Positive form#Positive line bundles

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 $\Lambda^{p,p}(M)\cap \Lambda^{2p}(M,{\mathbb R}).$ A real (1,1)-form $\omega$ is called semi-positiveHuybrechts (2005) (sometimes just positiveDemailly (1994)), respectively, positiveHuybrechts (2005) (or positive definiteDemailly (1994)) 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.Demailly (1994) 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

References

P. Griffiths and J. Harris (1978), Principles of Algebraic Geometry, Wiley. {{isbn|0-471-32792-1}}
{{cite web |url=https://hdl.handle.net/20.500.12111/7881 |hdl=20.500.12111/7881 |title=Positivity and Vanishing Theorems |date=3 January 2020 |last1=Griffiths |first1=Phillip }}
J.-P. Demailly, [https://arxiv.org/abs/alg-geom/9410022 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)].
{{Citation | author1-last=Huybrechts | author1-first=Daniel | author1-link=Daniel Huybrechts | title=Complex Geometry: An Introduction | publisher=Springer | year=2005 | isbn=3-540-21290-6 | mr=2093043}}
{{Citation | author1-last=Voisin | author1-first=Claire | author1-link=Claire Voisin | title=Hodge Theory and Complex Algebraic Geometry (2 vols.) | publisher=Cambridge University Press | year=2007 | orig-year=2002 | isbn=978-0-521-71801-1 | mr=1967689 | doi=10.1017/CBO9780511615344}}

Category:Complex manifolds

Category:Algebraic geometry

Category:Differential forms