zbMATH — the first resource for mathematics

On the moduli space of bundles on \(K3\) surfaces. I. (English) Zbl 0674.14023
Vector bundles on algebraic varieties, Pap. Colloq., Bombay 1984, Stud. Math., Tata Inst. Fundam. Res. 11, 341-413 (1987).
[For the entire collection see Zbl 0653.00006.]
The author has shown in a previous paper [Invent. Math. 77, 101–116 (1984; Zbl 0565.14002)] that the moduli space \(M_ s\) of stable sheaves on an abelian or a \(K3\) surface \(S\) is smooth and is endowed with a natural symplectic structure.
Here the author studies more closely the case of \(K3\) surfaces. Considering a natural analogue for \(K3\) surfaces of the notion of isogeny of abelian surfaces, one shows that all components of dimension 2 of \(M_ 2\) are isogenous to \(S\).
We note two applications: (1) Two K3 surfaces with Picard number \(\geq 11\) are isogenous iff their transcendental Hodge structures are isogenous; this partly answers a question posed by I. R. Shafarevich in his article in Acta Congr. Int. Math. 1970, Vol. 1, 413–417 (1971; Zbl 0236.14016)].
(2) A proof of a conjecture of D. R. Morrison [see Invent. Math. 75, 105–121 (1984; Zbl 0509.14034)].

14J10 Families, moduli, classification: algebraic theory
14C22 Picard groups
14J28 \(K3\) surfaces and Enriques surfaces
32J25 Transcendental methods of algebraic geometry (complex-analytic aspects)
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)