The monodromy group of an algebraic function wolfram. 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. An example of the schwarz reflection principle, where a. Monodromy theorem article about monodromy theorem by the. 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. In section 3 we show how this theorem can be applied to the context of monodromy. The idea of the proof of theorem 1 is to combine the continuity method. 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. Let fg ign i1 be a collection of nite discrete groups and k 0 be the 0simplicial complex on nvertices. 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 subgroup, fibration. Applications are numerous pluriharmonic free functions have globally defined pluriharmonic conjugates, locally invertible functions are globally invertible, and there is no nontrivial cohomology theory arising from analytic continuation on connected free sets.
Our proof of theorem 1 is based on ideas of 4, 7, 8 and 9. In complex analysis, the monodromy theorem is an important result about analytic continuation. Applications of a result of turrittin by nicholas m. We show that in the case of a log smooth degeneration over the standard log disc, this construction allows one to. 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. A adic local monodromy theorem annals of mathematics. Monodromy is the study of how objects \run round a singularity, and so the viewpoint of this paper shall be geometric. We go even further in extending isogonies of complex torii towers of modular curves modulartowers. 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. We shall prove various results concerning the existence and uniqueness of such lifts. Monodromy transformation encyclopedia of mathematics. Using this filtration, we deduce a conjecture of crew on padic differential equations, analogous to grothendiecks local monodromy theorem also. Let be the generic point of u and suppose moreover that a k has big monodromy.
Venkataramana tifr monodromy and arithmetic groups february 10, 2015 5 30. U be a family of principally polarized abelian varieties. It is a free z module of rank 1 by letting n n mod n2z act on t t n n2t g m by t tn n. Before stating and proving the monodromy theorem, we need two lemmas and a definition. We establish a generalization of the padic local monodromy theorem of andre, mebkhout, and the author in which differential equations on rigid analytic. The definition of analytic continuation along a curve is a bit technical, but the basic idea is that one starts. In section 4 we revisit the symmetry approach to monodromy presented in the works 30,40, and link it to the rotation.
Hejhal also conjectured that generic representations into sl2, are monodromy representations. 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. The following is a simpler version concerning the uniqueness of the monodromy theorem. Monodromy theorems in the affine setting where f g. Monodromy for the hypergeometric function monodromy and yinvariants walter d. Pdf on a monodromy theorem for sheaves of local fields and. Monodromy and log geometry piotr achinger arthur ogus y february 7, 2018 abstract a now classical construction due to kato and nakayama attaches a topological space the \betti realization to a log scheme over c. This openaccess thesis is brought to you for free and open access by. With these we introduce nielsen classes attached to g. Work of many mathematicians ritt, zariski, guralnick, thompson, aschbacher. Neumann received 3 april 1978 if x2i is a smooth closed oriented manifold and a. Let f,d be a function element which admits unrestricted continuation in the simply connected region g. We prove monodromy theorem for local vector fields be.
Main theorem theorem zbzywina let u be a nonempty open subset of pn k and let a. To point out relations between mondromy and spectrum preserving deformations. 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. Sc hub youtube channel me aapka sawagt hai, hamara m. 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. W e note that the present work is closely related to the works 30, 40, which demon. Pdf we prove a monodromy theorem for local vector fields belonging. For a nitedimensional gmodule v, r c is the following connection on the trivial vector bundle h reg v. As you might know, analytic continuation works as follows. Variations on a theorem of tate memoirs of the ams, vol. A sufficient criterion for the singlevaluedness of a branch of an analytic function. As the name implies, the fundamental meaning of monodromy comes from running round singly.
We study kedlayas slope filtration theorem in this context, and show that g. We discuss the history of the monodromy theorem, starting from. Outf n be the monodromy representation where f n is isomorphic to the kernel of the projection p. Recently, we used three such paradoxical ideas to prove the free will theorem strengthened here, which is the culmination of. From the 00 viewpoint, n is a locally trivial fibre. The isomonodromy method in the modern theory of painleve equations was. Monodromy groups some naturally occurring subgroups of slnz are monodromy groups. Oct 24, 2014 the monodromy theorem in as important theorem in the field of complex analysis. A padic local monodromy theorem annals of mathematics. In complex analysis, the monodromy theorem is an important result about analytic continuation of a complexanalytic function to a larger set. Grothendiecks monodromy theorem says that this local monodromy action is always quasiunipotent, i. Monodromy for the hypergeometric function theorem 6. We discuss in detail two examples and make a connection to the duistermaatheckman theorem 22. 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.
We produce a canonical filtration for locally free sheaves on an open padic annulus equipped with a frobenius structure. To introduce a new method for solving the initial value problem for equations such as 1. 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. The monodromy transformation in this case is also called a covering or. It is closely associated with covering maps and their degeneration into ramification. General method for building the riemann surface of a. One of the hardest cases is when the monodromy group is a d or s d for some d 6 degf. 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. Monodromy groups of indecomposable rational functions \random degreen rational function should have monodromy group a n or s n. 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. Conway and simon kochen t he two theories that revolutionized physics in the twentieth century, relativity and quantum mechanics, are full of predictions that defy common sense.
S is a morphism of varieties which is a c1locally trivial. Theorem nori the image of the monodromy representation of f3 on the. On monodromy representations in denhamsuciu fibrations 5 lemma 1. The idea of a monodromy transformation arose in the study of multivalued functions see monodromy theorem. Hamiltonian monodromy and morse theory springerlink. The grothendieck monodromy theorem raymond van bommel. X be a continuous map, and let wbe a point of x satisfying pw h0.
The monodromy theorem in as important theorem in the field of complex analysis. In this case the image of the local padic monodromy. Hurwitz monodromy and full number fields stanford mathematics. The actual configuration may depend on the choice of the branch cuts, but the group generated by the branch cycles is always the same. We show that the monodromy theorem holds on arbitrary connected free sets for noncommutative free analytic functions.
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