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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

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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:

  1. is the imaginary part of a positive semidefinite (respectively, 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 (respectively, positive).
  3. For any (1,0)-tangent vector , (respectively, ).
  4. For any real tangent vector , (respectively, ), where is the complex structure operator.

Positive line bundles

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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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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

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  1. ^ Huybrechts (2005)
  2. ^ Demailly (1994)
  3. ^ Huybrechts (2005)
  4. ^ Demailly (1994)
  5. ^ Demailly (1994)

References

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  • P. Griffiths and J. Harris (1978), Principles of Algebraic Geometry, Wiley. ISBN 0-471-32792-1
  • Griffiths, Phillip (3 January 2020). "Positivity and Vanishing Theorems". hdl:20.500.12111/7881.
  • J.-P. Demailly, L2 vanishing theorems for positive line bundles and adjunction theory, Lecture Notes of a CIME course on "Transcendental Methods of Algebraic Geometry" (Cetraro, Italy, July 1994).
  • Huybrechts, Daniel (2005), Complex Geometry: An Introduction, Springer, ISBN 3-540-21290-6, MR 2093043
  • Voisin, Claire (2007) [2002], Hodge Theory and Complex Algebraic Geometry (2 vols.), Cambridge University Press, doi:10.1017/CBO9780511615344, ISBN 978-0-521-71801-1, MR 1967689