Volume 2015, Issue 706

Two Dehn surgeries on a knot are called purely cosmetic, if they yield manifolds that are homeomorphic as oriented manifolds. Suppose there exist purely cosmetic surgeries on a knot in S 3 , we show that the two surgery slopes must be the opposite of each other. One ingredient of our proof is a Dehn surgery formula for correction terms in Heegaard Floer homology.

Let K be a number field, let f:ℙ1→ℙ1$f: {\mathbb {P}}_1 \rightarrow {\mathbb {P}}_1$ be a nonconstant rational map of degree greater than 1, let S be a finite set of places of K , and suppose that u,w∈ℙ1(K)$u, w \in {\mathbb {P}}_1(K)$ are not preperiodic under f . We prove that the set of (m,n)∈ℕ2$(m,n) \in \mathbb {N}^2$ such that f∘m(u)$f^{\circ m}(u)$ is S -integral relative to f∘n(w)$f^{\circ n}(w)$ is finite and effectively computable. This may be thought of as a two-parameter analog of a result of Silverman on integral points in orbits of rational maps. This issue can be translated in terms of integral points on an open subset of ℙ12${\mathbb {P}}_1^2$; then one can apply a modern version of the method of Runge, after increasing the number of components at infinity by iterating the rational map. Alternatively, an ineffective result comes from a well-known theorem of Vojta.

We construct the moduli space of Enriques surfaces in positive characteristic and eventually over the integers, and determine its local and global structure. As an application, we show lifting of Enriques surfaces to characteristic zero. The key observation is that the canonical double cover of an Enriques surface is birational to the complete intersection of three quadrics in ℙ 5 , even in characteristic 2.

In this paper, we study the amoeba volume of a given k -dimensional generic analytic variety V of the complex algebraic torus (ℂ*)n$(\mathbb {C}^*)^n$. When n≥2k$n\ge 2k$, we show that V is algebraic if and only if the volume of its amoeba is finite. Moreover, in this case, we establish a comparison theorem for the volume of the amoeba and the coamoeba. Examples and applications to the k -dimensional affine linear spaces are also given.

We explore a relationship between the classical representation theory of a complex, semisimple Lie algebra 𝔤 and the resonance varieties ℛ(V,K)⊂V*${\mathcal {R}}(V,K)\subset V^*$ attached to irreducible 𝔤-modules V and submodules K⊂V∧V$K\subset V\wedge V$. In the process, we give a precise roots-and-weights criterion insuring the vanishing of these varieties, or, equivalently, the finite-dimensionality as ℂ-vector spaces of certain modules 𝒲(V,K)$\mathcal {W}(V,K)$ over the symmetric algebra on V . In the case when 𝔤=𝔰𝔩2(ℂ)${\mathfrak {g}}={\mathfrak {sl}}_2(\mathbb {C})$, our approach sheds new light on the modules studied by Weyman and Eisenbud in the context of Green's conjecture on free resolutions of canonical curves. In the case when 𝔤=𝔰𝔩n(ℂ)${\mathfrak {g}}={\mathfrak {sl}}_n(\mathbb {C})$ or 𝔰𝔭2g(ℂ)${\mathfrak {sp}}_{2g}(\mathbb {C})$, our approach yields a unified proof of two vanishing results for the resonance varieties of the (outer) Torelli groups of surface groups, results which arose in recent work by Dimca, Hain, and the authors on homological finiteness in the Johnson filtration of mapping class groups and automorphism groups of free groups.

We give a new notion of angle in general metric spaces; more precisely, given three points p , x , q in a metric space ( X ,d), we introduce the notion of angle cone ∠𝐩𝐱𝐪${{\angle _{\mathbf {pxq}}}}$ as being an interval ∠𝐩𝐱𝐪:=[∠pxq-,∠pxq+]${{\angle _{\mathbf {pxq}}}:=[\angle ^-_{pxq},\angle ^+_{pxq}]}$, where the quantities ∠pxq±${\angle ^\pm _{pxq}}$ are defined in terms of the distance functions from p and q via a duality construction of differentials and gradients holding for locally Lipschitz functions on a general metric space. Our definition in the Euclidean plane gives the standard angle between three points and in a Riemannian manifold coincides with the usual angle between the geodesics if x is not in the cut locus of p or q . We show that in general the angle cone is not single-valued (even in case the metric space is a smooth Riemannian manifold, if x is in the cut locus of p or q ), but if we endow the metric space with a positive Borel measure 𝔪 obtaining the metric measure space ( X ,d,𝔪), then under quite general assumptions (which include many fundamental examples as Riemannian manifolds, finite dimensional Alexandrov spaces with curvature bounded from below, Gromov–Hausdorff limits of Riemannian manifolds with Ricci curvature bounded from below, and normed spaces with strictly convex norm), fixed p , q ∈ X , the angle cone at x is single-valued for 𝔪-a.e. x ∈ X . We prove some basic properties of the angle cone (such as the invariance under homotheties of the space) and we analyze in detail the case ( X ,d,𝔪) is a measured-Gromov–Hausdorff limit of a sequence of Riemannian manifolds with Ricci curvature bounded from below, showing the consistency of our definition with a recent construction of Honda [“A weakly second order differential structure on rectifiable metric measure spaces”, preprint 2012].

In the following article we study the limiting properties of the Yang–Mills flow associated to a holomorphic vector bundle E over an arbitrary compact Kähler manifold ( X ,ω). In particular we show that the flow is determined at infinity by the holomorphic structure of E . Namely, if we fix an integrable unitary reference connection A 0 defining the holomorphic structure, then the Yang–Mills flow with initial condition A 0 converges (away from an appropriately defined singular set) in the sense of the Uhlenbeck compactness theorem to a holomorphic vector bundle E ∞ , which is isomorphic to the associated graded object of the Harder–Narasimhan–Seshadri filtration of ( E , A 0 ). Moreover, E ∞ extends as a reflexive sheaf over the singular set as the double dual of the associated graded object. This is an extension of previous work in the cases of one and two complex dimensions and proves the general case of a conjecture of Bando and Siu.

Let G be a reductive complex Lie group acting holomorphically on normal Stein spaces X and Y , which are locally G -biholomorphic over a common categorical quotient Q . When is there a global G -biholomorphism X → Y ? If the actions of G on X and Y are what we, with justification, call generic, we prove that the obstruction to solving this local-to-global problem is topological and provide sufficient conditions for it to vanish. Our main tool is the equivariant version of Grauert's Oka principle due to Heinzner and Kutzschebauch. We prove that X and Y are G -biholomorphic if X is K -contractible, where K is a maximal compact subgroup of G , or if X and Y are smooth and there is a G -diffeomorphism ψ : X → Y over Q , which is holomorphic when restricted to each fibre of the quotient map X → Q . We prove a similar theorem when ψ is only a G -homeomorphism, but with an assumption about its action on G -finite functions. When G is abelian, we obtain stronger theorems. Our results can be interpreted as instances of the Oka principle for sections of the sheaf of G -biholomorphisms from X to Y over Q . This sheaf can be badly singular, even for a low-dimensional representation of SL 2 (ℂ). Our work is in part motivated by the linearisation problem for actions on ℂ n . It follows from one of our main results that a holomorphic G -action on ℂ n , which is locally G -biholomorphic over a common quotient to a generic linear action, is linearisable.

We introduce and give numerical characterizations of two notions of rigidity for a class of regular singular q -difference equations. A special attention is devoted to the generalized q -hypergeometric equations: we show their rigidity and we proceed with a detailed “monodromic” study of these equations.