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| {{for|the linguistics term |Positive (linguistics)}}
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| In [[complex geometry]], the term ''positive form''
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| refers to several classes of real differential forms
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| of [[Hodge decomposition#Hodge_decomposition|Hodge type]] ''(p, p)''.
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| == (1,1)-forms ==
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| Real (''p'',''p'')-forms on a complex manifold ''M''
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| are forms which are of type (''p'',''p'') and real,
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| that is, lie in the intersection
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| :<math>\Lambda^{p,p}(M)\cap \Lambda^{2p}(M,{\Bbb R}).</math>
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| A real (1,1)-form <math>\omega</math>
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| is called '''positive''' if any of the
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| following equivalent conditions hold
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| #<math>\sqrt{-1}\omega</math> is an imaginary part of a positive (not necessarily positive definite) [[Hermitian form]].
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| #For some basis <math>dz_1, ... dz_n</math> in the space <math>\Lambda^{1,0}M</math> of (1,0)-forms,<math>\sqrt{-1}\omega</math> can be written diagonally, as <math> \sqrt{-1}\omega = \sum_i \alpha_i dz_i\wedge d\bar z_i,</math> with <math>\alpha_i</math> real and non-negative.
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| #For any (1,0)-tangent vector <math>v\in T^{1,0}M</math>, <math>-\sqrt{-1}\omega(v, \bar v) \geq 0</math>
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| #For any real tangent vector <math>v\in TM</math>, <math>\omega(v, I(v)) \geq 0</math>, where <math>I:\; TM\mapsto TM</math> is the [[complex structure]]{{dn|date=September 2012}} operator.
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| == Positive line bundles ==
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| In algebraic geometry, positive (1,1)-forms arise as curvature
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| forms of [[ample line bundle]]s (also known as
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| ''positive line bundles''). Let ''L'' be a holomorphic Hermitian line
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| bundle on a complex manifold,
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| :<math> \bar\partial:\; L\mapsto L\otimes \Lambda^{0,1}(M)</math>
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| its complex structure operator. Then ''L'' is equipped with a unique connection preserving the Hermitian structure and satisfying
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| :<math>\nabla^{0,1}=\bar\partial</math>.
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| This connection is called ''the [[Hermitian connection|Chern connection]]''.
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| The curvature <math>\Theta</math> of a Chern connection is always a
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| purely imaginary (1,1)-form. A line bundle ''L'' is called ''positive'' if
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| :<math>\sqrt{-1}\Theta</math>
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| is a positive definite (1,1)-form. The [[Kodaira embedding theorem]] claims that a positive line bundle is ample, and conversely, any [[ample line bundle]] admits a Hermitian metric with <math>\sqrt{-1}\Theta</math> positive.
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| == Positivity for ''(p, p)''-forms ==
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| Positive (1,1)-forms on ''M'' form a [[convex cone]].
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| When ''M'' is a compact [[complex surface]],
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| <math>dim_{\Bbb C}M=2</math>, this cone is
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| [[Convex cone#Dual_cone|self-dual]], with respect
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| to the Poincaré pairing
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| :<math> \eta, \zeta \mapsto \int_M \eta\wedge\zeta</math> | |
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| For ''(p, p)''-forms, where <math>2\leq p \leq dim_{\Bbb C}M-2</math>,
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| there are two different notions of positivity. A form is called
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| '''strongly positive''' if it is a linear combination of
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| products of positive forms, with positive real coefficients.
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| A real ''(p, p)''-form <math>\eta</math> on an ''n''-dimensional
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| complex manifold ''M'' is called '''weakly positive'''
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| if for all strongly positive ''(n-p, n-p)''-forms
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| ζ with compact support, we have
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| <math>\int_M \eta\wedge\zeta\geq 0 </math>.
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| Weakly positive and strongly positive forms
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| form convex cones. On compact manifolds
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| these cones are [[Convex cone#Dual_cone|dual]] | |
| with respect to the Poincaré pairing.
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| ==References==
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| *Phillip Griffiths and Joseph Harris (1978), ''Principles of Algebraic Geometry'', Wiley. ISBN 0-471-32792-1
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| *J.-P. Demailly, ''[http://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)]''.
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| [[Category:Complex manifolds]]
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| [[Category:Algebraic geometry]]
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| [[Category:Differential forms]]
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