Meyer's signature cocycle and hyperelliptic fibrations

We show that the cohomology class represented by Meyer's signature cocycle is of order in the 2-dimensional cohomology group of the hyperelliptic mapping class group of genus . By using the -cochain cobounding the signature cocycle, we extend the local signature for singular fibers of genus 2 fibrations due to Y. Matsumoto [18] to that for singular fibers of hyperelliptic fibrations of arbitrary genus and calculate its values on Lefschetz singular fibers. Finally, we compare our local signature with another local signature which arises from algebraic geometry. Received: 6 August 1998 / in final form: 24 February 1999     

On the fundamental group of some open manifolds

We study fundamental groups of noncompact Riemannian manifolds. We find conditions which ensure that the fundamental group is trivial, finite or finitely generated.     

Bounds for the relative Euler-Poincaré characteristic of certain hyperelliptic fibrations

In this paper, we find the lower bound for the relative Euler-Poincaré characteristic of a relatively minimal hyperelliptic fibration with slope four. We prove the existence of hyperelliptic fibrations over an elliptic curve, which attain our bound.     

On the purely irregular fundamental group

Based on the (not yet fully understood analogy) between irregular connections and wild ramification, we define a purely irregular fundamental group for complex algebraic varieties and prove some results about this fundamental group which are analogous to the p‐adic étale fundamental group of algebraic varieties over fields of characteristic p (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Cohomology of the hyperelliptic Torelli group

Let SI(S _g ) denote the hyperelliptic Torelli group of a closed surface S _g of genus g. This is the subgroup of the mapping class group of S _g consisting of elements that act trivially on H ₁(S _g ; ?) and that commute with some fixed hyperelliptic involution of S _g . We prove that the cohomological dimension of SI(S _g ) is g ? 1 when g ≥ 1. We also show that H _g?1(SI(S _g ); ?) is infinitely generated when g ≥ 2. In particular, SI(S ₃) is not finitely presentable. Finally, we apply our main results to show that the kernel of the Burau representation of the braid group B _n at t = ?1 has cohomological dimension equal to the integer part of n/2, and it has infinitely generated homology in this top dimension.     

On base change of the fundamental group scheme

We provide for all prime numbers $p$ examples of smooth projective curves over a field of characteristic $p$ for which base change of the fundamental group scheme fails. This is intimately related to how $F$ -trivial vector bundles, i.e. bundles trivialized by a power of the Frobenius morphism, behave in (trivial) families. We conclude with a study of the behavior of $F$ -triviality in (not necessarily trivial) families.     

On the fundamental group of real toric varieties

LetX (Δ) be the real toric variety associated to a smooth fan Δ. The main purpose of this article is: (i) to determine the fundamental group and the universal cover ofX (Δ), (ii) to give necessary and sufficient conditions on Δ under which π₁(X(Δ)) is abelian, (iii) to give necessary and sufficient conditions on Δ under whichX(Δ) is aspherical, and when Δ is complete, (iv) to give necessary and sufficient conditions forC _Δ to be aK (π, 1) space whereC _Δ is the complement of a real subspace arrangement associated to Δ.     

On some fundamental partial integral inequalities

This paper presents several new partial integral inequalities in two independent variables which can be used in the analysis of various problems in the theory of partial differential and integral equations as ready and powerful tools. An elementary technique of reducing the integral inequality to second order partial differential inequality and then integrating it by Riemann's method is used to establish our results.     

On the slope of bielliptic fibrations

Let be a bielliptic fibration. We prove is, up to base change, a rational double cover of an elliptic fibration and that is isotrivial provided it is smooth. Finally, we prove that the slope of is at least four provided the genus of the fibre is at least six.