scholarly journals Periods for holomorphic maps via Lefschetz numbers

2005 ◽  
Vol 2005 (6) ◽  
pp. 575-579 ◽  
Author(s):  
Jaume Llibre ◽  
Michael Todd
Keyword(s):  
Fixed Points ◽  
Homology Group ◽  
Main Tool ◽  
Lefschetz Number ◽  
Complex Manifolds ◽  
Holomorphic Maps ◽  
Lefschetz Numbers

We characterise the set of fixed points of a class of holomorphic maps on complex manifolds with a prescribed homology. Our main tool is the Lefschetz number and the action of maps on the first homology group.

Fixed Points of Automorphisms of Compact Riemann Surfaces

1970 ◽  
Vol 22 (5) ◽  
pp. 922-932 ◽  
Author(s):  
M. J. Moore
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Riemann Surfaces ◽  
Homology Group ◽  
Group Of Automorphisms ◽  
Quartic Curve ◽  
Lefschetz Fixed Point Formula ◽  
Compact Riemann Surfaces ◽  
Klein's Quartic Curve ◽  
Fundamental Paper

In his fundamental paper [3], Hurwitz showed that the order of a group of biholomorphic transformations of a compact Riemann surface S into itself is bounded above by 84(g – 1) when S has genus g ≧ 2. This bound on the group of automorphisms (as we shall call the biholomorphic self-transformations) is attained for Klein's quartic curve of genus 3 [4] and, from this, Macbeath [7] deduced that the Hurwitz bound is attained for infinitely many values of g.After genus 3, the next smallest genus for which the bound is attained is the case g = 7. The equations of such a curve of genus 7 were determined by Macbeath [8] who also gave the equations of the transformations. The equations of these transformations were found by using the Lefschetz fixed point formula. If the number of fixed points of each element of a group of automorphisms is known, then the Lefschetz fixed point formula may be applied to deduce the character of the representation given by the group acting on the first homology group of the surface.

scholarly journals On holomorphic maps with only fold singularities

2001 ◽  
Vol 164 ◽  
pp. 147-184
Author(s):  
Yoshifumi Ando
Keyword(s):  
Homotopy Class ◽  
Homotopy Equivalent ◽  
Line Bundles ◽  
Jet Space ◽  
Complex Manifolds ◽  
Holomorphic Maps ◽  
Regular Map ◽  
Holomorphic Map ◽  
Fold Map ◽  
Tangent Bundles

Let f : N ≡ P be a holomorphic map between n-dimensional complex manifolds which has only fold singularities. Such a map is called a holomorphic fold map. In the complex 2-jet space J2(n,n;C), let Ω10 denote the space consisting of all 2-jets of regular map germs and fold map germs. In this paper we prove that Ω10 is homotopy equivalent to SU(n + 1). By using this result we prove that if the tangent bundles TN and TP are equipped with SU(n)-structures in addition, then a holomorphic fold map f canonically determines the homotopy class of an SU(n + 1)-bundle map of TN ⊕ θN to TP⊕ θP, where θN and θP are the trivial line bundles.

scholarly journals Fixed points and periodic points of semiflows of holomorphic maps

2003 ◽  
Vol 2003 (4) ◽  
pp. 217-260
Author(s):  
Edoardo Vesentini
Keyword(s):  
Banach Space ◽  
Fixed Points ◽  
Complex Banach Space ◽  
Periodic Points ◽  
General Question ◽  
Open Unit ◽  
Holomorphic Maps ◽  
Open Unit Ball ◽  
Spin Factor ◽  
Cartan Factor

Letϕbe a semiflow of holomorphic maps of a bounded domainDin a complex Banach space. The general question arises under which conditions the existence of a periodic orbit ofϕimplies thatϕitself is periodic. An answer is provided, in the first part of this paper, in the case in whichDis the open unit ball of aJ∗-algebra andϕacts isometrically. More precise results are provided when theJ∗-algebra is a Cartan factor of type one or a spin factor. The second part of this paper deals essentially with the discrete semiflowϕgenerated by the iterates of a holomorphic map. It investigates how the existence of fixed points determines the asymptotic behaviour of the semiflow. Some of these results are extended to continuous semiflows.

scholarly journals RIGIDITY OF HOLOMORPHIC MAPS BETWEEN FIBER SPACES

2014 ◽  
Vol 25 (01) ◽  
pp. 1450006 ◽  
Author(s):  
GAUTAM BHARALI ◽  
INDRANIL BISWAS
Keyword(s):  
Riemann Surface ◽  
Special Form ◽  
General Class ◽  
Specific Information ◽  
Complex Manifolds ◽  
Holomorphic Maps ◽  
Fiber Space ◽  
Rigidity Result ◽  
Holomorphic Map ◽  
Genus 2

In the study of holomorphic maps, the term "rigidity" refers to certain types of results that give us very specific information about a general class of holomorphic maps owing to the geometry of their domains or target spaces. Under this theme, we begin by studying when, given two compact connected complex manifolds X and Y, a degree-one holomorphic map f : Y → X is a biholomorphism. Given that the real manifolds underlying X and Y are diffeomorphic, we provide a condition under which f is a biholomorphism. Using this result, we deduce a rigidity result for holomorphic self-maps of the total space of a holomorphic fiber space. Lastly, we consider products X = X1 × X2 and Y = Y1 × Y2 of compact connected complex manifolds. When X1 is a Riemann surface of genus ≥ 2, we show that any non-constant holomorphic map F : Y → X is of a special form.

scholarly journals Lefschetz Numbers for C*-Algebras

2011 ◽  
Vol 54 (1) ◽  
pp. 82-99 ◽  
Author(s):  
Heath Emerson
Keyword(s):  
Classical Case ◽  
Lefschetz Number ◽  
Poincaré Duality ◽  
Poincare Duality ◽  
C Algebras ◽  
Lefschetz Numbers ◽  
The Matrix ◽  
Lefschetz Type ◽  
Polynomial Formula ◽  
Infinite Set

AbstractUsing Poincaré duality, we obtain a formula of Lefschetz type that computes the Lefschetz number of an endomorphism of a separable nuclear C*-algebra satisfying Poincaré duality and the Kunneth theorem. (The Lefschetz number of an endomorphism is the graded trace of the induced map on K-theory tensored with ℂ, as in the classical case.) We then examine endomorphisms of Cuntz–Krieger algebras OA. An endomorphism has an invariant, which is a permutation of an infinite set, and the contracting and expanding behavior of this permutation describes the Lefschetz number of the endomorphism. Using this description, we derive a closed polynomial formula for the Lefschetz number depending on the matrix A and the presentation of the endomorphism.

Families of smooth hypersurfaces on certain compact homogeneous complex manifolds

1983 ◽  
Vol 93 (2) ◽  
pp. 315-321
Author(s):  
Ciprian Borcea
Keyword(s):  
Complex Manifold ◽  
Betti Number ◽  
Homology Group ◽  
Cell Decomposition ◽  
Homogeneous Manifold ◽  
Holomorphic Maps ◽  
Algebraic Sets ◽  
Homogeneous Complex ◽  
Homogeneous Complex Manifold ◽  
Lower Dimensional

Let X be a compact connected homogeneous complex manifold, which is Kāhlerian and has the second Betti number equal to one: b2(X) = 1; dimcX ≥ 3.It is known that these conditions imply the following: X is a projective-rational homogeneous manifold (see (3)); X has an ‘algebraic cell-decomposition’: the 2s-dimensional closed cells are s-dimensional irreducible algebraic sets in X and they form a basis for the 2s-homology group of X, s = 1, 2, …, dimcX (see (1)); there are no holomorphic maps of X on lower dimensional (normal) analytic spaces except constants (see (9)).

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