A new proof of the global Torelli theorem for K3 surfaces. (English) Zbl 0559.14004

Friedman, Robert

Ann. Math. (2) 120, 237-269 (1984).

The aim of this paper is to give a new proof of the global Torelli theorem for algebraic K3 surfaces. The method of proof consists in interpreting the theorem of Kulikov-Persson-Pinkham [cf. V. S. Kulikov, Math. USSR, Izv. 11, 957-989 (1977); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 41, 1008-1042 (1977; Zbl 0367.14014) and U. Persson and H. Pinkham, Ann. Math., II. Ser. 113, 45-66 (1981; Zbl 0426.14015)] on degenerations of K3 surfaces as a way to construct partial compactifications of the moduli spaces of polarized K3 surfaces. Analogous constructions of partial compactifications of the appropriate period spaces have been given by Mumford. The proof then reduces to a degree calculation, which is done via an analysis of the limiting mixed Hodge structures.

MSC:

14C30	Transcendental methods, Hodge theory (algebro-geometric aspects)
14J15	Moduli, classification: analytic theory; relations with modular forms
32J05	Compactification of analytic spaces
14J25	Special surfaces
32J25	Transcendental methods of algebraic geometry (complex-analytic aspects)

Keywords:

global Torelli theorem for algebraic K3 surfaces; compactifications of the moduli spaces of polarized K3 surfaces; mixed Hodge structures

Citations:

Zbl 0387.14007; Zbl 0447.14007; Zbl 0367.14014; Zbl 0426.14015