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 ...
, a Hopf surface is a compact complex surface obtained as a quotient of the complex
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
(with zero deleted)
by a
free action
In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism g ...
of a discrete group. If this group is the integers the Hopf surface is called primary, otherwise it is called secondary. (Some authors use the term "Hopf surface" to mean "primary Hopf surface".) The first example was found by , with the discrete group isomorphic to the integers, with a generator acting on
by multiplication by 2; this was the first example of a compact complex surface with no
Kähler metric Kähler may refer to:
;People
*Alexander Kähler (born 1960), German television journalist
*Birgit Kähler (born 1970), German high jumper
*Erich Kähler (1906–2000), German mathematician
*Heinz Kähler (1905–1974), German art historian and arc ...
.
Higher-dimensional analogues of Hopf surfaces are called
Hopf manifold In complex geometry, a Hopf manifold is obtained
as a quotient of the complex vector space
(with zero deleted) (^n\backslash 0)
by a free action of the group \Gamma \cong of
integers, with the generator \gamma
of \Gamma acting by holomorphic co ...
s.
Invariants
Hopf surfaces are
surfaces of class VII and in particular all have
Kodaira dimension In algebraic geometry, the Kodaira dimension ''κ''(''X'') measures the size of the canonical ring, canonical model of a projective variety ''X''.
Igor Shafarevich, in a seminar introduced an important numerical invariant of surfaces with the ...
, and all their plurigenera vanish. The geometric genus is 0. The
fundamental group
In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of ...
has a normal central infinite cyclic subgroup of finite index. The
Hodge diamond
Homological mirror symmetry is a mathematical conjecture made by Maxim Kontsevich. It seeks a systematic mathematical explanation for a phenomenon called mirror symmetry first observed by physicists studying string theory.
History
In an address ...
is
In particular the first
Betti number
In algebraic topology, the Betti numbers are used to distinguish topological spaces based on the connectivity of ''n''-dimensional simplicial complexes. For the most reasonable finite-dimensional spaces (such as compact manifolds, finite simplicia ...
is 1 and the second Betti number is 0.
Conversely showed that a compact complex surface with vanishing the second Betti number and whose fundamental group contains an infinite cyclic subgroup of finite index is a Hopf surface.
Primary Hopf surfaces
In the course of
classification of compact complex surfaces, Kodaira classified the primary Hopf surfaces.
A primary Hopf surface is obtained as
:
where
is a group generated by
a polynomial contraction
.
Kodaira has found a normal form for
.
In appropriate coordinates,
can be written as
:
where
are complex numbers
satisfying
, and either
or
.
These surfaces contain an elliptic curve (the image of the ''x''-axis) and if
the image of the ''y''-axis is a second elliptic curve. When
, the Hopf surface is an elliptic fiber space over the projective line if
for some positive integers ''m'' and ''n'', with the map to the projective line given by
, and otherwise the only curves are the two images of the axes.
The
Picard group
In mathematics, the Picard group of a ringed space ''X'', denoted by Pic(''X''), is the group of isomorphism classes of invertible sheaves (or line bundles) on ''X'', with the group operation being tensor product. This construction is a global ve ...
of any primary Hopf surface is isomorphic to the non-zero complex numbers
.
has proven that a complex surface
is diffeomorphic to
if and only if it is a primary Hopf surface.
Secondary Hopf surfaces
Any secondary Hopf surface has a finite unramified cover that is a primary Hopf surface. Equivalently, its fundamental group has a subgroup of finite index in its center that is isomorphic to the integers. classified them by finding the finite groups acting without fixed points on primary Hopf surfaces.
Many examples of secondary Hopf surfaces can be constructed with underlying space a product of a
spherical space form
In mathematics, a spherical 3-manifold ''M'' is a 3-manifold of the form
:M=S^3/\Gamma
where \Gamma is a finite subgroup of SO(4) acting freely by rotations on the 3-sphere S^3. All such manifolds are prime, orientable, and closed. Spherical 3-m ...
s and a circle.
References
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*{{eom, id=H/h110270, first=Liviu , last=Ornea, title=Hopf manifold
Complex surfaces