scholarly journals Deformations of a Holomorphic Map and Its Degeneracy Locus

2012 ◽  
Vol 35 (2) ◽  
pp. 253-277
Author(s):  
Madoka EBIHARA
Keyword(s):  
Holomorphic Map ◽  
Degeneracy Locus

scholarly journals Families of abelian surfaces with real multiplication over Hilbert modular surfaces

1982 ◽  
Vol 88 ◽  
pp. 17-53 ◽  
Author(s):  
G. van der Geer ◽  
K. Ueno
Keyword(s):  
Endomorphism Ring ◽  
Symplectic Group ◽  
Abelian Surface ◽  
Quadratic Relation ◽  
Holomorphic Map ◽  
Real Multiplication ◽  
Compact Complex Surfaces ◽  
Hilbert Modular Surfaces ◽  
Hilbert Modular ◽  
Abelian Functions

Around the beginning of this century G. Humbert ([9]) made a detailed study of the properties of compact complex surfaces which can be parametrized by singular abelian functions. A surface parametrized by singular abelian functions is the image under a holomorphic map of a singular abelian surface (i.e. an abelian surface whose endomorphism ring is larger than the ring of rational integers). Humbert showed that the periods of a singular abelian surface satisfy a quadratic relation with integral coefficients and he constructed an invariant D of such a relation with respect to the action of the integral symplectic group on the periods.

scholarly journals Type II degenerations of K3 surfaces

1985 ◽  
Vol 99 ◽  
pp. 11-30 ◽  
Author(s):  
Shigeyuki Kondo
Keyword(s):  
Number Field ◽  
Complex Manifold ◽  
Complex Number ◽  
Three Dimensional ◽  
K3 Surfaces ◽  
Type Ii ◽  
Holomorphic Map ◽  
Canonical Class ◽  
Proper Holomorphic Map ◽  
Complex Number Field

A degeneration of K3 surfaces (over the complex number field) is a proper holomorphic map π: X→Δ from a three dimensional complex manifold to a disc, such that, for t ≠ 0, the fibres Xt = π-1(t) are smooth K3 surfaces (i.e. surfaces Xt with trivial canonical class KXt = 0 and dim H1(Xt, Oxt) = 0).

scholarly journals Moduli Space of Brody Curves, Energy and Mean Dimension

2008 ◽  
Vol 192 ◽  
pp. 27-58 ◽  
Author(s):  
Masaki Tsukamoto
Keyword(s):  
Complex Plane ◽  
Moduli Space ◽  
Hermitian Manifold ◽  
Value Distribution ◽  
Holomorphic Map ◽  
Infinite Dimensional ◽  
Mean Dimension ◽  
The Mean ◽  
Moduli Theory ◽  
Bounded Derivative

AbstractA Brody curve is a holomorphic map from the complex plane ℂ to a Hermitian manifold with bounded derivative. In this paper we study the value distribution of Brody curves from the viewpoint of moduli theory. The moduli space of Brody curves becomes infinite dimensional in general, and we study its “mean dimension”. We introduce the notion of “mean energy” and show that this can be used to estimate the mean dimension.

scholarly journals On the size of linearization domains

2008 ◽  
Vol 145 (2) ◽  
pp. 443-456
Author(s):  
XAVIER BUFF ◽  
CARSTEN L. PETERSEN
Keyword(s):  
Power Series ◽  
Formal Power Series ◽  
Radius Of Convergence ◽  
Holomorphic Map ◽  
Root Of Unity ◽  
Rotation Numbers ◽  
Formal Power

AbstractAssume $f{:}\,U\subset \C\to \C$ is a holomorphic map fixing 0 with derivative λ, where 0 < |λ| ≤ 1. If λ is not a root of unity, there is a formal power series φf(z) = z + ${\cal O}$(z2) such that φf(λ z) = f(φf(z)). This power series is unique and we denote by Rconv(f) ∈ [0,+∞] its radius of convergence. We denote by Rgeom(f) the largest radius r ∈ [0, Rconv(f)] such that φf(D(0,r)) ⊂ U. In this paper, we present new elementary techniques for studying the maps f ↦ Rconv(f) and f ↦ Rgeom(f). Contrary to previous approaches, our techniques do not involve studying the arithmetical properties of rotation numbers.

scholarly journals Diagrams and Essential Sets for Signed Permutations

10.37236/8106 ◽  
2018 ◽  
Vol 25 (3) ◽  
Author(s):  
David Anderson
Keyword(s):  
Schubert Variety ◽  
Signed Permutation ◽  
Signed Permutations ◽  
Degeneracy Locus ◽  
Essential Set ◽  
Diagrammatic Method ◽  
Essential Sets ◽  
Rank Conditions ◽  
Theoretic Version

We introduce diagrams and essential sets for signed permutations, extending the analogous notions for ordinary permutations.  In particular, we show that the essential set provides a minimal list of rank conditions defining the Schubert variety or degeneracy locus corresponding to a signed permutation.  Our essential set is in bijection with the poset-theoretic version defined by Reiner, Woo, and Yong, and thus gives an explicit, diagrammatic method for computing the latter.

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.

The multiplicity of a holomorphic map

1966 ◽  
Vol 2 (1) ◽  
pp. 15-58 ◽  
Author(s):  
Wilhelm Stoll
Keyword(s):  
Holomorphic Map

scholarly journals ON ALGEBRAIC HYPERBOLICITY OF LOG VARIETIES

2004 ◽  
Vol 06 (04) ◽  
pp. 513-559 ◽  
Author(s):  
XI CHEN
Keyword(s):  
Algebraic Curve ◽  
Holomorphic Map

Kobayashi conjecture says that every holomorphic map [Formula: see text] is constant for a very general hypersurface D⊂ℙn of degree deg D≥2n+1. As a corollary of our main theorem, we show that f is constant if [Formula: see text] is contained in an algebraic curve.

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.

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