Positive form

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,{\mathbb R}). 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^{1,0}M of (1,0)-forms, \omega can be written diagonally, as \omega = \sqrt{-1} \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^{1,0}M, -\sqrt{-1}\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^{0,1}(M) its complex structure operator. Then L is equipped with a unique connection preserving the Hermitian structure and satisfying :\nabla^{0,1}=\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{-1}\Theta is a positive (1,1)-form. (Note that the de Rham cohomology class of \sqrt{-1}\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{-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, dim_{\mathbb C}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_{\mathbb C}M-2, there are two different notions of positivity. 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. == Notes ==