Volume 2015, Issue 698

We give conditions for the Mayer–Vietoris property to hold for the algebraic K -theory of blow-up squares of toric varieties and schemes, using the theory of monoid schemes. These conditions are used to relate algebraic K -theory to topological cyclic homology in characteristic p . To achieve our goals, we develop many notions for monoid schemes based on classical algebraic geometry, such as separated and proper maps and resolution of singularities.

For a smooth and proper variety X over a local field K with residue characteristic p the reciprocity map ρ X : SK 1 ( X ) → π 1 ab ( X ) is a well-defined map from its class group to the corresponding abelianized étale fundamental group of X . In this paper we show that its kernel is the direct sum of a finite group and a group which is ℓ-divisible for all primes ℓ ≠ p . This result generalizes the work of Shuji Saito and Uwe Jannsen for curves and surfaces to arbitrary dimensions.

We introduce the tropicalization of closed sub-schemes of a torus defined over a higher dimensional local field. We study the basic invariants of such tropicalizations. This is a generalization of the fundamental theorem of tropical geometry to higher dimensional local fields.

We give a direct proof that the Mazur–Tate and Coleman–Gross heights on elliptic curves coincide. The main ingredient is to extend the Coleman–Gross height to the case of divisors with non-disjoint support and, doing some p -adic analysis, show that, in particular, its component above p gives, in the special case of an ordinary elliptic curve, the p -adic sigma function. We use this result to give a short proof of a theorem of Kim characterizing integral points on elliptic curves in some cases under weaker assumptions. As a further application, we give new formulas to compute double Coleman integrals from tangential basepoints.

The main conjecture of non-commutative Iwasawa theory is shown to be equivalent, modulo an appropriate torsion hypothesis, to a family of main conjectures over cyclotomic ℤ p -extensions together with suitable compatibility relations between the associated abelian p -adic L -functions. This result combines with techniques of Ritter and Weiss and of Kakde to give a concrete strategy for deducing main conjectures for (non-abelian) compact p -adic Lie extensions of arbitrarily large rank from main conjectures in the classical sense. As a first application of this approach we combine it with a recent result of Ritter and Weiss to prove, modulo an appropriate vanishing hypothesis on μ-invariants, the main conjecture of non-commutative Iwasawa theory for Tate motives over compact p -adic Lie extensions of totally real number fields of arbitrarily large rank. We then also deduce several interesting consequences of this result concerning a range of related conjectures including special cases of the equivariant Tamagawa number conjecture.

We derive an Ahlfors–Weill type extension for a class of holomorphic mappings defined in the ball 𝔹 n , generalizing the formula for Nehari mappings in the disk. The class of mappings holomorphic in the ball is defined in terms of the Schwarzian operator. Convexity relative to the Bergman metric plays an essential role, as well as the concept of a weakly linearly convex domain. The extension outside the ball takes values in the projective dual to ℂ n , that is, in the set of complex hyperplanes.

In this paper we investigate Moishezon twistor spaces which have a structure of double covering over a very simple rational threefold. These spaces can be regarded as a direct generalization of the twistor spaces studied in [J. Differential Geom. 36 (1992), 451–491] and [Compos. Math. 82 (1992), 25–55] to the case of arbitrary signature. In particular, the branch divisor of the double covering is a cut of the rational threefold by a single quartic hypersurface. We determine a defining equation of the hypersurface in an explicit form. We also show that these twistor spaces interpolate LeBrun twistor spaces and the twistor spaces constructed in [J. Differential Geom. 82 (2009), 411–444].

We study some symplectic geometric aspects of rationally connected 4-folds. As a corollary, we prove that any smooth projective 4-fold symplectic deformation equivalent to a Fano 4-fold of pseudo-index at least 2 or a rationally connected 4-fold whose second Betti number is 2 is rationally connected.