Authors:Javier Fern�ndez de Bobadilla Pages: 659 - 688 Abstract: Abstract We introduce the concept of topological finite-determinacy for germs of analytic functions within a fixed ideal I, which provides a notion of topological finite-determinacy of functions with non-isolated singularities. We prove the following statement which generalizes classical results of Thom and Varchenko: let A be the complement in the ideal I of the space of germs whose topological type remains unchanged under a deformation within the ideal that only modifies sufficiently large order terms of the Taylor expansion. Then A has infinite codimension in I in a suitable sense. We also prove the existence of generic topological types of families of germs of I parametrized by an irreducible analytic set. PubDate: 2004-12-01 DOI: 10.1007/s00014-004-0813-1 Issue No:Vol. 79, No. 4 (2004)

Authors:Fabien Morel Pages: 689 - 703 Abstract: Abstract Nous reformulons un résultat récent de Arason et Elman en donnant une présentation très simple des puissances de l’idéal fondamental de l’anneau de Witt d’un corps de caractéristique ≠ 2. PubDate: 2004-12-01 DOI: 10.1007/s00014-004-0815-z Issue No:Vol. 79, No. 4 (2004)

Authors:A. A. Glutsyuk Pages: 704 - 752 Abstract: Abstract We consider a two-dimensional linear foliation on torus of arbitrary dimension. For any smooth family of complex structures on the leaves we prove existence of smooth family of uniformizing (conformal complete flat) metrics on the leaves. We extend this result to linear foliations on \(\mathbb T^2\times\mathbb R\) and families of complex structures with bounded derivatives C 3-close to the standard complex structure. We prove that the analogous statement for arbitrary C ∞ two-dimensional foliation on compact manifold is wrong in general, even for suspensions over \(\mathbb T^2:\) in dimension 3 the uniformizing metric can be nondifferentiable at some points; in dimension 4 the uniformizing metric of each noncompact leaf can be unbounded. PubDate: 2004-12-01 DOI: 10.1007/s00014-004-0818-9 Issue No:Vol. 79, No. 4 (2004)

Authors:Christian Bonatti; Carlos Matheus; Marcelo Viana; Amie Wilkinson Pages: 753 - 757 Abstract: Abstract We consider the set \(\mathcal P\mathcal H_\omega(M)\) of volume preserving partially hyperbolic diffeomorphisms on a compact manifold having 1-dimensional center bundle. We show that the volume measure is ergodic, and even Bernoulli, for any C 2 diffeomorphism in an open and dense subset of \(\mathcal P\mathcal H_\omega(M).\) This solves a conjecture of Pugh and Shub, in this setting. PubDate: 2004-12-01 DOI: 10.1007/s00014-004-0819-8 Issue No:Vol. 79, No. 4 (2004)

Authors:Alejandro Adem; James F. Davis; �zg�n �nl� Pages: 758 - 778 Abstract: Abstract A representation G ⊂ U(n) of degree n has fixity equal to the smallest integer f such that the induced action of G on U(n) /U(n-f-1) is free. Using bundle theory we show that if G admits a representation of fixity one, then it acts freely and smoothly on \(\mathbb S^{2n-1}\times\mathbb S^{4n-5}.\) We use this to prove that a finite p-group (for p > 3) acts freely and smoothly on a product of two spheres if and only if it does not contain (ℤ /p)3 as a subgroup. We use propagation methods from surgery theory to show that a representation of fixity f < n - 1 gives rise to a free action of G on a product of f + 1 spheres provided the order of G is relatively prime to (n - 1)!. We give an infinite collection of new examples of finite p-groups of rank r which act freely on a product of r spheres, hence verifying a strong form of a well-known conjecture for these groups. In addition we show that groups of fixity two act freely on a finite complex with the homotopy type of a product of three spheres. A number of examples are explicitly described. PubDate: 2004-12-01 DOI: 10.1007/s00014-004-0810-4 Issue No:Vol. 79, No. 4 (2004)

Authors:Serge Cantat Pages: 779 - 797 Abstract: Abstract We classify the holomorphic diffeomorphisms of complex projective varieties with an Anosov dynamics and holomorphic stable and unstable foliations: The variety is finitely covered by a compact complex torus and the diffeomorphism corresponds to a linear transformation of this torus. PubDate: 2004-12-01 DOI: 10.1007/s00014-004-0811-3 Issue No:Vol. 79, No. 4 (2004)

Authors:Martin Traizet Pages: 798 - 825 Abstract: Abstract We prove a balancing condition for weak limits of families of embedded minimal surfaces of finite total curvature. We use it to prove compactness theorems for certain families of minimal surfaces. PubDate: 2004-12-01 DOI: 10.1007/s00014-004-0805-1 Issue No:Vol. 79, No. 4 (2004)

Authors:Aldo Conca; J�rgen Herzog; Takayuki Hibi Pages: 826 - 839 Abstract: Abstract The Betti-numbers of a graded ideal I in a polynomial ring and the Betti-numbers of its generic initial ideal Gin(I) are compared. In characteristic zero it is shown that if these Betti-numbers coincide in some homological degree, then they coincide in all higher homological degrees. We also compare the Betti-numbers of componentwise linear ideals which are contained in each other and have the same Hilbert polynomial. PubDate: 2004-12-01 DOI: 10.1007/s00014-004-0812-2 Issue No:Vol. 79, No. 4 (2004)

Authors:Yik-Man Chiang; Walter K. Hayman Abstract: Abstract We give a pointwise estimate of meromorphic solutions of linear differential equations with coefficients meromorphic in a finite disk or in the open plane. Our results improve some earlier estimates of Bank and Laine. In particular we show that the growth of meromorphic solutions with δ(∞)>0 can be estimated in terms of initial conditions of the solution at or near the origin and the characteristic functions of the coefficients. Examples show that the estimates are sharp in a certain sense. Our results give an affirmative answer to a question of Milne Anderson. Our method consists of two steps. In Theorem 2.1 we construct a path Γ(θ0, ρ, t) consisting of the ray \(z=\tau e^{i\theta_0},\quad \rho\le\tau\le t, \) followed by the circle \(z=t e^{i\theta},\quad \theta_0\le \theta\le \theta_0+2\pi, \) on which the coefficients are all bounded in terms of the sum of their characteristic functions on a larger circle. In Theorem 2.2 we show how such an estimate for the coefficients leads to a corresponding bound for the solution on ∣z∣ = t. Putting these two steps together we obtain our main result, Theorem 2.3. PubDate: 2004-09-01 DOI: 10.1007/s00014-003-0792-7 Issue No:Vol. 79, No. 3 (2004)

Authors:C. De Concini; C. Procesi; M. Salvetti Abstract: Abstract In this paper we study the Schwarz genus for the covering of the space of polynomials with distinct roots by its roots. We show that, for the first unknown case (degree 6), the genus is strictly less than the one predicted by dimension arguments, contrary to what happens in all other reflection groups. PubDate: 2004-09-01 DOI: 10.1007/s00014-004-0809-x Issue No:Vol. 79, No. 3 (2004)

Authors:Tomohiro Okuma Abstract: Abstract Let π : X → T be a small deformation of a normal Gorenstein surface singularity X 0 = π-1(0) over the complex number field ℂ. Suppose that T is a neighborhood of the origin of ℂ and that X 0 is not log-canonical. We show that if a topological invariant -P t ⋅ P t of X t = π-1(t) is constant, then, after a suitable finite base change, π admits a simultaneous resolution f : M → X which induces a locally trivial deformation of each maximal string of rational curves at an end of the exceptional set of M 0 → X 0; in particular, if X 0 has a star-shaped resolution graph, then π admits a weak simultaneous resolution (in other words, π is an equisingular deformation). PubDate: 2004-09-01 DOI: 10.1007/s00014-003-0791-8 Issue No:Vol. 79, No. 3 (2004)

Authors:L� D?ng Tr�ng; Meral Tosun Abstract: Abstract A normal surface singularity is rational if and only if the dual intersection graph of a desingularization satisfies some combinatorial properties. In fact, the graphs defined in this way are trees. In this paper we give geometric features of these trees. In particular, we prove that the number of vertices of valency ≥ 3 in the dual intersection tree of the minimal desingularization of a rational singularity of multiplicity m ≥ 3 is at most m - 2. PubDate: 2004-09-01 DOI: 10.1007/s00014-004-0808-y Issue No:Vol. 79, No. 3 (2004)

Authors:Erwan Lanneau Abstract: Abstract Moduli spaces of quadratic differentials with prescribed singularities are not necessarily connected. We describe here all cases when they have a special hyperelliptic connected component. We announce the general classification theorem: up to the four exceptional cases in low dimensional stratum, any stratum of meromorphic quadratic differentials is either connected, or has exactly two connected components. In this last case, one component is hyperelliptic, the other not. PubDate: 2004-09-01 DOI: 10.1007/s00014-004-0806-0 Issue No:Vol. 79, No. 3 (2004)

Authors:Daniel Ruberman; Nikolai Saveliev Abstract: Abstract This is the first in a series of papers exploring the relationship between the Rohlin invariant and gauge theory. We discuss a Casson-type invariant of a 3-manifold Y with the integral homology of the 3-torus, given by counting projectively flat U(2)-connections. We show that its mod 2 evaluation is given by the triple cup product in cohomology, and so it coincides with a certain sum of Rohlin invariants of Y. Our counting argument makes use of a natural action of H 1 (Y;ℤ2) on the moduli space of projectively flat connections; along the way we construct perturbations that are equivariant with respect to this action. Combined with the Floer exact triangle, this gives a purely gauge-theoretic proof that Casson’s homology sphere invariant reduces mod 2 to the Rohlin invariant. PubDate: 2004-09-01 DOI: 10.1007/s00014-004-0816-y Issue No:Vol. 79, No. 3 (2004)

Authors:Kai Cieliebak; Viktor L. Ginzburg; Ely Kerman Abstract: Abstract We show that a small neighborhood of a closed symplectic submanifold in a geometrically bounded aspherical symplectic manifold has non-vanishing symplectic homology. As a consequence, we establish the existence of contractible closed characteristics on any thickening of the boundary of the neighborhood. When applied to twisted geodesic flows on compact symplectically aspherical manifolds, this implies the existence of contractible periodic orbits for a dense set of low energy values. PubDate: 2004-09-01 DOI: 10.1007/s00014-004-0814-0 Issue No:Vol. 79, No. 3 (2004)

Authors:Ko Honda; William H. Kazez; Gordana Matić Pages: 502 - 515 Abstract: Abstract We present a new, completely three-dimensional proof of the fact, due to the combined work of Gabai and Eliashberg-Thurston, that every closed, oriented, connected, irreducible 3-manifold with nonzero second homology carries a universally tight contact structure. PubDate: 2004-09-01 DOI: 10.1007/s00014-004-0804-2 Issue No:Vol. 79, No. 3 (2004)

Authors:A. S. Dzhumadil’daev Pages: 516 - 553 Abstract: Abstract The N-commutator \( s_N(X_1, \ldots, X_N) = \mathop{\sum}\limits_{\sigma\in {\mathfrak{S}}_N}\text{\rm sign}\, \sigma\, X_{\sigma(1)}\cdots X_{\sigma(N)} \) is conjecturally a well-defined nontrivial operation on \(W(n)=Der \ {\mathbb K}[x]\) for x = (x 1, ... , x n ) if and only if N = n 2 + 2n - 2. This is proved for n = 2 and confirmed by computer experiments for n < 5. Under 2- and 5-commutators the algebra of divergence-free vector fields in two dimensions is an sh-Lie (strong homotopic Lie) algebra in the sense of Stasheff. Similarly, W(2) is an sh-Lie algebra with respect to 2- and 6-commutators. PubDate: 2004-09-01 DOI: 10.1007/s00014-004-0807-2 Issue No:Vol. 79, No. 3 (2004)

Authors:Leonid Makar-Limanov; Peter van Rossum; Vladimir Shpilrain; Jie-Tai Yu Pages: 341 - 349 Abstract: Abstract Let K be an arbitrary field of characteristic 0, and $\mathbf{A}^n$ the n-dimensional affine space over K. A well-known cancellation problem asks, given two algebraic varieties $V_1, V_2 \subseteq \mathbf{A}^n$ with isomorphic cylinders $V_1 \times \mathbf{A}^1$ and $V_2 \times \mathbf{A}^1$, whether $V_1$ and $V_2$ themselves are isomorphic. In this paper, we focus on a related problem: given two varieties with equivalent (under an automorphism of $\mathbf{A}^{n+1}$) cylinders $V_1 \times \mathbf{A}^1$ and $V_2 \times \mathbf{A}^1$, are $V_1$ and $V_2$ equivalent under an automorphism of $\mathbf{A}^n$? We call this stable equivalence problem. We show that the answer is positive for any two curves $V_1, V_2 \subseteq \mathbf{A}^2$. For an arbitrary $n \ge 2$, we consider a special, arguably the most important, case of both problems, where one of the varieties is a hyperplane. We show that a positive solution of the stable equivalence problem in this case implies a positive solution of the cancellation problem. PubDate: 2004-04-01 DOI: 10.1007/s00014-003-0796-3 Issue No:Vol. 79, No. 2 (2004)

Authors:Christine Riedtmann; Grzegorz Zwara Pages: 350 - 361 Abstract: Abstract Let d be a prehomogeneous dimension vector for a finite tame quiver Q. We show that the common zeros of all non-constant semi-invariants for the variety of representations of Q with dimension vector $N\cdot\mathbf d$, under the product of the general linear groups at all vertices, is a complete intersection for $N\geq 3$. PubDate: 2004-04-01 DOI: 10.1007/s00014-003-0797-2 Issue No:Vol. 79, No. 2 (2004)