Monodromy Operator

The Monodromy Group

In singularity theory and algebraic geometry the monodromy group is embodied in the Picard-Lefschetz formula and the Picard-Fuchs equations. It has applications in the weakened 16th Hilbert problem and in mixed Hodge structures. In the theory of systems of linear differential equations one has the Riemann-Hilbert problem, the Stokes phenomena and the hypergeometric functions with their multidimensional generalizations. In the theory of homomorphic foliations there appear the Ecalle-Voronin-Martinet-Ramis moduli. On the other hand, there is a deep connection of monodromy theory with Galois theory of differential equations and algebraic functions. All this is presented in this book, underlining the unifying role of the monodromy group. The material is addressed to a wide audience, ranging from specialists in the theory of ordinary differential equations to algebraic geometers. The book contains a lot of results which are usually spread in many sources. Readers can quickly get introduced to modern and vital mathematical theories, such as singularity theory, analytic theory of ordinary differential equations, holomorphic foliations, Galois theory, and parts of algebraic geometry, without searching in vast literature.

The Local Monodromy Operator as an Algebraic Cycle

In this thesis we study a mapping of complex varieties whose special fibre has at worst four-fold singularities. We show that the local monodromy operator and its higher powers are represented by algebraic cycles. Let X be an n-dimensional connected complex-analytic manifold and let Delta be the unit disc. Let f : X & rarr; Delta denote a proper surjective morphism, which is submersive (smooth) over the complement of the origin. Then one can associate to f an important topological operator called the local monodromy operator N. The question of representing the monodromy operator by an algebraic cycle, i.e. by intersection with an algebraic cycle, was raised by Kato (cf. [6] page 3) following ground breaking work of Steenbrink (1976, [29]) and is the topic of this thesis, which extends earlier work of Consani (1999, [5]). Algebraic cycles have been a major subject of investigation in algebraic geometry, reflected in the "Standard Conjectures" of Grothendieck, the Hodge conjecture and the Tate conjecture. The monodromy operator corresponds to a class in a cohomology group of a space that is closely related to the generic fibre of f. This space arises as the limit of a spectral sequence that degenerates at the E2-term. The main problem is to find algebraic representatives for cohomology classes appearing in E1-terms of a spectral sequence that degenerates at the E2-term to the cohomology group above, and which induce powers Nr of the monodromy operator in the latter. In this thesis we consider the local monodromy operator at a point in the special fibre. The main problem that we study is to find algebraic representatives of the classes above when n = 4 and f : X & rarr; Delta has at worst four-fold singularities (so that every fibre has dimension 3). We find algebraic cocycles [Ni] & isin; H2n-2 i-2(S & tilde; (2i+2), Q) for each i (see below), which we show represent the monodromy operator and its higher powers except in two special cases. To do this we relate the spectral sequence above to a spectral sequence associated to a suitable resolution of singularities of the fibred product X xDelta X & rarr; Delta. The E1 terms of the latter spectral sequence are direct sums of cohomology groups of "strata" given by disjoint unions S & tilde; (k) of intersections of components of the special fibre S of the resolution of singularities of X xDelta X. We will find algebraic cycles that represent cohomology classes on the S & tilde;(k) which induce Nr in the E2-term. In order to find these algebraic cycle representatives we explicitly compute Chow rings of certain S & tilde;(k) using combinatorial constructions from toric geometry and use a theorem of Fulton relating them to the cohomology rings we are interested in.

Symplectic Geometry and Its Applications