Positive form
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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 A real (1,1)-form is called semi-positive[1] (sometimes just positive[2]), respectively, positive[3] (or positive definite[4]) 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.
Positive line bundles
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Perspective
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.
Positivity for (p, p)-forms
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Perspective
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.[5] 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 .
Weakly positive and strongly positive forms form convex cones. On compact manifolds these cones are dual with respect to the Poincaré pairing.
Notes
References
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