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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)].

MSC:
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)