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

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

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

(i) ${\sqrt {-1}}\omega$ is an imaginary part of a positive (not necessarily positive definite) Hermitian form.

(ii) 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.

(iii) For any (1,0)-tangent vector $v\in T^{1,0}M$, ${\sqrt {-1}}\omega (v,{\bar {v}})\geq 0$

(iv) 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 (1,1)-form. 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.

Positivity for (p,p)-forms

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

$\eta ,\zeta \mapsto \int _{M}\eta \wedge \zeta$

For (p,p)-forms, where $2\leq p\leq dim_{\mathbb {C}}M-2$, 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 Poincare pairing.

Reference

• Phillip Griffiths and Joseph Harris, Principles of Algebraic Geometry

• J.-P. Demailly, $L^2$ 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)