# Positive form

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

#$sqrt\left\{-1\right\}omega$ is an imaginary part of a positive (not necessarily positive definite) Hermitian form.
#For some basis $dz_1, ... dz_n$ in the space $Lambda^\left\{1,0\right\}M$ of (1,0)-forms,$sqrt\left\{-1\right\}omega$ can be written diagonally, as with $alpha_i$ real and non-negative.
#For any (1,0)-tangent vector $vin T^\left\{1,0\right\}M$,
#For any real tangent vector $vin TM$, $omega\left(v, I\left(v\right)\right) geq 0$, 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

:$sqrt\left\{-1\right\}Theta$

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

Positivity for "(p, p)"-forms

Positive (1,1)-forms on "M" form a convex cone.When "M" is a compact complex surface, $dim_\left\{Bbb C\right\}M=2$, this cone is
self-dual, with respectto the Poincaré pairing:$eta, zeta mapsto int_M etawedgezeta$

For "(p, p)"-forms, where $2leq p leq dim_\left\{Bbb C\right\}M-2$,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)] ".

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