Monodromy representation of elementary simple covers. A padic local monodromy theorem 95 example, it fails for the pushforward of the constant isocrystal on a family of ordinary elliptic curves degenerating to a supersingular elliptic curve and for the bessel isocrystal described in section 1. The monodromy conjecture for hyperplane arrangements. Our proof is based on takenss index theorem, which specifies how the energyh chern number changes when h passes a nondegenerate critical value, and a choice of admissible cycles in fomenkozieschang. On a monodromy theorem for sheaves of local fields and. Monodromy for the hypergeometric function theorem 6. Abstract we investigate an analogue of the grothendieck pcurvature conjecture, where the vanishing of the pcurvature is replaced by the stronger condition, that the module with connection mod p underlies a xdx module structure. Grothendiecks monodromy theorem says that this local monodromy action is always quasiunipotent, i. To understand how the fundamental group based at a point of the target of a covering map acts naturally on the fiber of the covering map over that point, the fiber being thought of as embedded inside the source of the covering map. Hamiltonian monodromy and morse theory springerlink. For a nitedimensional gmodule v, r c is the following connection on the trivial vector bundle h reg v. Pdf on a monodromy theorem for sheaves of local fields and.
Before stating and proving the monodromy theorem, we need two lemmas and a definition. Volume 324, issue 9, september 2009, pages 19531982. Mod07 lec28 proof of the first homotopy version of the monodromy theorem. The monodromy conjecture asserts that if cis a pole of the local topological zeta function of a hypersurface, then exp2. Our proof is based on takenss index theorem, which specifies how the energyh chern number changes when h passes a nondegenerate critical value, and a choice of. The hodge theoretic statement, amounted to the existence of what is now called a limit mixed hodge structure. It is closely associated with covering maps and their. Monodromy for the hypergeometric function download references. We discuss in detail two examples and make a connection to the duistermaatheckman theorem 21. These include the theorems of hurwitz and rouche, the open mapping theorem, the inverse and implicit function theorems, applications of those theorems, behaviour at a critical point, analytic branches, constructing riemann surfaces for functional inverses, analytic continuation and monodromy, hyperbolic geometry and the riemann mapping theorem. Reduction of the purity theorem to the monodromy theorem.
The celebrated arnoldliouville theorem gives the structure of a completely integrable system near a regular level in terms of actionangle coordinates. On monodromy representation of period integrals associated. Reduction of the target theorem to the purity theorem lecture iii. In section 4 we return to the behavior of the perturbed classical system described in section 2.
The actual configuration may depend on the choice of the branch cuts, but the group generated by the branch cycles is always the same. Sep 08, 2015 mod07 lec28 proof of the first homotopy version of the monodromy theorem. Applications of a result of turrittin by nicholas m. We consider one of the simplest possible systems with monodromy. This is especially true in view of recent results 12 connecting monodromy groups with quasi fuchsian and totally degenerate groups. Monodromy theorem article about monodromy theorem by the. A adic local monodromy theorem annals of mathematics.
I believe that a version of this is in joe harriss article, galois groups of enumerative problems. The following is a simpler version concerning the uniqueness of the monodromy theorem. This mapping, and also the homomorphisms induced by it on the homology and cohomology spaces of, is also called a monodromy transformation. As you might know, analytic continuation works as follows. If you have watched this lecture and know what it is about, particularly what mathematics topics are discussed, please help us by commenting on this video with your suggested description and title.
The correct analogue of the local monodromy theorem was formulated. Let f,d be a function element which admits unrestricted continuation in the simply connected region g. Path, lifting of a path, uniquepathlifting property, covering homotopy theorem, surjective local homeomorphism, universal covering space, injective group homomorphism, fundamental group, simply connected space, trivial group, fiber of a covering map, coset space, group action, orbit, orbit map, stabilizer. Monodromy is the simplest obstruction to the existence of global actionangle variables in integrable hamiltonian dynamical systems. The grothendieck monodromy theorem universiteit leiden. In mathematics, monodromy is the study of how objects from mathematical analysis, algebraic topology, algebraic geometry and differential geometry behave as they run round a singularity. He gives obstructions to the existence of global actionangle coordinates, one of which is the monodromy. As an application, we show how to recover from this classification the cohomological version of a classical theorem of hopf, and we extend it to the non abelian case. The first method that gives a full representation of the fundamental group of the complement to singular loci relies on the generalised picardlefschetz theorem. Monodromy groups of indecomposable rational functions. On a monodromy theorem for sheaves of local fields and applications. We shall introduce the basics of classical gauss hypergeometric equation and to study its monodromy problem.
We calculate two differenent monodromy representations of period integrals for the affine variety x 2,2 obtained by the dual polyhedron mirror variety construction from y. Oct 24, 2014 the monodromy theorem in as important theorem in the field of complex analysis. Nonetheless, there remain many fundamental questions where we do not even know what sort of answers to expect. We prove that the conductors computed by the arithmetic ramification filtrations on g k defined by abbes and saito amer. As the name implies, the fundamental meaning of monodromy comes from running round singly. General method for building the riemann surface of a. Let k be a complete discretely valued field of equal characteristic p 0 with possibly imperfect residue field, and let g k be its galois group.
Work of many mathematicians ritt, zariski, guralnick, thompson, aschbacher. The monodromy transformation in this case is also called a covering or. We establish a generalization of the padic local monodromy theorem of andre, mebkhout, and the author in which differential equations on rigid analytic. If is the riemann surface of such a function, then by eliminating the singular points of the function from the riemann sphere, an unbranched covering is obtained. Find out information about darbouxs monodromy theorem. This was solved by schmid s a couple of years later.
The monodromy group of an algebraic function wolfram. Monodromy theorems in the affine setting where f g. Monodromy groups and linearly polymorphic functions one should take another look at differential equations and their monodromy groups. This demonstration shows the structure of the branches of a multivalued function defined by a polynomial equation, illustrating the transitions between the branches along paths going around a branch point. The idea of a monodromy transformation arose in the study of multivalued functions see monodromy theorem. We prove a monodromy theorem for local vector fields belonging to a sheaf satisfying the unique continuation property. By the arnoldliouville theorem, actionangle coordinates exist on a neighbourhood of any regular fibre. Monodromy groups and linearly polymorphic functions. One obstruction to the existence of global actionangle coordinates is the monodromy, which.
We show that hamiltonian monodromy of an integrable two degrees of freedom system with a global circle action can be computed by applying morse theory to the hamiltonian of the system. Covering spaces and the monodromy theorem springerlink. In complex analysis, the monodromy theorem is an important result about analytic continuation of a complexanalytic function to a larger set. We show that 2 monodromy classifies locally constant stacks on a reasonably wellbehaved space x. Sc hub youtube channel me aapka sawagt hai, hamara m. On the monodromy theorem for isolated hypersurface singularities. Annals of mathematics, 160 2004, 93184 a padic local monodromy theorem by kiran s. If a complex function is analytic at a point of a bounded simply connected domain and can be continued analytically along every curve from the point, then.
On the monodromy theorem for isolated hypersurface. The past century in particular has seen huge progress in this study, from mordells theorem in 1922 to the work of wiles and taylorwiles in 1994. We show that 2monodromy classifies locally constant stacks on a reasonably wellbehaved space x. The definition of analytic continuation along a curve is a bit technical, but the basic idea is. This video lecture, part of the series advanced complex analysis i by prof. Nov 28, 2019 we show that hamiltonian monodromy of an integrable two degrees of freedom system with a global circle action can be computed by applying morse theory to the hamiltonian of the system. A number of important properties of can be inferred from the structure of the monodromy group. The padic local monodromy theorem in this chapter, we assert the padic local monodromy theorem, and sketch how it can be proved either using deep properties of padic di. In section 4 we revisit the symmetry approach to monodromy presented in the works 29,39, and link it to the rotation.
One of the hardest cases is when the monodromy group is a d or s d for some d 6 degf. In particular, in the case of admissible regular sheaves of local fields defined on a simply connected manifold, we obtain a global extension result for every local field of the sheaf. Dynamical manifestations of hamiltonian monodromy sciencedirect. In section 3 we show how this theorem can be applied to the context of monodromy. We shall prove various results concerning the existence and uniqueness of such lifts. Therefore, a simple, elementary cover with nonempty branch locus has monodromy group equal to the entire symmetric group. The correspondence of with gives a representation of the fundamental group on. Mod07 lec28 proof of the first homotopy version of the.
Monodromy of elliptic surfaces fedor bogomolov and yuri tschinkel january 25, 2006 1 introduction let e b be a nonisotrivial jacobian elliptic. The monodromyweight conjecture donu arapura deligne d1 formulated his conjecture in 1970, simultaneously in the adic and hodge theoretic settings. Monodromy transformation encyclopedia of mathematics. In each case we also derive the inverse mapping which allows one to reconstruct the solutions to 1. The idea is that one can extend a complexanalytic function from here on called simply analytic function along curves starting in the original domain of the function and ending in the larger set. We prove a monodromy theorem for local vector fields belonging to a sheaf satisfying the unique continuation.
Although taylors series have radii of convergence, the circle in which they are defined can sometimes be shifted, which means that sometimes it is possible to work ones way around singularities. Monodromy groups of indecomposable rational functions \random degreen rational function should have monodromy group a n or s n. An important element in the proof of the above results is a theorem of levelt, which gives a simple algebraic characterisation of the monodromy group of a hypergeometric differential equation le, thm. In complex analysis, the monodromy theorem is an important result about analytic continuation. Using this filtration, we deduce a conjecture of crew on padic differential equations, analogous to grothendiecks local monodromy theorem also a. A sufficient criterion for the singlevaluedness of a branch of an analytic function. Department of mathematics, university of toronto, m5s 1a1, toronto, canada. The monodromy theorem in as important theorem in the field of complex analysis. The fact stated below is not required for our proof of theorem 8. A potential problem of this analytic continuation along a. Duistermaat addressed the question of the existence of global actionangle coordinates in his seminal paper. The author apologises for all errors, unclarities, omissions of details and other imperfections and encourages the.
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