Volume 2009, Issue 637

There is a forgetful map from the mapping class group of a punctured surface to that of the surface with one fewer puncture. We prove that finitely generated purely pseudo-Anosov subgroups of the kernel of this map are convex cocompact in the sense of B. Farb and L. Mosher. In particular, we obtain an affirmative answer to their question of local convex cocompactness of K. Whittlesey's group. In the course of the proof, we obtain a new proof of a theorem of I. Kra. We also relate the action of this kernel on the curve complex to a family of actions on trees. This quickly yields a new proof of a theorem of J. Harer.

Performing local extension from pseudoconcave boundaries along Levi-Hartogs figures and building a Morse-theoretical frame for the global control of monodromy, we establish a version of the Hartogs extension theorem which is valid in singular complex spaces (and currently not available by means of techniques), namely: For every domain Ω of an ( n – 1)- complete normal complex space of pure dimension n ≧ 2, and for every compact set K ⊂ Ω such that Ω\ K is connected, holomorphic or meromorphic functions in Ω\ K extend holomorphically or meromorphically to Ω. Assuming that X is reduced and globally irreducible, but not necessarily normal, and that the regular part [Ω\ K ] reg is connected, we also show that meromorphic functions on Ω\ K extend meromorphically to Ω.

Let X be a connected normal complex space of dimension n ≧ 2 which is cohomologically ( n – 1)-complete, and let π : M → X be a resolution of singularities. By use of Takegoshi's generalization of the Grauert-Riemenschneider vanishing theorem, we deduce , which in turn implies Hartogs' extension theorem on X by the -technique of Ehrenpreis.

It is proved that every elementary amenable group of type FP ∞ admits a cocompact classifying space for proper actions.

Let 𝔬 be a complete DVR of fraction field K and algebraically closed residue field k . Let A be an 𝔬-adic domain which is smooth and topologically of finite type. Let 𝒟 be the ring of 𝔬-linear differential operators over A and let ℳ be a 𝒟-module which is finitely generated as A -module. Given an 𝔬-point of Spf( A ) we construct using a Tannakian theory of Bruguières-Nori, a faithfully flat 𝔬-group-scheme Π which is analogous—in the sense that its category of dualizable representations is equivalent to a category of 𝒟-modules—to the Tannakian group-scheme (the differential Galois or monodromy group) associated to a 𝒟-module over a field. We show that the differential Galois group G of the reduced 𝒟-module ℳ ⊗ k is a closed subgroup of Π ⊗ k , which coincides with (Π ⊗ k ) red when Π is finite, and gives back, in any case, the differential Galois group of ℳ ⊗ K upon tensorisation with K .

Finding appropriate notions of discrete holomorphic maps and, more generally, conformal immersions of discrete Riemann surfaces into 3-space is an important problem of discrete differential geometry and computer visualization. We propose an approach to discrete conformality that is based on the concept of holomorphic line bundles over “discrete surfaces”, by which we mean the vertex sets of triangulated surfaces with bi-colored set of faces. The resulting theory of discrete conformality is simultaneously Möbius invariant and based on linear equations. In the special case of maps into the 2-sphere we obtain a reinterpretation of the theory of complex holomorphic functions on discrete surfaces introduced by Dynnikov and Novikov. As an application of our theory we introduce a Darboux transformation for discrete surfaces in the conformal 4-sphere. This Darboux transformation can be interpreted as the space- and time-discrete Davey–Stewartson flow of Konopelchenko and Schief. For a generic map of a discrete torus with regular combinatorics, the space of all Darboux transforms has the structure of a compact Riemann surface, the spectral curve. This makes contact to the theory of algebraically completely integrable systems and is the starting point of a soliton theory for triangulated tori in 3- and 4-space devoid of special assumptions on the geometry of the surface.

We introduce the notion of biextensions of 1-motives over an arbitrary scheme S and we define bilinear morphisms between 1-motives as isomorphism classes of such biextensions. If S is the spectrum of a field of characteristic 0, we check that these biextensions define bilinear morphisms between the realizations of 1-motives. Generalizing we obtain the notion of multilinear morphisms between 1-motives.

We give a short and direct proof of Getzler and Pandharipande's λ g -conjecture. The approach is through the Ekedahl-Lando-Shapiro-Vainshtein theorem, which establishes the “polynomiality” of Hurwitz numbers, from which we pick off the lowest degree terms. The proof is independent of Gromov-Witten theory. We briefly describe the philosophy behind our general approach to intersection numbers and how it may be extended to other intersection number conjectures. Ideas from this paper feature in two independent recent enlightening proofs of Witten's conjecture by Kazarian [Adv. Math.] and Chen, Li, and Liu [Asian J. Math. 12: 511–518, 2009].

We show that deformations of a surjective morphism onto a Fano manifold of Picard number 1 are unobstructed and rigid modulo the automorphisms of the target, if the variety of minimal rational tangents of the Fano manifold is non-linear or finite. The condition on the variety of minimal rational tangents holds for practically all known examples of Fano manifolds of Picard number 1, except the projective space. When the variety of minimal rational tangents is non-linear, the proof is based on an earlier result of N. Mok and the author on the birationality of the tangent map. When the varieties of minimal rational tangents of the Fano manifold is finite, the key idea is to factorize the given surjective morphism, after some transformation, through a universal morphism associated to the minimal rational curves.

We give an effective upper bound on the h *-polynomial of a lattice polytope in terms of its degree and leading coefficient, confirming a conjecture of Batyrev. We deduce this bound as a consequence of a strong Cayley decomposition theorem which says, roughly speaking, that any lattice polytope with a large multiple that has no interior lattice points has a nontrivial decomposition as a Cayley sum of polytopes of smaller dimension. Polytopes with nontrivial Cayley decompositions correspond to projectivized sums of toric line bundles, and our approach is partially inspired by classification results of Fujita and others in algebraic geometry. In an appendix, we interpret our Cayley decomposition theorem in terms of adjunction theory for toric varieties.

In this paper we investigate some questions related with Siegel's trace problem in abelian fields, and present an application of this type of results to character values of finite groups.