Kummer structures on a K3 surface: An old question of T. Shioda

Abstract

We apply our earlier results on Fourier-Mukai partners to answer definitively a question about Kummer surface structures posed by T. Shioda twenty-five years ago.where π :Â × A → and π A :Â × A → A are the natural projections. In this equivalence, the structure sheaf Oâ of the pointâ ∈ is mapped to the invertible

Some Remarks about the FM-partners of K3 Surfaces with Picard Numbers 1 and 2

“…Lemma 3.1 is true also when L ¼ U È U È U. The period map is onto also for abelian surfaces (see [16]) and so, using the lattices S d;n , the following proposition (similar to a result given in [6]) can be proved in a similar manner. PROPOSITION 3.5.…”

Derived categories and Kummer varieties

“…This is one of the main differences with the untwisted case treated by Hosono et al [11] (see (A) and (C) in Sect. 1).…”

“…Reasoning as before and using the surjectivity of the period map and the Torelli Theorem for abelian surfaces [28], we get an isometry ϕ 1 : T(E 1 × F) → U 2 fitting in the following commutative diagram: The main result in [11] proves that the preimage of [(Km(A), 1)] is finite, for any abelian surface A and 1 ∈ Br(A) the trivial class (see [11,Theorem 0.1]). On the other hand [11] shows that the cardinality of the preimages of can be arbitrarily large. This answered an old question by Shioda.…”

“…Using Theorem 1.2, the picture in [11] can be completely generalized to the twisted case. In other words, Proposition 4.5 shows that on a twisted K3 surface we can put just a finite number of non-isomorphic twisted Kummer structures.…”

Construction of Nikulin configurations on some Kummer surfaces and applications

“…A Kummer structure on a K3 surface X is an isomorphism class of Abelian surfaces B such that X ≃ Km(B). The following Proposition is stated in [12]; we give here a proof for completeness:…”