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. Applications of a result of turrittin by nicholas m. General method for building the riemann surface of a. There are a few di erent ways to prove the geometric monodromy theorem. Ma525 on cauchys theorem and greens theorem 2 we see that the integrand in each double integral is identically zero. Picards little theorem monodromy theorem is an important ingredient in one of the proofs of picards little theorem. In complex analysis, the monodromy theorem is an important result about analytic continuation. It is therefore evident that in order to solve the aforementioned k ahlerproblem, a profound understanding of the representa. For instance, the theorem can be proven by looking at the variation of the rotation. Using this filtration, we deduce a conjecture of crew on padic differential equations, analogous to grothendiecks local monodromy theorem also a consequence of results of andre and of mebkhout.
Monodromy is the study of how objects \run round a singularity, and so the viewpoint of this paper shall be geometric. U be a family of principally polarized abelian varieties. In complex analysis, the monodromy theorem is an important result about analytic continuation of a complexanalytic function to a larger set. The definition of analytic continuation along a curve is a bit technical, but the basic idea is that. According to ullrich, the full statement of the monodromy theorem for simple. We shall develop the results of this chapter in the context of manifolds definition 1 in 2. It is closely associated with covering maps and their degeneration into ramification. General method for building the riemann surface of a function representable by radicals 6 3. The papers center is the solution of davenports problem.
We note that the geometric monodromy theorem is valid also in the quantum case 95. Weildeligne representations and padic hodge theory. As the name implies, the fundamental meaning of monodromy comes from running round singly. Monodromy invariants in symplectic topology denis auroux thistextisasetoflecturenotesforaseriesoffourtalksgivenati. One of the hardest cases is when the monodromy group is a d or s d for some d 6 degf. Work of many mathematicians ritt, zariski, guralnick, thompson, aschbacher. Monodromy theorems in the affine setting where f g. The riemann zeta function and the prime numbers theorem 97 10. The following is a simpler version concerning the uniqueness of the monodromy theorem. The monodromyweight conjecture donu arapura deligne d1 formulated his conjecture in 1970, simultaneously in the adic and hodge theoretic settings.
General measures, outer measures, lebesgue measure, measurable functions, integration, product measures, fubinis theorem. We show that orbits with t 1 isotropy are associated to nontrivial monodromy and we give a simple formula for computing the monodromy matrix in this case. Oct 11, 2001 we produce a canonical filtration for locally free sheaves on an open padic annulus equipped with a frobenius structure. University of groningen rotation forms and local hamiltonian. The same conclusion already follows if only the monodromy h 11 at in nity of f is trivial h1 is the monodromy around a very large circle in c. We give a quick statement here, deferring the precise formulation to section 5. A sufficient criterion for the singlevaluedness of a branch of an analytic function. Finally, we apply margulis superrigidity theorem to show that 0k, where k is the kernel of the natural monodromy representation, is zariskidense. Monodromy various strong results like burnsides paqbtheorem, frobenius theorem or haboushs theorem impressively demonstrate the close ties between various classes of groups and their representations. Big monodromy theorem for abelian varieties over finitely generated fields big monodromy theorem for abelian varieties over finitely generated fields ariasdereyna, sara. On the monodromy theorem for isolated hypersurface singularities john scherk 1 inventiones mathematicae volume 58, pages 289 301 1980 cite this article. Click, drag, and drop to reorder files or press delete to remove any content you dont want.
Monodromy of hamiltonian systems with complexity 1 torus. Complex analysis additional class notes webpage faculty. In this sense, cauchys theorem is an immediate consequence of greens theorem. On the monodromy theorem for isolated hypersurface.
First, inspired by a construction of colin and honda 9, we show that any open book decomposition with pseudoanosov monodromy, can be. Monodromy groups of indecomposable rational functions. Finally i wish to thank trfing and bjsrk for interesting discussions on the subject. Main theorem theorem zbzywina let u be a nonempty open subset of pn k and let a. In fact, greens theorem is itself a fundamental result in mathematics the fundamental theorem of calculus in higher dimensions. Let f,d be a function element which admits unrestricted continuation in the simply connected region g. Hyperbolic monodromy groups for the hypergeometric equation and cartan involutions elena fuchs chen meiri peter sarnak y september 10, 20 to nicholas katz with admiration abstract we give a criterion which ensures that a group generated by cartan involutions in the automorph group of a rational quadratic form of signature n 1.
The classical one is easier to understand, it revolves around the concept of function, and distinguishes between a monodromic i. Pdf we prove a monodromy theorem for local vector fields belonging to a sheaf satisfying the unique continuation property. Variables separated equations, contrasts the contributions of the simple group classification and of the branch cycle lemma for figuring the defining. X be a continuous map, and let wbe a point of x satisfying pw h0.
Covering spaces and the monodromy theorem springerlink. Big monodromy theorem for abelian varieties over finitely. Monodromy transformation encyclopedia of mathematics. Lie local subgroupoids and their holonomy and monodromy lie. Holomorphic functions we begin by recalling the basic facts about the eld of the complex numbers c and the power. The proofs of theorems files were prepared in beamer. More precisely, there exists an integer e such that. For each nonarithmetic lattice arising in theorem i i. So the eigenvalues of t are roots of unity and the size of the jordan blocks is bounded by q c1.
Y is the complement of the branch locus, with its complex topology. You can merge pdfs or a mix of pdf documents and other files. A focusfocus ber with one singular point a pinched torus. Let be the unit disc in the complex plane, let be an analytic space, let be a proper holomorphic mapping cf. From the 00 viewpoint, n is a locally trivial fibre. Stabilizing the monodromy of an open book decomposition.
Monodromy and asymptotic properties of certain multiple integrals. Monodromy theorem ensures that there exists a holomorphic function extending gz logfz in u. C has trivial global monodromy group if and only if fis rational of simple type, in the sense of miyanishi and sugie 12. Davenports problem is to figure out the nature of two polynomials over a number field having the same ranges on almost all residue class fields of the number field. This is the starting point of our proof of theorem 1. Analysis qualifying exam syllabus, 20 real analysis text. Before stating and proving the monodromy theorem, we need two lemmas and a definition. In the case q is a lie groupoid we expect to deal with lie local subgroupoids s and the globalization of local smooth morphisms to a smooth morphism ms k on a. The author apologises for all errors, unclarities, omissions of details and other imperfections and encourages the. Solving this problem initiated the monodromy method.
We consider the monodromy of ntorus bundles in n degree of freedom integrable hamiltonian systems with a complexity 1 torus action, that is, a hamiltonian t n. The hodge theoretic statement, amounted to the existence of what is now called a limit mixed hodge structure. 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. Pdf on a monodromy theorem for sheaves of local fields and. Monodromy theorem article about monodromy theorem by the. For an html and pdf or ppt file with the same name, the html is an exposition. As you might know, analytic continuation works as follows. 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. If m is the monodromy matrix of the fundamental solution xt, x0 i of 1. 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. Monodromy and spectrumpreserving deformations 69 only. The idea of a monodromy transformation arose in the study of multivalued functions see monodromy theorem. The monodromy transformation arises most frequently in the following situation.
Monodromy groups of indecomposable rational functions \random degreen rational function should have monodromy group a n or s n. We prove a monodromy theorem for local vector fields belonging to a sheaf satisfying the unique continuation. In this case the image of the local padic monodromy. The field of definition of function fields and a problem in the reducibility of polynomials in two variables, illinois journal of math. Oct 24, 2014 the monodromy theorem in as important theorem in the field of complex analysis. It is closely associated with covering maps and their. Monodromy and asymptotic properties of certain multiple integrals 183 tion of singularities. The monodromy theorem in as important theorem in the field of complex analysis. Namely, given a finite locally free sheaf on an open padic annulus with a. Let be the generic point of u and suppose moreover that a k has big monodromy. On a monodromy theorem for sheaves of local fields and applications. What links here related changes upload file special pages permanent link page information wikidata item cite this page. We establish a generalization of the padic local monodromy theorem of andre, mebkhout, and the author in which differential equations on rigid analytic. The printout of proofs are printable pdf files of the beamer slides without the pauses.
The monodromy transformation in this case is also called a covering or deck transformation. In each case we also derive the inverse mapping which allows one to reconstruct the solutions to 1. The monodromy transformation in this case is also called a covering or. Discrimanant complements and kernels of monodromy reps. A adic local monodromy theorem annals of mathematics. The subject of padic hodge theory can to an outsider like myself sometimes seem like it su ers from. The monodromy group of an algebraic function wolfram.
Talk at uci irvine in 1995, weils decomposition theorem. The main topic of 1 is the introduction of ramification data for wild ramification, extending the notion of higher ramification groups for not necessarily galois covers. This was solved by schmid s a couple of years later. University of groningen on monodromy in integrable. The outstanding point about the monodromy version of the schur property is the significance of distinguishing between the arithmetic and geometric monodromy of a cover. A read is counted each time someone views a publication summary such as the title, abstract, and list of authors, clicks on a figure, or views or downloads the fulltext. The monodromy theorem curiously enough, there are several famous monodromy theorems, the classical one and some modern ones. We also remark that the present theorem 3 is an improvement of theorem 1 of the authors paper 4. We produce a canonical filtration for locally free sheaves on an open padic annulus equipped with a frobenius structure. Grothendiecks monodromy theorem says that this local monodromy action is always quasiunipotent, i. 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.
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. This theorem can be proved 1991 mathematics subject classi. How to merge pdfs and combine pdf files adobe acrobat dc. Click add files and select the files you want to include in your pdf.
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