Positive form

In complex geometry, the term positive form refers to several classes of real differential forms of Hodge type (p, p). 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 A real (1,1)-form is called semi-positive (sometimes just positive), respectively, positive (or positive definite) if any of the following equivalent conditions holds: is the imaginary part of a positive semidefinite (respectively, positive definite) Hermitian form. For some basis in the space of (1,0)-forms, can be written diagonally, as with real and non-negative (respectively, positive). For any (1,0)-tangent vector , (respectively, ). For any real tangent vector , (respectively, ), where is the complex structure operator. 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, its complex structure operator. Then L is equipped with a unique connection preserving the Hermitian structure and satisfying This connection is called the Chern connection. The curvature of the Chern connection is always a purely imaginary (1,1)-form. A line bundle L is called positive if is a positive (1,1)-form. (Note that the de Rham cohomology class of is 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 positive. Semi-positive (1,1)-forms on M form a convex cone. When M is a compact complex surface, , this cone is self-dual, with respect to the Poincaré pairing : For (p, p)-forms, where , 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 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 .