Positive form


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 sqrt{-1}omega = sum_i alpha_i dz_iwedge dar z_i, with alpha_i real and non-negative.
#For any (1,0)-tangent vector vin T^{1,0}M, -sqrt{-1}omega(v, ar v) geq 0
#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,

: arpartial:; Lmapsto Lotimes Lambda^{0,1}(M)

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

: abla^{0,1}=arpartial.

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 haveint_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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