Notes for a Talk on Resolvent Degree
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Arxiv:1509.06617V2 [Math.GT] 12 May 2016 Point a Maps
LOCAL MONODROMY OF BRANCHED COVERS AND DIMENSION OF THE BRANCH SET MARTINA AALTONEN AND PEKKA PANKKA Abstract. We show that, if the local dimension of the branch set of a discrete and open mapping f : M → N between n-manifolds is less than (n − 2) at a point y of the image of the branch set fBf , then the local monodromy of f at y is perfect. In particular, for generalized branched covers between n-manifolds the dimension of fBf is exactly (n − 2) at the points of abelian local monodromy. As an application, we show that a generalized branched covering f : M → N of local multiplicity at most three between n-manifolds is either a covering or fBf has local dimension (n − 2). 1. Introduction A continuous mapping f : X → Y between topological spaces is a (gener- alized) branched cover if f is discrete and open, that is, pre-image f −1(y) of a point y ∈ Y is a discrete set and f maps open sets to open sets. The name branched cover for these maps stems from the Chernavskii–Väisälä theo- rem [5,18]: the branch set of a branched cover between (generalized) manifolds has codimension at least two. It is an easy consequence of the Chernavskii– Väisälä theorem that branched covers between (generalized) manifolds are, at least locally, completions of covering maps. We follow here the typical naming convention in this context and say that a point x ∈ X is a branch point of f if f is not a local homeomorphism at x. The branch set of the mapping f, i.e. -
On the Topology of Simply-Connected Algebraic Surfaces by Richard Mandelbaum and Boris Moishezon
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 260, Number 1, July 1980 ON THE TOPOLOGY OF SIMPLY-CONNECTED ALGEBRAIC SURFACES BY RICHARD MANDELBAUM AND BORIS MOISHEZON Abstract. Suppose A" is a smooth simply-connected compact 4-manifold. Let P = CP2 and Q = -CP2 be the complex projective plane with orientation oppo- site to the usual. We shall say that X is completely decomposable if there exist integers a, b such that X is diffeomorphic to aP %bQ. By a result of Wall [Wl] there always exists an integer k such that X Jf (A: + \)P 8 kQ is completely decomposable. If X # P is completely decomposable we shall say that X is almost completely decomposable. In [MM] we demonstrated that any nonsingular hypersurface of CP3 is almost completely decomposable. In this paper we generalize this result in two directions as follows: Theorem 3.5. Suppose W is a simply-connected nonsingular complex projective 3-fold. Then there exists an integer m„ > 1 such that any hypersurface section Vm of W of degree m > mg which is nonsingular will be almost completely decomposable. Theorem 5.3. Let V be a nonsingular complex algebraic surface which is a complete intersection. Then V is almost completely decomposable. Introduction. Suppose A" is a simply-connected compact 4-manifold. Let P = CP2 and Q — — CP2 be the complex projective plane with orientation opposite to the usual. We shall say that X is completely decomposable if there exist integers a, b such that X s» aP # bQ. (Read '» ' as 'is diffeomorphic to'.) By a result of Wall [Wl], [W2] there always exists an integer k such that X # (k + l)P # kQ is completely decomposable. -
A Barth-Type Theorem for Branched Coverings of Projective Space
lilm llulm Math. Ann. 249, 153-162 (1980) Amlm © by Springer-Verlag1 980 A Barth-Type Theorem for Branched Coverings of Projective Space Robert Lazarsfeld Department of Mathematics, Brown University, Providence, RI 02912, USA Introduction Let X be a non-singular connected complex projective variety of dimension n. In 1970, Barth [B1] discovered that if X admits an embedding X"~IP "+~ of codimension e, then the restriction mappings H~(IP"+~,~)--+Ht(X, (E) are isomor- phisms for i < n - e . Our main result is an analogue of Barth's theorem for branched coverings of projective space: Theorem 1. Let f :X" ~ IW be a finite mapping of degree d. Then the induced maps f* : H~(1W,~ ) ~ Hi(X, ~) are isomorphisms for i < n + 1 - d. Observe that the conclusion is vacuous for d > n + 1. On the other hand, as the degree d becomes small compared to n, one obtains progressively stronger topological obstructions to expressing a variety as a d-sheeted covering of IP". The proof of the theorem relies on a basic construction which clarifies somewhat the connection between subvarieties and branched coverings. Canonically associated to a finite morphism f : X " ~ F " of degree d, there exists a vector bundle E~IW of rank d - 1 having the property that f factors through an embedding of X in the total space of E (Sect. 1). An important fact about coverings of projective space is that these bundles are always ample. This leads one to consider quite generally a smooth n-dimensional projective variety Y, an ample vector bundle E ~ Y of rank e, and a non-singular projective variety X of dimension n embedded in the total space of E : X ~ , E Y. -
Non-Manifold Monodromy Spaces of Branched Coverings Between Manifolds
NON-MANIFOLD MONODROMY SPACES OF BRANCHED COVERINGS BETWEEN MANIFOLDS MARTINA AALTONEN 1 2 Abstract. By a construction of Berstein and Edmonds every proper branched cover f between manifolds is a factor of a branched covering orbit map from a locally connected and locally compact Hausdorff space called the monodromy space of f to the target manifold. For proper branched covers between 2-manifolds the monodromy space is known to be a manifold. We show that this does not generalize to dimension 3 by constructing a self-map of the 3-sphere for which the monodromy space is not a locally contractible space. 1. Introduction A map f : X ! Y between topological spaces is a branched covering, if f is open, continuous and discrete map.The branch set Bf ⊂ X of f is the set of points in X for which f fails to be a local homeomorphism. The map f is proper, if the pre-image in f of every compact set is compact. Let f : X ! Y be a proper branched covering between manifolds. Then the codimension of Bf ⊂ X is at least two by Väisälä [14] and the restriction map 0 −1 −1 f := fjX n f (f(Bf )): X n f (f(Bf )) ! Y n f(Bf ) is a covering map between open connected manifolds, see Church and Hem- mingsen [6]. Thus there exists, by classical theory of covering maps, an open 0 manifold Xf and a commutative diagram of proper branched covering maps 0 Xf p0 q0 ~ f X / Y arXiv:1612.00665v1 [math.GT] 2 Dec 2016 0 0 −1 0 0 where p : Xf ! X nf (f(Bf )) and q : Xf ! Y nf(Bf ) are normal covering 0 0 maps and the deck-transformation group of the covering map q : Xf ! 0 Y n f(Bf ) is isomorphic to the monodromy group of f : Further, by Berstein and Edmonds [3], there exists a locally compact 0 and locally connected second countable Hausdorff space Xf ⊃ Xf so that 0 0 0 Xf nXf does not locally separate Xf and the maps p and q extend to proper Date: October 10, 2018. -
Introduction to Complex Analysis Michael Taylor
Introduction to Complex Analysis Michael Taylor 1 2 Contents Chapter 1. Basic calculus in the complex domain 0. Complex numbers, power series, and exponentials 1. Holomorphic functions, derivatives, and path integrals 2. Holomorphic functions defined by power series 3. Exponential and trigonometric functions: Euler's formula 4. Square roots, logs, and other inverse functions I. π2 is irrational Chapter 2. Going deeper { the Cauchy integral theorem and consequences 5. The Cauchy integral theorem and the Cauchy integral formula 6. The maximum principle, Liouville's theorem, and the fundamental theorem of al- gebra 7. Harmonic functions on planar regions 8. Morera's theorem, the Schwarz reflection principle, and Goursat's theorem 9. Infinite products 10. Uniqueness and analytic continuation 11. Singularities 12. Laurent series C. Green's theorem F. The fundamental theorem of algebra (elementary proof) L. Absolutely convergent series Chapter 3. Fourier analysis and complex function theory 13. Fourier series and the Poisson integral 14. Fourier transforms 15. Laplace transforms and Mellin transforms H. Inner product spaces N. The matrix exponential G. The Weierstrass and Runge approximation theorems Chapter 4. Residue calculus, the argument principle, and two very special functions 16. Residue calculus 17. The argument principle 18. The Gamma function 19. The Riemann zeta function and the prime number theorem J. Euler's constant S. Hadamard's factorization theorem 3 Chapter 5. Conformal maps and geometrical aspects of complex function the- ory 20. Conformal maps 21. Normal families 22. The Riemann sphere (and other Riemann surfaces) 23. The Riemann mapping theorem 24. Boundary behavior of conformal maps 25. Covering maps 26. -
On Algebraic Functions
On Algebraic Functions N.D. Bagis Stenimahou 5 Edessa Pella 58200, Greece [email protected] Abstract In this article we consider functions with Moebius-periodic rational coefficients. These functions under some conditions take algebraic values and can be recovered by theta functions and the Dedekind eta function. Special cases are the elliptic singular moduli, the Rogers- Ramanujan continued fraction, Eisenstein series and functions asso- ciated with Jacobi symbol coefficients. Keywords: Theta functions; Algebraic functions; Special functions; Peri- odicity; 1 Known Results on Algebraic Functions The elliptic singular moduli kr is the solution x of the equation 1 1 2 2F1 , ;1;1 x 2 2 − = √r (1) 1 1 2 2F1 2 , 2 ; 1; x where 1 2 π/2 ∞ 1 1 2 2 n 2n 2 2 dφ 2F1 , ; 1; x = 2 x = K(x)= (2) 2 2 (n!) π π 2 n=0 Z0 1 x2 sin (φ) X − q The 5th degree modular equation which connects k25r and kr is (see [13]): 5/3 1/3 krk25r + kr′ k25′ r +2 (krk25rkr′ k25′ r) = 1 (3) arXiv:1305.1591v3 [math.GM] 27 Mar 2014 The problem of solving (3) and find k25r reduces to that solving the depressed equation after named by Hermite (see [3]): u6 v6 +5u2v2(u2 v2)+4uv(1 u4v4) = 0 (4) − − − 1/4 1/4 where u = kr and v = k25r . The function kr is also connected to theta functions from the relations 2 θ (q) ∞ 2 ∞ 2 k = 2 , where θ (q)= θ = q(n+1/2) and θ (q)= θ = qn r θ2(q) 2 2 3 3 3 n= n= X−∞ X−∞ (5) 1 π√r q = e− . -
Symmetries and Polynomials
Symmetries and Polynomials Aaron Landesman and Apurva Nakade June 30, 2018 Introduction In this class we’ll learn how to solve a cubic. We’ll also sketch how to solve a quartic. We’ll explore the connections between these solutions and group theory. Towards the end, we’ll hint towards the deep connection between polynomials and groups via Galois Theory. This connection allows one to convert the classical question of whether a general quintic can be solved by radicals to a group theory problem, which can be relatively easily answered. The 5 day outline is as follows: 1. The first two days are dedicated to studying polynomials. On the first day, we’ll study a simple invariant associated to every polynomial, the discriminant. 2. On the second day, we’ll solve the cubic equation by a method motivated by Galois theory. 3. Days 3 and 4 are a “practical” introduction to group theory. On day 3 we’ll learn about groups as symmetries of geometric objects. 4. On day 4, we’ll learn about the commutator subgroup of a group. 5. Finally, on the last day we’ll connect the two theories and explain the Galois corre- spondence for the cubic and the quartic. There are plenty of optional problems along the way for those who wish to explore more. Tricky optional problems are marked with ∗, and especially tricky optional problems are marked with ∗∗. 1 1 The Discriminant Today we’ll introduce the discriminant of a polynomial. The discriminant of a polynomial P is another polynomial Q which tells you whether P has any repeated roots over C. -
THE ÉTALE FUNDAMENTAL GROUP of an ELLIPTIC CURVE Contents Introduction 1 1. Fundamental Group 2 2. Curves 3 3. Elliptic Curves
THE ETALE´ FUNDAMENTAL GROUP OF AN ELLIPTIC CURVE ARNAB KUNDU Abstract. We first look at the fundamental group, and try to find a suitable definition that can be simulated for algebraic varieties. In the next couple of sections we give a summary of the theory of Elliptic curves and ´etalemaps. In the last section we define the finite ´etalesite and define the ´etalefundamental group. Finally we compute the ´etalefundamental group of an Elliptic curve. Contents Introduction 1 1. Fundamental Group 2 2. Curves 3 3. Elliptic Curves 5 4. Etale´ Maps 7 5. Etale´ Fundamental Group 10 Acknowledgments 15 References 15 Introduction The fundamental group is a topological invariant that measures how well con- nected a space is. For a well behaved topological space there is a space called the universal cover whose group of deck transformation gives the fundamental group. This nice topological construction can be used to classify all finite covers of the space; in fact there is a correspondence between finite quotients of the fundamental group and finite covers of the space. Varieties over complex numbers have a natu- ral complex manifold structure and hence along with the algebraic structure they reveal topological structure as well. It is natural to ask if the fundamental group in the case of varieties over complex number can be computed algebraically. The answer is that the finite covers which can be defined algebraically reveal almost all of the fundamental group. The strength of this approach is in the fact that this could be generalized to other fields, whatever their characteristic may be. -
Ample Vector Bundles and Branched Coverings, II
Ample Vector Bundles and Branched Coverings, II Thomas Peternell Andrew J. Sommese November 5, 2002 Introduction To a degree d covering f : X → Y of projective manifolds there is naturally attached a holomorphic rank (d−1)−vector bundle Ef , just by dualizing f∗(OX ) and dividing by the trivial factor. This bundle reflects interesting properties of f or X. It has certain positivity properties: it is nef on the generic curve, as observed in [PS00], the first part to this paper, and even more, it is nef on every curve not contained in the branch locus of f (Lazarsfeld). Often it has a lot of sections ([PS00]). For special Y much more is true: if Y = Pn, then E is ample and spanned (Lazarsfeld); the same being true for some rational homogeneous manifolds (Kim, Manivel) and conjecturally for all (as long as b2 = 1). In the first part to this paper we investigated Fano manifolds and proved nefness in certain cases (del Pezzo manifolds of higher degrees). We suspected that we might have ampleness as well. The first result in this paper is that E is not always ample (2.1): we provide a triple covering over the Fano threefold V5, appearing as a linear section of the Grassman- nian G(2, 5) ⊂ P9, for which E is not ample. However the only curves on which E fails to be ample, are lines (not necessarily in the branch locus). Then (2.2) we give a criterion for ampleness, and as a consequence we see that given a covering X → V5, then E is ample if and only if E is ample on every line. -
Homeomorphism of S^ 1 and Factorization
HOMEOMORPHISMS OF S1 AND FACTORIZATION MARK DALTHORP AND DOUG PICKRELL Abstract. For each n> 0 there is a one complex parameter family of home- omorphisms of the circle consisting of linear fractional transformations ‘con- jugated by z → zn’. We show that these families are free of relations, which determines the structure of ‘the group of homeomorphisms of finite type’. We next consider factorization for more robust groups of homeomorphisms. We refer to this as root subgroup factorization (because the factors correspond to root subgroups). We are especially interested in how root subgroup fac- torization is related to triangular factorization (i.e. conformal welding), and correspondences between smoothness properties of the homeomorphisms and decay properties of the root subgroup parameters. This leads to interesting comparisons with Fourier series and the theory of Verblunsky coefficients. 0. Introduction In this paper we consider the question of whether it is possible to factor an orientation preserving homeomorphism of the circle, belonging to a given group, as a composition of ‘linear fractional transformations conjugated by z zn’. What we mean by factorization depends on the group of homeomorphisms we→are considering. In the introduction we will start with the simplest classes of homeomorphisms and build up. For algebraic homeomorphisms, factorization is to be understood in terms of generators and relations. For less regular homeomorphisms factorization involves limits and ordering, and in particular is highly asymmetric with respect to inversion. 0.1. Diffeomorphisms of Finite Type. Given a positive integer n and wn ∆ := w C : w < 1 , define a function φ : S1 S1 by ∈ { ∈ | | } n → n 1/n (1+w ¯nz− ) (0.1) φn(wn; z) := z n 1/n , z =1 (1 + wnz ) | | arXiv:1408.5402v4 [math.GT] 22 Jun 2019 1 It is straightforward to check that φn Diff(S ), the group of orientation pre- 1 1∈ serving diffeomorphisms of S , and φ− (z) = φ ( w ; z). -
On Generalized Integrals and Ramanujan-Jacobi Special Functions
On Generalized Integrals and Ramanujan-Jacobi Special Functions N.D.Bagis Department of Informatics, Aristotele University Thessaloniki, Greece [email protected] Keywords: Elliptic Functions; Ramanujan; Special Functions; Continued Fractions; Generalization; Evaluations Abstract In this article we consider new generalized functions for evaluating integrals and roots of functions. The construction of these general- ized functions is based on Rogers-Ramanujan continued fraction, the Ramanujan-Dedekind eta, the elliptic singular modulus and other similar functions. We also provide modular equations of these new generalized functions and remark some interesting properties. 1 Introduction Let ∞ η(τ)= eπiτ/12 (1 e2πinτ ) (1) − n=1 Y denotes the Dedekind eta function which is defined in the upper half complex plane. It not defined for real τ. Let for q < 1 the Ramanujan eta function be | | ∞ arXiv:1309.7247v3 [math.GM] 15 Nov 2015 f( q)= (1 qn). (2) − − n=1 Y The following evaluation holds (see [12]): 1/3 1/2 1/24 1/12 1/3 1/2 f( q)=2 π− q− k k∗ K(k) (3) − 2 where k = kr is the elliptic singular modulus, k∗ = √1 k and K(x) is the complete elliptic integral of the first kind. − The Rogers-Ramanujan continued fraction is (see [8],[9],[14]): q1/5 q1 q2 q3 R(q) := ... (4) 1+ 1+ 1+ 1+ 1 which have first derivative (see [5]): 1 5/6 4 6 5 5 R′(q)=5− q− f( q) R(q) R(q) 11 R(q) (5) − − − − and also we can write p dR(q) 1 1/3 2/3 6 5 5 =5− 2 (kk∗)− R(q) R(q) 11 R(q) . -
Differential Algebra and Liouville's Theorem
Math 4010/5530 Elementary Functions and Liouville's Theorem April 2016 Definition: Let R be a commutative ring with 1. We call a map @ : R ! R a derivation if @(a+b) = @(a)+@(b) and @(ab) = @(a)b + a@(b) for all a; b 2 R. A ring equipped with a particular derivative is called a differential ring. For convenience, we write @(a) = a0 (just like in calculus). If R is a differential ring and an integral domain, it is a differential integral domain. If R is a differential ring and a field, it is a differential field. Let R be a differential ring. • 10 = (1·1)0 = 10 ·1+1·10 so 10 = 10 +10 and so 0 = 10. Notice that @ is a group homomorphism (consider R as an abelian group under addition), so @(na) = n@(a) for all a 2 R and n 2 Z. Therefore, @(n1) = n@(1) = n0 = 0. Thus everything in the prime subring is a constant (i.e. has derivative zero). • Given a 2 R, notice that (a2)0 = (a · a)0 = a0a + aa0 = 2aa0. In fact, we can prove (using induction), that n n−1 0 a = na a for any n 2 Z≥0 (The power rule) • Let a 2 R× (a is a unit). Then 0 = 10 = (aa−1)0 = a0a−1 + a(a−1)0 so a(a−1)0 = −a−1a0. Dividing by a, we get that (a−1)0 = −a−2a0. Again, using induction, we have that an = nan−1 for all n 2 Z (if a−1 exists).