Normal forms of real matrices with respect to complex similarity

This paper gives a complete classification of real linear transformations between two complex vector spaces in terms of matrices.     

Complex hyperbolic Fenchel-Nielsen coordinates

Let Σ be a closed, orientable surface of genus g. It is known that the representation variety of π₁(Σ) has 2g−3 components of (real) dimension 16g−16 and two components of dimension 8g−6. Of special interest are the totally loxodromic, faithful (that is quasi-Fuchsian) representations. In this paper we give global real analytic coordinates on a subset of the representation variety that contains the quasi-Fuchsian representations. These coordinates are a natural generalisation of Fenchel-Nielsen coordinates on the Teichmüller space of Σ and complex Fenchel-Nielsen coordinates on the (classical) quasi-Fuchsian space of Σ.     

Local metric properties of semi-quasihomogeneous real surfaces

We show that a generic semi-quasihomogeneous surface germ in R³ with isolated singularity is bi-Lipschitz homeomorphic, with respect to the inner metric, to its quasihomogeneous approximation.     

Zur Klassifikation glatter kompakter ℂ*-Flächen

Ohne Zusammenfassung

---

     

Pointlike sets with respect to R and J

We present an algorithm to compute the pointlike subsets of a finite semigroup with respect to the pseudovariety of all finite R-trivial semigroups. The algorithm is inspired by Henckell’s algorithm for computing the pointlike subsets with respect to the pseudovariety of all finite aperiodic semigroups. We also give an algorithm to compute -pointlike sets, where denotes the pseudovariety of all finite J-trivial semigroups. We finally show that, in contrast with the situation for , the natural adaptation of Henckell’s algorithm to computes pointlike sets, but not all of them.     

Monodromy eigenvalues and zeta functions with differential forms

For a complex polynomial or analytic function f, there is a strong correspondence between poles of the so-called local zeta functions or complex powers ∫|f|^2sω, where the ω are C^∞ differential forms with compact support, and eigenvalues of the local monodromy of f. In particular Barlet showed that each monodromy eigenvalue of f is of the form , where s₀ is such a pole. We prove an analogous result for similar p-adic complex powers, called Igusa (local) zeta functions, but mainly for the related algebro-geometric topological and motivic zeta functions.     

Tube domain and an orbit of a complex triangular group

Let w be a complex symmetric matrix of order r, and Δ ₁(w), . . . , Δ _r (w) the principal minors of w. If w belongs to the Siegel right half-space, then it is known that Re (Δ _k (w)/Δ _k-1(w)) > 0 for k = 1, . . . , r. In this paper we study this property in three directions. First we show that this holds for general symmetric right half-spaces. Second we present a series of non-symmetric right half-spaces with this property. We note that case-by-case verifications up to dimension 10 tell us that there is only one such irreducible non-symmetric tube domain. The proof of the property reduces to two lemmas. One is entirely generalized to non-symmetric cases as we prove in this paper. This is the third direction. As a byproduct of our study, we show that the basic relative invariants associated to a homogeneous regular open convex cone Ω studied earlier by the first author are characterized as the irreducible factors of the determinant of right multiplication operators in the complexification of the clan associated to Ω.     

Separators of points on algebraic surfaces

For a finite set of points X⊆Pⁿ and for a given point P∈X, the notion of a separator of P in X (a hypersurface containing all the points in X except P) and of the degree of P in X, (the minimum degree of these separators) has been largely studied. In this paper we extend these notions to a set of points X on a projectively normal surface S⊆Pⁿ, considering as separators arithmetically Cohen-Macaulay curves and generalizing the case S=P² in a natural way. We denote the minimum degree of such curves as and we study its relation to . We prove that if S is a variety of minimal degree these two terms are explicitly related by a formula, whereas only an inequality holds for other kinds of surfaces.     

Lichnerowicz-Obata theorem for Kohn Laplacian on the real ellipsoid

We give the sharp lower bound for Ricci curvature on the real ellipsoid in Cⁿ⁺¹, and prove the Lichnerowicz-Obata theorem for Kohn Laplacian.     

Upper semicontinuity of automorphism groups

The second author was partially supported by NSF#DMS-9101826