Kodaira surface
Encyclopedia
In mathematics, a Kodaira surface is a compact
complex surface
of Kodaira dimension
0 and odd first Betti number
.
These are never algebraic, though they have non-constant meromorphic functions. They are usually divided into two subtypes: primary Kodaira surfaces with trivial canonical bundle, and secondary Kodaira surfaces which are quotients of these by finite groups of orders 2, 3, 4, or 6, and which have non-trivial canonical bundles. The secondary Kodaira surfaces have the same relation to primary ones that Enriques surfaces have to K3 surfaces, or bielliptic surfaces have to abelian surfaces.
Invariants: If the surface is the quotient of a primary Kodaira surface by a group of order k=1,2,3,4,6, then the plurigenera Pn are 1 if n is divisible by k and 0 otherwise.
Hodge diamond:
Examples: Take a non-trivial line bundle over an elliptic curve, remove the zero section, then quotient out the fibers by Z acting as multiplication by powers of some complex number z.
This gives a primary Kodaira surface.
Compact space
In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
complex surface
Algebraic surface
In mathematics, an algebraic surface is an algebraic variety of dimension two. In the case of geometry over the field of complex numbers, an algebraic surface has complex dimension two and so of dimension four as a smooth manifold.The theory of algebraic surfaces is much more complicated than that...
of Kodaira dimension
Kodaira dimension
In algebraic geometry, the Kodaira dimension κ measures the size of the canonical model of a projective variety V.The definition of Kodaira dimension, named for Kunihiko Kodaira, and the notation κ were introduced in the seminar.-The plurigenera:...
0 and odd first Betti number
Betti number
In algebraic topology, a mathematical discipline, the Betti numbers can be used to distinguish topological spaces. Intuitively, the first Betti number of a space counts the maximum number of cuts that can be made without dividing the space into two pieces....
.
These are never algebraic, though they have non-constant meromorphic functions. They are usually divided into two subtypes: primary Kodaira surfaces with trivial canonical bundle, and secondary Kodaira surfaces which are quotients of these by finite groups of orders 2, 3, 4, or 6, and which have non-trivial canonical bundles. The secondary Kodaira surfaces have the same relation to primary ones that Enriques surfaces have to K3 surfaces, or bielliptic surfaces have to abelian surfaces.
Invariants: If the surface is the quotient of a primary Kodaira surface by a group of order k=1,2,3,4,6, then the plurigenera Pn are 1 if n is divisible by k and 0 otherwise.
Hodge diamond:
| 1 | |||||
|---|---|---|---|---|---|
| 1 | 2 | ||||
| 1 | 2 | 1 | (Primary) | ||
| 2 | 1 | ||||
| 1 |
| 1 | |||||
|---|---|---|---|---|---|
| 0 | 1 | ||||
| 0 | 0 | 0 | (Secondary) | ||
| 1 | 0 | ||||
| 1 |
Examples: Take a non-trivial line bundle over an elliptic curve, remove the zero section, then quotient out the fibers by Z acting as multiplication by powers of some complex number z.
This gives a primary Kodaira surface.