Positive form

For the linguistics term, see Positive (linguistics).

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 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 $dz_{1},...dz_{n}$ 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 (respectively, positive).
3. For any (1,0)-tangent vector $v\in T^{1,0}M$, $-{\sqrt {-1}}\omega (v,{\bar {v}})\geq 0$ (respectively, $>0$).
4. For any real tangent vector $v\in TM$, $\omega (v,I(v))\geq 0$ (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,

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

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 the 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. (Note that the de Rham cohomology class of ${\sqrt {-1}}\Theta$ 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 ${\sqrt {-1}}\Theta$ positive.

Positivity for (p, p)-forms

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

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

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

References

• 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". hdl: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).
• Huybrechts, Daniel (2005), Complex Geometry: An Introduction, Springer, ISBN 3-540-21290-6, MR 2093043
• Voisin, Claire (2007) [2002], Hodge Theory and Complex Algebraic Geometry (2 vols.), Cambridge University Press, doi:10.1017/CBO9780511615344, ISBN 978-0-521-71801-1, MR 1967689