- Positive form
complex geometry, the term "positive form"refers to several classes of real differential formsof Hodge type "(p, p)".
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 :A real (1,1)-form is called positive if any of thefollowing equivalent conditions hold
# is an imaginary part of a positive (not necessarily positive definite)
#For some basis in the space of (1,0)-forms, can be written diagonally, as with real and non-negative.
#For any (1,0)-tangent vector ,
#For any real tangent vector , , where is the
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
The curvature of a Chern connection is always apurely imaginary (1,1)-form. A line bundle "L" is called "positive" if
is a positive (1,1)-form. The
Kodaira vanishing theoremclaims that a positive line bundle is ample, and conversely, any ample line bundleadmits a Hermitian metric with positive.
Positivity for "(p, p)"-forms
Positive (1,1)-forms on "M" form a
convex cone.When "M" is a compact complex surface, , this cone is
self-dual, with respectto the Poincaré pairing:
For "(p, p)"-forms, where ,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 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.
Weakly positive and strongly positive formsform convex cones. On compact manifoldsthese cones are dualwith respect to the Poincaré pairing.
*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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