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 Λ p , p ( M ) ∩ Λ 2 p ( M , R ) . {\displaystyle \Lambda ^{p,p}(M)\cap \Lambda ^{2p}(M,{\mathbb {R} }).} A real (1,1)-form ω {\displaystyle \omega } is called semi-positive (sometimes just positive), respectively, positive (or positive definite) if any of the following equivalent conditions holds:
− ω {\displaystyle -\omega } is the imaginary part of a positive semidefinite (respectively, positive definite) Hermitian form. For some basis d z 1 , . . . d z n {\displaystyle dz_{1},...dz_{n}} in the space Λ 1 , 0 M {\displaystyle \Lambda ^{1,0}M} of (1,0)-forms, ω {\displaystyle \omega } can be written diagonally, as ω = − 1 ∑ i α i d z i ∧ d z ¯ i , {\displaystyle \omega ={\sqrt {-1}}\sum _{i}\alpha _{i}dz_{i}\wedge d{\bar {z}}_{i},} with α i {\displaystyle \alpha _{i}} real and non-negative (respectively, positive). For any (1,0)-tangent vector v ∈ T 1 , 0 M {\displaystyle v\in T^{1,0}M} , − − 1 ω ( v , v ¯ ) ≥ 0 {\displaystyle -{\sqrt {-1}}\omega (v,{\bar {v}})\geq 0} (respectively, > 0 {\displaystyle >0} ). For any real tangent vector v ∈ T M {\displaystyle v\in TM} , ω ( v , I ( v ) ) ≥ 0 {\displaystyle \omega (v,I(v))\geq 0} (respectively, > 0 {\displaystyle >0} ), where I : T M ↦ T M {\displaystyle 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,
∂ ¯ : L ↦ L ⊗ Λ 0 , 1 ( M ) {\displaystyle {\bar {\partial }}:\;L\mapsto L\otimes \Lambda ^{0,1}(M)}
its complex structure operator. Then L is equipped with a unique connection preserving the Hermitian structure and satisfying
∇ 0 , 1 = ∂ ¯ {\displaystyle \nabla ^{0,1}={\bar {\partial }}} . This connection is called the Chern connection. The curvature Θ {\displaystyle \Theta } of the Chern connection is always a purely imaginary (1,1)-form. A line bundle L is called positive if − 1 Θ {\displaystyle {\sqrt {-1}}\Theta } is a positive (1,1)-form. (Note that the de Rham cohomology class of − 1 Θ {\displaystyle {\sqrt {-1}}\Theta } is 2 π {\displaystyle 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 − 1 Θ {\displaystyle {\sqrt {-1}}\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, d i m C M = 2 {\displaystyle dim_{\mathbb {C} }M=2} , this cone is self-dual, with respect to the Poincaré pairing : η , ζ ↦ ∫ M η ∧ ζ {\displaystyle \eta ,\zeta \mapsto \int _{M}\eta \wedge \zeta }
… excerpt ends here. Continue reading the full article.
