# Positive form

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{{for|the linguistics term |Positive (linguistics)}}

In [complex geometry](/source/complex_geometry), the term ''positive form'' refers to several classes of real [differential form](/source/differential_form)s of [Hodge type](/source/Hodge_decomposition) ''(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 <math>\Lambda^{p,p}(M)\cap \Lambda^{2p}(M,{\mathbb R}).</math> A real (1,1)-form <math>\omega</math> is called '''semi-positive'''<ref>Huybrechts (2005)</ref> (sometimes just ''positive''<ref>Demailly (1994)</ref>), respectively, '''positive'''<ref>Huybrechts (2005)</ref> (or ''positive definite''<ref>Demailly (1994)</ref>) if any of the following equivalent conditions holds:

#<math>-\omega</math> is the imaginary part of a positive semidefinite (respectively, positive definite) [Hermitian form](/source/Hermitian_form).
#For some basis <math>dz_1, ... dz_n</math> in the space <math>\Lambda^{1,0}M</math> of (1,0)-forms, <math>\omega</math> can be written diagonally, as <math>\omega = \sqrt{-1} \sum_i \alpha_i dz_i\wedge d\bar z_i,</math> with <math>\alpha_i</math> real and non-negative (respectively, positive).
#For any (1,0)-tangent vector <math>v\in T^{1,0}M</math>, <math>-\sqrt{-1}\omega(v, \bar v) \geq 0</math> (respectively, <math>>0</math>).
#For any real tangent vector <math>v\in TM</math>, <math>\omega(v, I(v)) \geq 0</math> (respectively, <math>>0</math>), where <math>I:\; TM\mapsto TM</math> is the [complex structure](/source/Complex_manifold) operator.

== Positive line bundles ==

In algebraic geometry, positive definite (1,1)-forms arise as curvature forms of [ample line bundle](/source/ample_line_bundle)s (also known as ''positive line bundles''). Let ''L'' be a holomorphic Hermitian line bundle on a complex manifold,

:<math> \bar\partial:\; L\mapsto L\otimes \Lambda^{0,1}(M)</math>

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

:<math>\nabla^{0,1}=\bar\partial</math>.

This connection is called ''the [Chern connection](/source/Hermitian_connection)''.

The curvature <math>\Theta</math> of the Chern connection is always a
purely imaginary (1,1)-form. A line bundle ''L'' is called '''positive''' if <math>\sqrt{-1}\Theta</math> is a positive (1,1)-form. (Note that the de Rham cohomology class of <math>\sqrt{-1}\Theta</math> is <math>2\pi</math> times the first [Chern class](/source/Chern_class) of ''L''.) The [Kodaira embedding theorem](/source/Kodaira_embedding_theorem) claims that a positive line bundle is ample, and conversely, any [ample line bundle](/source/ample_line_bundle) admits a Hermitian metric with <math>\sqrt{-1}\Theta</math> positive.

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

Semi-positive (1,1)-forms on ''M'' form a [convex cone](/source/convex_cone). When ''M'' is a compact [complex surface](/source/complex_surface), <math>dim_{\mathbb C}M=2</math>, this cone is [self-dual](/source/Convex_cone), with respect to the Poincaré pairing :<math> \eta, \zeta \mapsto \int_M \eta\wedge\zeta</math>

For ''(p, p)''-forms, where <math>2\leq p \leq dim_{\mathbb C}M-2</math>, there are two different notions of positivity.<ref>Demailly (1994)</ref> 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 <math>\eta</math> 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 <math>\int_M \eta\wedge\zeta\geq 0 </math>.

Weakly positive and strongly positive forms form convex cones. On compact manifolds these cones are [dual](/source/Convex_cone) with respect to the Poincaré pairing.

== Notes ==
{{Reflist}}
== References ==
*[P. Griffiths](/source/Phillip_Griffiths) and [J. Harris](/source/Joe_Harris_(mathematician)) (1978), ''Principles of Algebraic Geometry'', Wiley. {{isbn|0-471-32792-1}}
* {{cite web |url=https://hdl.handle.net/20.500.12111/7881 |hdl=20.500.12111/7881 |title=Positivity and Vanishing Theorems |date=3 January 2020 |last1=Griffiths |first1=Phillip }}
*[J.-P. Demailly](/source/Jean-Pierre_Demailly), ''[https://arxiv.org/abs/alg-geom/9410022 L<sup>2</sup> 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=Huybrechts | author1-first=Daniel | author1-link=Daniel Huybrechts | title=Complex Geometry: An Introduction | publisher=[Springer](/source/Springer_Science%2BBusiness_Media) | year=2005 | isbn=3-540-21290-6 | mr=2093043}}
*{{Citation | author1-last=Voisin | author1-first=Claire | author1-link=Claire Voisin | title=Hodge Theory and Complex Algebraic Geometry (2 vols.) | publisher=[Cambridge University Press](/source/Cambridge_University_Press) | year=2007 | orig-year=2002 | isbn=978-0-521-71801-1 | mr=1967689 | doi=10.1017/CBO9780511615344}}

Category:Complex manifolds
Category:Algebraic geometry
Category:Differential forms

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