Volume 2010, Issue 648

Kuga and Satake associate with every polarized complex K3 surface ( X , ℒ) a complex abelian variety called the Kuga-Satake abelian variety of ( X , ℒ). We use this construction to define morphisms between moduli spaces of polarized K3 surfaces with certain level structures and moduli spaces of polarized abelian varieties with level structure over ℂ. In this note we study these morphisms. We prove first that they are defined over finite extensions of ℚ. This is done by proving analogues of the main theorems of complex multiplication for abelian varieties for K3 surfaces. Then we show that they extend in positive characteristic. In this way we give an indirect construction of Kuga-Satake abelian varieties over an arbitrary base. We also give some applications of this construction to canonical lifts of ordinary K3 surfaces.

We construct knotted proper holomorphic embeddings of the unit disc in .

We prove that Lipschitz intrinsic graphs in the Heisenberg groups , with n > 1, which are vanishing viscosity solutions of the minimal surface equation, are smooth and satisfy the PDE in a strong sense.

We prove that for a noetherian semilocal ring R with exactly k isomorphism classes of simple right modules the monoid V *( R ) of isomorphism classes of countably generated projective right (left) modules, viewed as a submonoid of V *( R / J ( R )), is isomorphic to the monoid of solutions in (ℕ 0 ∪ {∞}) k of a system consisting of congruences and diophantine linear equations. The converse also holds, that is, if M is a submonoid of (ℕ 0 ∪ {∞}) k containing an order unit ( n 1 , . . . , n k ) of which is the set of solutions of a system of congruences and linear diophantine equations then it can be realized as V *( R ) for a noetherian semilocal ring such that R / J ( R ) ≅ M n 1 ( D 1 ) × ⋯ × M n k ( D k ) for suitable division rings D 1 , . . . , D k .

We discuss the Capitulation Problem for Néron-Raynaud class groups of tori over global fields F and obtain generalizations of the main results of [González-Avilés, J. reine angew. Math. 613: 75–97, 2007]. We also show that short exact sequences of F -tori induce long exact sequences involving the corresponding Néron-Raynaud class groups. For example, we show that the Néron-Raynaud class group of any F -torus which is split by a metacyclic extension of F can be “resolved” in terms of classical ideal class groups of global fields.

This paper extends the comparison results for cohomological support varieties for Chevalley groups of the form G (𝔽 p ) established by J. Carlson, Z. Lin, and D. Nakano to Chevalley groups of the form G (𝔽 p d ). Other topics considered are complexity and non-maximal support varieties for modules. Techniques developed by J. Pevtsova and the author, together with the use of Weil restriction functors, are employed.

We give a geometric definition of smooth toric Deligne-Mumford stacks using the action of a “torus”. We show that our definition is equivalent to the one of Borisov, Chen and Smith in terms of stacky fans. In particular, we give a geometric interpretation of the combinatorial data contained in a stacky fan. We also give a bottom up classification in terms of simplicial toric varieties and fiber products of root stacks.