Positive form

In complex geometry, the term "positive form"refers to several classes of real differential formsof 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 thefollowing equivalent conditions hold

#$sqrt\{-1\}omega$ is an imaginary part of a positive (not necessarily positive definite) Hermitian form.
#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 with $alpha\_i$ real and non-negative.
#For any (1,0)-tangent vector $vin\; T^\{1,0\}M$,
#For any real tangent vector $vin\; TM$, $omega(v,\; I(v))\; geq\; 0$, where $I:;\; TMmapsto\; TM$ is the complex structure operator.

Positive line bundles

In algebraic geometry, positive (1,1)-forms arise as curvatureforms of ample line bundles (also known as "positive line bundles"). Let "L" be a holomorphic Hermitian linebundle on a complex manifold,

:

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

:.

This connection is called "the Chern connection".

The curvature $Theta$ of a Chern connection is always apurely 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.

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 respectto the Poincaré pairing:$eta,\; zeta\; mapsto\; int\_M\; etawedgezeta$

For "(p, p)"-forms, where $2leq\; p\; leq\; dim\_\{Bbb\; C\}M-2$,there are two different notions of positivity. A form is calledstrongly positive if it is a linear combination ofproducts of positive forms, with positive real coefficients.A real "(p, p)"-form $eta$ on an "n"-dimensionalcomplex manifold "M" is called weakly positiveif for all strongly positive "(n-p, n-p)"-forms ζ with compact support, we have$int\_M\; etawedgezetageq\; 0$.

Weakly positive and strongly positive formsform convex cones. On compact manifoldsthese cones are dualwith respect to the Poincaré pairing.

References

*Phillip Griffiths and Joseph Harris (1978), "Principles of Algebraic Geometry", Wiley. ISBN 0471327921

*J.-P. Demailly, " [http://arxiv.org/abs/alg-geom/9410022 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)] ".