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

1. $-\omega$ is the imaginary part of a positive semidefinite (respectively, positive definite) Hermitian form.
2. For some basis $d{z}_1,...d{z}_n$ in the space ${\mathrm{\Lambda }}^{1,0}M$ of (1,0)-forms, $\omega$ can be written diagonally, as $\omega =\sqrt{-1}\sum _i{\alpha }_{i}d{z}_i\wedge d{\overline{z}}_i,$ with ${\alpha }_i$ real and non-negative (respectively, positive).
3. For any (1,0)-tangent vector $v\in T^{1,0}M$, $-\sqrt{-1}\omega (v,\overline{v})\ge 0$ (respectively, $>0$).
4. For any real tangent vector $v\in TM$, $\omega (v,I(v))\ge 0$ (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,

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

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

${\mathrm{\nabla }}^{0,1}=\overline{\partial }$.

This connection is called the Chern connection.

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

Positivity for (p, p)-forms

Semi-positive (1,1)-forms on M form a convex cone. When M is a compact complex surface, $di{m}_{ℂ}M=2$, this cone is self-dual, with respect to the Poincaré pairing :$\eta ,\zeta \mapsto \underset{M}{\int }\eta \wedge \zeta$

For (p, p)-forms, where $2\le p\le di{m}_{ℂ}M-2$, there are two different notions of positivity. A form is called strongly positive if it is a linear combination of products of semi-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 $\underset{M}{\int }\eta \wedge \zeta \ge 0$.

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. Template:Isbn
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• 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).
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