Positive form

For the linguistics term, see Positive (linguistics).

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 positive if any of the following equivalent conditions hold

  1. is an imaginary part of a positive (not necessarily positive definite) Hermitian form.
  2. For some basis in the space of (1,0)-forms, can be written diagonally, as with real and non-negative.
  3. For any (1,0)-tangent vector ,
  4. For any real tangent vector , , where is the complex structure operator.

Positive line bundles

In algebraic geometry, positive (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 a Chern connection is always a purely imaginary (1,1)-form. A line bundle L is called positive if

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 positive.

Positivity for (p, p)-forms

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 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.

References

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