On the Torelli problem for kählerian $K-3$ surfaces

Two algebraic deformations of a K3 surface

“…This result, existence of 37, is in the spirit of part of the work [3] of Burns and Rapoport where (besides the main theorem) it is shown that elementary operations are "the main reason for the phenomenon of nonseparatedness in the moduli of unpolarized non-ruled algebraic surfaces over C."…”

“…We assume the characteristic Φ2. The generalization of X is a nonsingular quartic surface Q in P 3 : we wish to show in § 1 that there is an irreducible algebraic family of surfaces over the aίϊine line, in which X is a member and in which Q is a general member. The specialization of X is a surface Y having a birational model which is a ramified double cover of a quadric cone in P\ It has been observed [19] that such a specialization exists; what we propose to show is that there are two different ways to get to it, that is there are two non-isomorphic irreducible algebraic families S?, £f* of surfaces over the affine line, each having a surface Y as a member and having X as a general member.…”

“…§ 2) in the known family £?, along a single, nonextending nodal curve R on Y (i.e. R is a non-singular rational curve with self-intersection -2 on Y), exists algebraically-it is always defined analytically, [3], [9]-and defines a birational transformation η of Sf. The image &** -rjζS?)…”

Moduli spaces of Einstein metrics on 4-manifolds

“…The proof of [10] follows the classical proofs of the Torelli and Surjectivity theorems for K3 surfaces, and uses heavily the results of [7,15] and especially [13].…”

“…There is a natural period map [4] As a consequence, one obtains immediately and naturally the Surjectivity Theorem of [15] and a form of the Global Torelli Theorem of [7], for Kàhlerian K3 surfaces. The proof of these corollaries does not require, for instance, results on the density of the period map or the Torelli theorem for Kummer or algebraic K3 surfaces.…”