The term “K3” is (Weil 79, p. 546)
in honor of Kummer, Kähler, Kodaira, and the beautiful K2 mountain in Kashmir
A cyclic cover branched over a curve of degree
All K3 surfaces are simply connected.
The Hodge diamond? is completely determined (even in positive characteristic) and hence the Hodge-de Rham spectral sequence degenerates at . This also implies that the Betti numbers are completely determined as .
The formal Brauer group is
or its height is , and every value may occur
|Calabi-Cau n-fold||line n-bundle||moduli of line n-bundles||moduli of flat/degree-0 n-bundles||Artin-Mazur formal group of deformation moduli of line n-bundles||complex oriented cohomology theory||modular functor/self-dual higher gauge theory of higher dimensional Chern-Simons theory|
|unit in structure sheaf||multiplicative group/group of units||formal multiplicative group||complex K-theory|
|elliptic curve||line bundle||Picard group/Picard scheme||Jacobian||formal Picard group||elliptic cohomology||3d Chern-Simons theory/WZW model|
|K3 surface||line 2-bundle||Brauer group||intermediate Jacobian||formal Brauer group||K3 cohomology|
|Calabi-Yau 3-fold||line 3-bundle||intermediate Jacobian||CY3 cohomology||7d Chern-Simons theory/M5-brane|
Original sources include
Michael Artin, Supersingular K3 Surfaces, Annal. Sc. d, l’Éc Norm. Sup. 4e séries, T. 7, fasc. 4, 1974, pp. 543-568
Andre Weil, Final report on contract AF 18 (603)-57. In Scientic works. Collected papers. Vol. II (1951-1964). 1979.
Lecture notes include
David Morrison, The geometry of K3 surfaces Lecture notes (1988)
Viacheslav Nikulin, Elliptic fibrations on K3 surfaces (arXiv:1010.3904)