The symplectomorphism group of a blow up

We study the relation between the symplectomorphism group Symp M of a closed connected symplectic manifold M and the symplectomorphism and diffeomorphism groups Symp and Diff of its one point blow up . There are three main arguments. The first shows that for any oriented M the natural map from to is often injective. The second argument applies when M is simply connected and detects nontrivial elements in the homotopy group that persist into the space of self-homotopy equivalences of . Since it uses purely homological arguments, it applies to c-symplectic manifolds (M, a), that is, to manifolds of dimension 2n that support a class such that . The third argument uses the symplectic structure on M and detects nontrivial elements in the (higher) homology of BSymp, M using characteristic classes defined by parametric Gromov–Witten invariants. Some results about many point blow ups are also obtained. For example we show that if M is the four-torus with k-fold blow up (where k > 0) then is not generated by the groups as ranges over the set of all symplectic forms on . Partially supported by NSF grants DMS 0305939 and 0604769.     

Regular quantizations and covering maps

Let $\tilde{M} \rightarrow MLet be a holomorphic (unbranched) covering map between two compact complex manifolds, with . We prove that if and M both admit regular K?hler forms and ω respectively then, up to homotheties, and (M, ω) are biholomorphically isometric. This work was supported by the M.I.U.R. Project “Geometric Properties of Real and Complex Manifolds”.     

On the Limit Set of Discrete Subgroups of PU(2,1)

Let G be a discrete subgroup of PU(2,1); G acts on preserving the unit ball , equipped with the Bergman metric. Let be the limit set of G in the sense of Chen–Greenberg, and let be the limit set of the G-action on in the sense of Kulkarni. We prove that L(G) = Λ(G) ∩ S ³ and Λ(G) is the union of all complex projective lines in which are tangent to S ³ at a point in L(G).     

A local rigidity theorem for minimal surfaces in Minkowski 3-space of Randers type

Let be a Minkowski 3-space of Randers type with , where is the Euclidean metric and . We consider minimal surfaces in and prove that if a connected surface M in is minimal with respect to both the Busemann–Hausdorff volume form and the Holmes–Thompson volume form, then up to a parallel translation of , M is either a piece of plane or a piece of helicoid which is generated by lines screwing about the x ³-axis.      

The Action of the Mapping Class Group on Maximal Representations

Let Γ _g be the fundamental group of a closed oriented Riemann surface Σ _g , g ≥ 2, and let G be a simple Lie group of Hermitian type. The Toledo invariant defines the subset of maximal representations Rep_max(Γ _g , G) in the representation variety Rep(Γ _g , G). Rep_max(Γ _g , G) is a union of connected components with similar properties as Teichmüller space . We prove that the mapping class group acts properly on Rep_max(Γ _g , G) when , SU(n,n), SO*(4n), Spin(2,n).     

Surfaces with many solitary points

It is classically known that a real cubic surface in cannot have more than one solitary point (or -singularity, locally given by x ² + y ² + z ² = 0) whereas it can have up to four nodes (or -singularity, locally given by x ² + y ² − z ² = 0). We show that on any surface of degree d ≥ 3 in the maximum possible number of solitary points is strictly smaller than the maximum possible number of nodes. Conversely, we adapt a construction of Chmutov to obtain surfaces with many solitary points by using a refined version of Brusotti’s Theorem. Combining lower and upper bounds, we deduce: , where denotes the maximum possible number of solitary points on a real surface of degree d in . Finally, we adapt this construction to get real algebraic surfaces in with many singular points of type for all k ≥ 1.      

Rigidity of higher rank abelian cocycles with values in diffeomorphism groups

We consider cocycles over certain hyperbolic actions, , and show rigidity properties for cocycles with values in a Lie group or a diffeomorphism group, which are close to identity on a set of generators, and are sufficiently smooth. The actions we consider are Cartan actions of or , for , and Γ torsion free cocompact lattice. The results in this paper rely on a technique developed recently by D. Damjanović and A. Katok.      

A speciality theorem for curves in P<Superscript>5</Superscript>

Let be an integral projective curve. One defines the speciality index e(C) of C as the maximal integer t such that , where ω _C denotes the dualizing sheaf of . Extending a classical result of Halphen concerning the speciality of a space curve, in the present paper we prove that if is an integral degree d curve not contained in any surface of degree  < s, in any threefold of degree  < t, and in any fourfold of degree  < u, and if , then Moreover equality holds if and only if C is a complete intersection of hypersurfaces of degrees u, , and . We give also some partial results in the general case , .      

Symplectic structure and reduction on the space of Riemannian metrics

The space of Riemannian metrics ${\mathfrak{Met}}MThe space of Riemannian metrics on an oriented compact manifold M of dimension n = 4k − 2 is endowed with a canonical presymplectic structure and a moment map [cf. Ferreiro Pérez and Mu?oz Masqué, Preprint (arXiv: math.DG/0507075)]. The fiber is characterized as the space of solutions to a differential equation. In dimension 2, the symplectic reduction of is analyzed and the construction presented here is compared with that introduced in Donaldson (Fields Medallists’ Lectures, 1997) and Fujiki (Sugaku Expositions 5(2):173–191, 1992). Finally, conformally flat metrics and, for n = 6, K?hler metrics of constant holomorphic sectional curvature are shown to be contained in .      

A Generalization of the prime geodesic theorem to counting conjugacy classes of free subgroups

The classical prime geodesic theorem (PGT) gives an asymptotic formula (as x tends to infinity) for the number of closed geodesics with length at most x on a hyperbolic manifold M. Closed geodesics correspond to conjugacy classes of π₁(M) = Γ where Γ is a lattice in G = SO(n,1). The theorem can be rephrased in the following format. Let be the space of representations of into Γ modulo conjugation by Γ. is defined similarly. Let be the projection map. The PGT provides a volume form vol on such that for sequences of subsets {B _t }, satisfying certain explicit hypotheses, |π⁻¹(B _t )| is asymptotic to vol(B _t ). We prove a statement having a similar format in which is replaced by a free group of finite rank under the additional hypothesis that n = 2 or 3.      

Non-free isometric immersions of Riemannian manifolds

Let (V, g) be a Riemannian manifold and let be the isometric immersion operator which, to a map , associates the induced metric on V, where denotes the Euclidean scalar product in . By Nash–Gromov implicit function theorem is infinitesimally invertible over the space of free maps. In this paper we study non-free isometric immersions . We show that the operator (where denotes the space of C ^∞- smooth quadratic forms on ) is infinitesimally invertible over a non-empty open subset of and therefore is an open map in the respective fine topologies.      

The Poisson boundary of lamplighter random walks on trees

Let be the homogeneous tree with degree q + 1 ≥ 3 and a finitely generated group whose Cayley graph is . The associated lamplighter group is the wreath product , where is a finite group. For a large class of random walks on this group, we prove almost sure convergence to a natural geometric boundary. If the probability law governing the random walk has finite first moment, then the probability space formed by this geometric boundary together with the limit distribution of the random walk is proved to be maximal, that is, the Poisson boundary. We also prove that the Dirichlet problem at infinity is solvable for continuous functions on the active part of the boundary, if the lamplighter “operates at bounded range”. Supported by ESF program RDSES and by Austrian Science Fund (FWF) P15577.     

Arboreal Galois representations

Let be the absolute Galois group of , and let T be the complete rooted d-ary tree, where d ≥ 2. In this article, we study “arboreal” representations of into the automorphism group of T, particularly in the case d = 2. In doing so, we propose a parallel to the well-developed and powerful theory of linear p-adic representations of . We first give some methods of constructing arboreal representations and discuss a few results of other authors concerning their size in certain special cases. We then discuss the analogy between arboreal and linear representations of . Finally, we present some new examples and conjectures, particularly relating to the question of which subgroups of Aut(T) can occur as the image of an arboreal representation of .      

Projective curves of degree=codimension+2

In this article we study nondegenerate projective curves of degree d which are not arithmetically Cohen-Macaulay. Note that for a rational normal curve and a point . Our main result is about the relation between the geometric properties of X and the position of P with respect to . We show that the graded Betti numbers of X are uniquely determined by the rank of P with respect to . In particular, X satisfies property N _2,p if and only if . Therefore property N _2,p of X is controlled by and conversely can be read off from the minimal free resolution of X. This result provides a non-linearly normal example for which the converse to Theorem 1.1 in (Eisenbud et al., Compositio Math 141:1460–1478, 2005) holds. Also our result implies that for nondegenerate projective curves of degree d which are not arithmetically Cohen–Macaulay, there are exactly distinct Betti tables.     

Diameter preserving bijections between Grassmann spaces over Bezout domains

Let R, S be Bezout domains. Assume that n is an integer ≥ 3, 1 ≤ k ≤ n − 2. Denoted by the k-dimensional Grassmann space on . Let be a map. This paper proves the following are equivalent: (i) is an adjacency preserving bijection in both directions. (ii) is a diameter preserving bijection in both directions. Moreover, Chow’s theorem on Grassmann spaces over division rings is extended to the case of Bezout domains: If is an adjacency preserving bijection in both directions, then is induced by either a collineation or the duality of a collineation. Project 10671026 supported by National Natural Science Foundation of China.     

The Extremal Truncated Moment Problem

For a degree 2n real d-dimensional multisequence to have a representing measure μ, it is necessary for the associated moment matrix to be positive semidefinite and for the algebraic variety associated to β, , to satisfy rank card as well as the following consistency condition: if a polynomial vanishes on , then . We prove that for the extremal case , positivity of and consistency are sufficient for the existence of a (unique, rank -atomic) representing measure. We also show that in the preceding result, consistency cannot always be replaced by recursiveness of . The first-named author’s research was partially supported by NSF Research Grants DMS-0099357 and DMS-0400741. The second-named author’s research was partially supported by NSF Research Grant DMS-0201430 and DMS-0457138.     

Bubble accumulations in an elliptic Neumann problem with critical Sobolev exponent

We consider the following critical elliptic Neumann problem on , Ω; being a smooth bounded domain in is a large number. We show that at a positive nondegenerate local minimum point Q ₀ of the mean curvature (we may assume that Q ₀ = 0 and the unit normal at Q ₀ is − e _N ) for any fixed integer K ≥ 2, there exists a μ _K > 0 such that for μ > μ _K , the above problem has K−bubble solution u _μ concentrating at the same point Q ₀. More precisely, we show that u _μ has K local maximum points Q ₁^μ, ... , Q _K ^μ ∈∂Ω with the property that and approach an optimal configuration of the following functional (*) Find out the optimal configuration that minimizes the following functional: where are two generic constants and φ (Q) = Q ^T G Q with G = (∇ _ij H(Q ₀)). Research supported in part by an Earmarked Grant from RGC of HK.     

Scales for co-compact embeddings of virtually free groups

Let Γ be a group which is virtually free of rank at least 2 and let be the family of totally disconnected, locally compact groups containing Γ as a co-compact lattice. We prove that the values of the scale function with respect to groups in evaluated on the subset Γ have only finitely many prime divisors. This can be thought of as a uniform property of the family .      

The geometric invariants $${\Omega^{n}}$$ of a product of groups

We compute the geometric invariants of a product G × H of groups in terms of and . This gives a sufficient condition in terms of and for a normal subgroup of G × H with abelian quotient to be of type F _n . We give an example involving the direct product of the Baumslag–Solitar group BS_1,2 with itself.