Finite Monodromy Research Articles

Bases for certain cohomology representations of the symmetric group

We give a combinatorial description (including explicit differential-form bases) for the cohomology groups of the space of n distinct nonzero complex numbers, with coefficients in rank-one local systems which are of finite monodromy around the coordinate hyperplanes and trivial monodromy around all other hyperplanes. In the case where the local system is equivariant for the symmetric group, we write the cohomology groups as direct sums of inductions of one-dimensional characters of subgroups. This relies on an equivariant description of the Orlik-Solomon algebras of full monomial reflection groups (wreath products of the symmetric group with a cyclic group). The combinatorial models involved are certain representations of these wreath products which possess bases indexed by labelled trees.

CONNECTION FORMULAS FOR ALGEBRAIC HYPERGEOMETRIC FUNCTIONS

When the hypergeometric differential equation has finite monodromy, algebraic expressions of the connection coefficients, which are originally given by Gamma values, are presented.

A negatively curved Kähler threefold not covered by the ball

We construct examples of three-dimensional compact Kahler manifolds with negative curvature, not covered by the ball. Our manifolds are obtained as a natural generalization of the two-dimensional examples discovered by Mostow and Siu, using their description in terms of monodromy covers of hypergeometric functions. Each example is obtained from a hypergeometric monodromy group in PU(3,1) that is not discrete but has finite local monodromy. We describe a manifold on which the group acts discretely, and check that it has a compact quotient with the above features. The examples are locally described as branched covers of the ball, with totally geodesic branch locus.

Lamé operators with finite monodromy—a combinatorial approach

We are describing Lamé differential operators with a full set of algebraic solutions. For each finite group G, we are describing the possible values of the degree parameter n such that the Lamé operator L n has the projective monodromy group G. The main technical tool is the combinatorics associated to Belyi functions, ideas that we already used in (Rend. Sem. Mat. Univ. Padova 107 (2002) 191–208) for describing the case n = 1 . We also supply proofs to some finiteness properties conjectured by Baldassarri and by Dwork, and we work out an explicit formula for the number of essentially different Lamé equations when n = 2 . This approach can be generalized for arbitrary degree n (see (Counting Integral Lamé Equations by Means of Dessins d'Enfants, arXiv:math.CA/0311510) for n integer).

Morphisms of F-isocrystals and the finite monodromy theorem for unit-root F-isocrystals

We discuss Tate-type problems for F-isocrystals, that is, the full faithfulness of the natural restriction functors between categories of overconvergent F-isocrystals on schemes of positive characteristic. We prove it in the cases of unit-root F-isocrystals. Using this result, we prove that an overconvergent unit-root F-isocrystal has a finite monodromy.

F

We discuss Tate-type problems for F-isocrystals, that is, the full faithfulness of the natural restriction functors between categories of overconvergent F-isocrystals on schemes of positive characteristic. We prove it in the cases of unit-root F-isocrystals. Using this result, we prove that an overconvergent unit-root F-isocrystal has a finite monodromy.

On a characterization of finite vector bundles as vector bundles admitting a flat connection with finite monodromy group

We prove that a holomorphic vector bundle E E over a compact connected Kähler manifold admits a flat connection, with a finite group as its monodromy, if and only if there are two distinct polynomials f f and g g , with nonnegative integral coefficients, such that the vector bundle f ( E ) f(E) is isomorphic to g ( E ) g(E) . An analogous result is proved for vector bundles over connected smooth quasi-projective varieties, of arbitrary dimension, admitting a flat connection with finite monodromy group. When the base space is a connected projective variety, or a connected smooth quasi-projective curve, the above characterization of vector bundles admitting a flat connection with finite monodromy group was established by M. V. Nori.

The local index and the Swan conductor

In this paper we study local indices of systems of p-adic linearly differential equations which arise from p-adic representations of the absolute Galois group of local field of characteristic p with finite monodromy. We show the induction formula of the local index of p-adic differential equations and prove the equality between the local index of differential equations and the Swan conductor of p-adic Galois representations by inductive methods.

On Lamé operators which are pull-backs of hypergeometric ones

We give a method that would allow one to calculate the number of Lamé operators, L n {\mathcal {L}_n} , n ∈ N n \in {\mathbf {N}} , with prescribed finite monodromy and do the calculation for the case n = 1 n = 1 . We find a bound for the degree over Q {\mathbf {Q}} of the field of definition of the coefficients of a Lamé operator with prescribed finite monodromy and give examples of Lamé operators with finite monodromy. Finally we study Lamé operators with infinite monodromy and generic second order differential operators which are pull-backs of hypergeometric ones under algebraic maps.

Abel’s theorem for twisted Jacobians

A twisted version of the Abel-Jacobi map, associated to a local system with finite monodromy on a smooth projectve complex curve, is introduced. An analogue of Abel’s theorem characterizing the kernel of this map is proved. The proof, which is new even in the classical case, involves reinterpreting the Abel-Jacobi map in the language of mixed Hodge structures and their extensions.

Finite monodromy of Pochhammer equation

Finite monodromy of Pochhammer equation

On the determinant of periods with finite monodromy.

On the determinant of periods with finite monodromy.

On a Certain Class of Generalized Hypergeometric Functions with Finite Monodromy Groups

On a Certain Class of Generalized Hypergeometric Functions with Finite Monodromy Groups

The monodromy of reducible plane curves

Let f : C "+1~ C be a polynomial function such that f ( 0 ) = 0 and f has an isolated singularity at 0. The local geometry of f 1 (0) near 0 is completely described by the algebraic link L 2"-1 of the singularity, where L 2" l = f l ( 0 ) n S 2"+1 is a smooth (n-2)-connected submanifold of the small E-sphere S 2"+i about the origin [13]. Moreover, the link complement X = S 2 " + I L fibers over S 1 with fiber F2"~_S"v ... v S". Let X denote the infinite cyclic covering space of X associated with the epimorphism H 1 ( X ) ~ HI(S 1) induced by the fibration. The action of the multiplicative generator t of 171(S 1) on H,(F;Z) is called the local algebraic monodromy transformation of f at O. Since )( ~ F x R, t can be viewed as the generator of the infinite cyclic covering translation group, and the algebraic monodromy is an automorphism t: H,( ) ) ; Z ) ~ Hn0); Z). The polynomial invariants of H, (X; Z) as a A = Z [t, t 1] module are therefore invariants of the monodromy. For example, the Alexander polynomial A t (t) is the characteristic polynomial of the monodromy. We are interested in the following problem: Given ) , determine whether the algebraic monodromy is of finite or infinite order. The problem is most amenable to solution in the case n= 1, because the geometry is completely understood [3,9, 15]. In the classical case (n=l) , f l ( 0 ) is called a plane curve, and the algebraic link L 1 has r components, one corresponding to each of the branches of f 1(0) at the origin. Each branch corresponds to a distinct analytically irreducible factor of f which maps 0 to 0. Each component of L is an iterated torus knot, and both the iteration on each component and the linking among the various components is completely specified by the Puiseux developments [1, i0] corresponding to the branches. In the case that f is analytically irreducible (r = 1), f always has finite monodromy [10, 1, 19]. If r>2 , the monodromy can be infinite [1, 20]. This paper

Some criteria for finite and infinite monodromy of plane algebraic curves

Some criteria for finite and infinite monodromy of plane algebraic curves