 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 $\int_M \eta\wedge\zeta\geq 0$.

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

References

• Phillip Griffiths and Joseph Harris (1978), Principles of Algebraic Geometry, Wiley. ISBN 0-471-32792-1
• 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).