Positive Form

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 $\Lambda^(M)\cap \Lambda^(M,).$ 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:

#$-\omega$ is the imaginary part of a positive semidefinite (respectively, positive definite) Hermitian form.
#For some basis $dz_1, ... dz_n$ in the space $\Lambda^M$ of (1,0)-forms, $\sqrt\omega$ can be written diagonally, as $\sqrt\omega = \sum_i \alpha_i dz_i\wedge d\bar z_i,$ with $\alpha_i$ real and non-negative (respectively, positive).
#For any (1,0)-tangent vector $v\in T^M$, $-\sqrt\omega(v, \bar v) \geq 0$ (respectively, $>0$).
#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,

:$\bar\partial:\; L\mapsto L\otimes \Lambda^(M)$

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

:$\nabla^=\bar\partial$.

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 $\sqrt\Theta$ is a positive (1,1)-form. (Note that the de Rham cohomology class of $\sqrt\Theta$ 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 $\sqrt\Theta$ positive.

Positivity for ''(p, p)''-forms

Semi-positive (1,1)-forms on ''M'' form a convex cone. When ''M'' is a compact complex surface, $dim_M=2$, this cone is self-dual, with respect to the Poincaré pairing :$\eta, \zeta \mapsto \int_M \eta\wedge\zeta$

For ''(p, p)''-forms, where $2\leq p \leq dim_M-2$, there are two different notions of positivity.Demailly (1994) 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.
* 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)
'.
*
*{{Citation , author1-last=Voisin , author1-first=Claire , author1-link=Claire Voisin , title=Hodge Theory and Complex Algebraic Geometry (2 vols.) , publisher=Cambridge University Press , year=2007 , orig-year=2002 , isbn=978-0-521-71801-1 , mr=1967689 , doi=10.1017/CBO9780511615344

Complex manifolds
Algebraic geometry
Differential forms