Kodaira surface

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 P_n 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.

References

• Barth, Wolf P.; Hulek, Klaus; Peters, Chris A.M.; Van de Ven, Antonius (2004), Compact Complex Surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. 4, Springer-Verlag, Berlin, ISBN 978-3-540-00832-3, MR 2030225 – the standard reference book for compact complex surfaces
• Kodaira, Kunihiko (1964), "On the structure of compact complex analytic surfaces. I", American Journal of Mathematics 86 (4): 751–798, doi:10.2307/2373157, ISSN 0002-9327, JSTOR 2373157, MR 0187255
• Kodaira, Kunihiko (1966), "On the structure of compact complex analytic surfaces. II", American Journal of Mathematics 88 (3): 682–721, doi:10.2307/2373150, ISSN 0002-9327, JSTOR 2373150, MR 0205280
• Kodaira, Kunihiko (1968), "On the structure of compact complex analytic surfaces. III", American Journal of Mathematics 90 (1): 55–83, doi:10.2307/2373426, ISSN 0002-9327, JSTOR 2373426, MR 0228019
• Kodaira, Kunihiko (1968), "On the structure of complex analytic surfaces. IV", American Journal of Mathematics 90 (4): 1048–1066, doi:10.2307/2373289, ISSN 0002-9327, JSTOR 2373289, MR 0239114