In
complex geometry
In mathematics, complex geometry is the study of geometric structures and constructions arising out of, or described by, the complex numbers. In particular, complex geometry is concerned with the study of spaces such as complex manifolds and c ...
, the term ''positive form'' refers to several classes of real
differential form
In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, ...
s 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 (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
In mathematics, a sesquilinear form is a generalization of a bilinear form that, in turn, is a generalization of the concept of the dot product of Euclidean space. A bilinear form is linear in each of its arguments, but a sesquilinear form allow ...
.
#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
In algebraic geometry, positive definite (1,1)-forms arise as curvature forms of
ample line bundle In mathematics, a distinctive feature of algebraic geometry is that some line bundles on a projective variety can be considered "positive", while others are "negative" (or a mixture of the two). The most important notion of positivity is that of an ...
s (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
In mathematics, in particular in algebraic topology, differential geometry and algebraic geometry, the Chern classes are characteristic classes associated with complex vector bundles. They have since found applications in physics, Calabi–Yau ma ...
of ''L''.) The
Kodaira embedding theorem
In mathematics, the Kodaira embedding theorem characterises non-singular projective varieties, over the complex numbers, amongst compact Kähler manifolds. In effect it says precisely which complex manifolds are defined by homogeneous polynomial ...
claims that a positive line bundle is ample, and conversely, any
ample line bundle In mathematics, a distinctive feature of algebraic geometry is that some line bundles on a projective variety can be considered "positive", while others are "negative" (or a mixture of the two). The most important notion of positivity is that of an ...
admits a Hermitian metric with
positive.
Positivity for ''(p, p)''-forms
Semi-positive (1,1)-forms on ''M'' form a
convex cone
In linear algebra, a ''cone''—sometimes called a linear cone for distinguishing it from other sorts of cones—is a subset of a vector space that is closed under scalar multiplication; that is, is a cone if x\in C implies sx\in C for every .
...
. When ''M'' is a compact
complex surface
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
,
, this cone is
self-dual
In mathematics, a duality translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a Injective function, one-to-one fashion, often (but not always) by means of an Involution (mathematics), involutio ...
, with respect to the Poincaré pairing :
For ''(p, p)''-forms, where
, 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
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
*
P. Griffiths and
J. Harris (1978), ''Principles of Algebraic Geometry'', Wiley.
*
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)'.
*
*{{Citation , author1-last=Voisin , author1-first=Claire , author1-link=Claire Voisin , title=Hodge Theory and Complex Algebraic Geometry (2 vols.) , publisher=
Cambridge University Press
Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press
A university press is an academic publishing hou ...
, year=2007 , orig-year=2002 , isbn=978-0-521-71801-1 , mr=1967689 , doi=10.1017/CBO9780511615344
Complex manifolds
Algebraic geometry
Differential forms