Real analytic elliptic surfaces

Here, we give an upper bound for the number of connected components of the real locus of several smooth complex compact elliptic surfaces defined over R in terms of the type of the singular fibers of their elliptic fibration.     

Real analytic convex surfaces with positive topological entropy and rigid body dynamics

Following Knieper and Weiss [9] we exhibit explicit real analytic metrics onS ² andR P ² with positive curvature and positive topological entropy using the dynamics of the rigid body. Supported by the Max-Planck-Institut für Mathematik and by a travel grant from CDE.     

Logarithmic foliations on compact algebraic surfaces

LetF be a holomorphic singular foliation with an invariant compact curveS on a compact algebraic surfaceX. In this work we prove thatF is logarithmic under some generic conditions in the singularities ofF inS.Supported by CNPq (Brazil).     

Multiple supersingular elliptic fibers on elliptic surfaces

Elliptic surfaces over an algebraically closed field in characteristic p>0 with multiple supersingular elliptic fibers, that is, multiple fibers of a supersingular elliptic curve, are investigated. In particular, it is shown that for an elliptic surface with q=g+1 and a supersingular elliptic curve as a general fiber, where q is the dimension of an Albanese variety of the surface and g is the genus of the base curve, the multiplicities of the multiple supersingular elliptic fibers are not divisible by p². As an application of this result, the structure of false hyperelliptic surfaces is discussed on this basis.     

Semilinear elliptic equations admitting similarity transformations

In this paper we study the equation −Δu+ρ^−(α+2)h(ρ^αu)=0$-\mathrm{\Delta }u+{\rho }^{-(\alpha +2)}h({\rho }^{\alpha }u)=0$ in a smooth bounded domain Ω   where ρ(x)=dist(x,∂Ω)$\rho (x)=\mathrm{dist}(x,\partial \Omega )$, α>0$\alpha >0$ and h is a nondecreasing function which satisfies Keller–Osserman condition. We introduce a condition on h which implies that the equation is subcritical, i.e., the corresponding boundary value problem is well posed with respect to data given by finite measures. Under additional assumptions on h we show that this condition is necessary as well as sufficient. We also discuss b.v. problems with data given by positive unbounded measures. Our results extend results of [13] treating equations of the form −Δu+ρ^βu^q=0$-\mathrm{\Delta }u+{\rho }^{\beta }{u}^q=0$ with q>1$q>1$, β>−2$\beta >-2$.     

Logarithmic transformations for discrete-time,finite-horizon stochastic control problems

Using logarithmic transformations, we construct discrete-time stochastic control problems where the optimal value function (cost-to-go) belongs to a same parametrized class of functions that remains invariant under the dynamic programming operator. This extends a well-known property of the classical LQG problems, where the optimal value function is a quadratic. Some related questions are also discussed.     

Logarithmic moduli spaces for surfaces of class VII

In this paper we describe logarithmic moduli spaces of pairs (S, D) consisting of a minimal surface S of class VII with second Betti number b ₂ > 0 together with a reduced maximal divisor D of b ₂ rational curves. The special case of Enoki surfaces has already been considered by Dloussky and Kohler. We use normal forms for the action of the fundamental group of S\D and for the associated holomorphic contraction . Part of this work was done while the first author visited the University of Osnabrück under the program “Globale Methoden in der komplexen Geometrie” of the DFG and while the second author visited the Max-Planck-Institut für Mathematik in Bonn and the LATP, Université de Provence. We thank these institutions for their hospitality and for financial support. Furthermore the authors wish to thank Georges Dloussky for numerous discussions on surfaces of class VII.     

On semianalytic subsets of rigid analytic varieties

We show some topological properties of semianalytic subsets of rigid analytic varieties: curve selection lemma, the closure of a semianalytic subsetS is semianalytic, for every quasi-compact morphismf. As an application we show that a morphismf: X Y of rigid analytic varieties is open at a pointx X if and only if SpecO _X,x SpecO _Y,f(x) is surjective.     

Torsion sections of elliptic surfaces

Given a torsion section of a semistable elliptic surface we prove equidistribution results for the components of singular fibers which are hit by the section and for the root of unity (identifying the zero component withC) which is hit by the section in case the section hits the zero component Research supported in part by the NSF under grant DMS 9104058 and the NSA under grant MDA 904-92 H 3022     

Linear systems on rational elliptic surfaces and elliptic fibrations on K3 surfaces

We consider K3 surfaces which are double covers of rational elliptic surfaces. The former are endowed with a natural elliptic fibration, which is induced by the latter. There are also other elliptic fibrations on such K3 surfaces, which are necessarily induced by special linear systems on the rational elliptic surfaces. We describe these linear systems. In particular, we observe that every conic bundle on the rational surface induces a genus 1 fibration on the K3 surface and we classify the singular fibers of the genus 1 fibration on the K3 surface it terms of singular fibers and special curves on the conic bundle on the rational surface.     

A rigid analytic version of M. Artin's theorem on analytic equations

Mathematische Annalen -     

Enriques surfaces and Jacobian elliptic K3 surfaces

This paper proposes a new geometric construction of Enriques surfaces. Its starting point are K3 surfaces with Jacobian elliptic fibration which arise from rational elliptic surfaces by a quadratic base change. The Enriques surfaces obtained in this way are characterised by elliptic fibrations with a rational curve as bisection which splits into two sections on the covering K3 surface. The construction has applications to the study of Enriques surfaces with specific automorphisms. It also allows us to answer a question of Beauville about Enriques surfaces whose Brauer groups show an exceptional behaviour. In a forthcoming paper, we will study arithmetic consequences of our construction.     

KAM-tori near an analytic elliptic fixed point

We study the accumulation of an elliptic fixed point of a real analytic Hamiltonian by quasi-periodic invariant tori. We show that a fixed point with Diophantine frequency vector ω ₀ is always accumulated by invariant complex analytic KAM-tori. Indeed, the following alternative holds: If the Birkhoff normal form of the Hamiltonian at the invariant point satisfies a Rüssmann transversality condition, the fixed point is accumulated by real analytic KAM-tori which cover positive Lebesgue measure in the phase space (in this part it suffices to assume that ω ₀ has rationally independent coordinates). If the Birkhoff normal form is degenerate, there exists an analytic subvariety of complex dimension at least d + 1 passing through 0 that is foliated by complex analytic KAM-tori with frequency ω ₀. This is an extension of previous results obtained in [1] to the case of an elliptic fixed point.