It also provides a local proof of the irreducibility. Background material on hamiltonian monodromy and chern classes is presented in the appendix. Note that the rank of the free group in this case depends only on the order. Grothendieck proved the following theorem which implies the previous theorem, but i wont go into the implication. Monodromy theorem article about monodromy theorem by the. Harmonic functions, maximum principles, schwarz lemma and uniqueness of riemann mappings.
Pdf on the monodromy of the hitchin connection yves. We discuss the history of the monodromy theorem, starting from weierstras, and the concept of a monodromy group. Periodicity and the monodromy theorem project euclid. The theorem states that quantum groups can be used to describe the monodromy of certain rst order fuchsian. Rn is topologically trivial if and only if the monodromy and the chernduistermaat class of the tnbundle are trivial. Our proof of theorem 1 is based on ideas of 4, 7, 8 and 9. An example point of singularity on the boundary of the. An ek, f isocrystal is a finitely generated free ek,f module m equipped with an. We will see that the monodromy pairing is the valuation of the analogue of the riemann form in the nonarchimedean setting. Also, it is evident that this map is a solution of our system. 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. Around 1990, kohno ko and drinfeld d2 proved a rather astonishing result, now known as the kohnodrinfeld theorem. 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.
In these expository notes, we describe results of cauchy, fuchs and pochhammer on differential equations. 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. Lectures on differential equations in complex domains university of. Hyperbolic monodromy groups for the hypergeometric.
Monodromy theorem the monodromy theorem the monodromy theorem gives a suf. The definition of analytic continuation along a curve is a bit technical, but the basic idea is that one starts. We give an analytic greeneliouville theorem, an entire free plurisubharmonic function is a sum of hereditary and. For kof characteristic zero, grothendiecks theorem says that the etale fundamental group of an npunctured curve of genus gover an algebraically closed eld kof characteristic zero is the same as over c. Hamiltonian monodromy and morse theory springerlink. We compute explicitly the monodromy representations of cyclotomic analogs of the knizhnikzamolodchikov differential system.
Variations on a theorem of tate memoirs of the ams, vol. We also give a brief overview, without proofs, of results of beukers and heckman. In section 14, we tie everything together, and give the final statement of the local monodromy theorem 14. Let sbe a smooth complex algebraic variety with a base point 0. It is enough to show the above theorem in the context of conformal field theory as defined by tsuchiyauenoyamada tuy. By way o contrastf th, e invers oe f legendres modular functio hans als a freeo group of rank two as its monodromy group bu, t its riemann surfac hae s countably. 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. Moreover if the symplectic form is exact then the existence of. Monodromy of hamiltonian systems with complexity 1 torus. Moreover, to obtain dynamical stability regions within the parameter space of interest, we have to. Proof of the first homotopy version of the monodromy theorem proof of the algebraic nature of analytic branches of the functional inverse of an analytic function at a critical point unit 8. Noncommutative free universal monodromy, pluriharmonic. There is an important relation between local and global monodromy. From this viewpoint we compare then the weierstras, the legendre and other normal forms for elliptic curves, explaining their geometric meaning and distinguishing them by their stabilizer in.
If ais an abelian variety over a nitely generated eld kof characteristic zero with enda z and dima 2. 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. The following is a simpler version concerning the uniqueness of the monodromy theorem. Hejhal also conjectured that generic representations into sl2, are monodromy representations. 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. I be a compact smooth manifold with a finite open cover ci and with a smooth free. In this thesis we study the monodromy groups of lisse sheaves and isocrystals in. The grothendieck monodromy theorem raymond van bommel.
The monodromy matrix computation is also a very difficult and timeconsuming task if the dimension n of eq. A secondary goal of the notes is to explicate the relation of grothendiecks monodromy pairing 17 to the nonarchimedean uniformization of abelian varieties. Monodromy for the hypergeometric function theorem 6. In mathematics, monodromy is the study of how objects from mathematical analysis, algebraic. Y t, which induces a cohomological monodromy transformation t. The class ix is a characteristic class of a possibly twisted re exive coherent sheaf ex of rank 2n 2 on x, which is locally free away from a single point xof x. Our proof of the main theorem follows halls proof of to some extent. For each nonarithmetic lattice arising in theorem i i. We generalize their results to compatible systems of semisimple lisse sheaves and overconvergent fisocrystals over arbitrary smooth varieties. Grothendiecks monodromy theorem says that this local monodromy action is always quasiunipotent, i. An important theorem 7, iii,2 says this transformation is quasiunipotent, i. The definition of analytic continuation along a curve is a bit technical, but the basic idea is th. As the name implies, the fundamental meaning of monodromy comes from running round singly.
Thus, sumnerss unknotting theorem holds for free z, actions as well. This theorem was originally conjectured by grothendieck and proved by him for a rankone equation on p1. The author apologises for all errors, unclarities, omissions of details and other imperfections and encourages the. We prove monodromy theorem for local vector fields be. The global monodromy representation depends only on the basis of.
Then we focus on the birth of the concept of the jacobian. Sheaves, covering spaces, monodromy and an application. We show that in the case of a log smooth degeneration over the standard log disc, this construction allows one to. Assume that kis nite, then there is an open subgroup of j. With the notation of the previous theorem, the torus bundle. Under these assump tions theorem 1 was proven by d. Nptel mathematics advanced complex analysis part 1. Theorem nori the image of the monodromy representation of f3 on the.
If you want to get a hold directly of the associated sheaf. If xsis a family of smooth projective varieties, one obtains the associated monodromy representation. On delignes monodromy finiteness theorem mimuw ag seminar, dec, 2018 piotr achinger abstract. A adic local monodromy theorem annals of mathematics. In this paper, we begin by giving a new proof of the unknotting theorem for the special case. X be a continuous map, and let wbe a point of x satisfying pw h0. It is closely associated with covering maps and their degeneration into ramification. Monodromy groups in positive characteristic refubium. By the definition of analytic continuation we can conclude. In particular, in theorem 3, we give a class of dual resolution graphs and divisors y. A padic local monodromy theorem annals of mathematics. We then apply these results to hypergeometric differential equation of type nfn. The proof of this theorem is a wellknown argument in picardlefschetz theory, see for instance 37 5, 6 theorem 1. One may also consider inverse questions, that is, which groups appear as monodromy or picardvessiot groups.
Needless to say, that conjecture was the starting point of the work presented here. There is a function whose monodromy group is free of rank two and whose riemann surface has a single infinite spiral ramp two over of its three singular points. Certainly, the most prominent result on computing monodromy groups is the classical theorem of serre cf. We discuss the history of the monodromy theorem, starting from. In this section, we present some techniques that can be used to factorize the monodromy of certain milnor open books. The paper is concluded with a discussion in section 5. 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.
Big monodromy theorem for abelian varieties over nitely. Moreov er, since the action is assumed to be free outside the. The author apologises for all errors, unclarities, omissions of. Local monodromy of drinfeld modules and tmotives eth zurich. Pdf periodicity and the monodromy theorem monica nicolau. The local monodromy elements depend on the choice of the system of paths. Nonarchimedean uniformization and monodromy pairing. Complex analysis maharshi dayanand university, rohtak. If you want to get a hold directly of the associated sheaf, its given. We also strengthen falks result, by showing that the 2complex modeled on the arvola presentation is homotopy equivalent to c2 na. 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. May 01, 2017 in zung gave a proof of the monodromy theorem based on the fact that all focusfocus singularities of the same complexity are semilocally c 0equivalent and on the existence of a unique hamiltonian s 1 action as described in the following result. An algebraic differential equation on u is by defi nition a pair m, v consisting of a coherent sheaf m on u with an integrable connec tion the existence of v implies that m is, in fact, locally free. Freely comaclaurin, trivial isometries for a tate, reducible monodromy h.
Let f,d be a function element which admits unrestricted continuation in the simply connected region g. The monodromy groups of lisse sheaves and overconvergent f. We strengthen these results in our main theorem as follows. Explore thousands of free applications across science, mathematics, engineering, technology, business, art, finance, social sciences, and more. In complex analysis, the monodromy theorem is an important result about analytic continuation. Jan 19, 2001 therefore we use numerical schemes, instead, to determine the monodromy matrix x t 0. Relative minimality is also a necessary assumption in theorem 1. In section 4 we revisit the symmetry approach to monodromy presented in the works 30,40, and link it to the rotation number 15. Before stating and proving the monodromy theorem, we need two lemmas and a definition.
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