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In this paper, we investigate the existence of singular directions, singular radii and indirect singular points of holomorphic curves in P n ⁢ ( ℂ ) {P^{n}(\mathbb{C})} . [ABSTRACT FROM AUTHOR]

We propose a coordinate-free approach to study the holomorphic maps between the real hyperquadrics in complex projective spaces. It is based on a notion of orthogonality on the projective spaces induced by the Hermitian structures that define the hyperquadrics. There are various kinds of special linear subspaces associated to this orthogonality which are well respected by the relevant holomorphic maps and we obtain rigidity theorems by analyzing the images of these linear subspaces, together with techniques in projective geometry. Our method allows us to recover and generalize a number of well-known results in the field with simpler arguments. [ABSTRACT FROM AUTHOR]

Let 핌m ⊂ ℂm be a unit ball centered at the origin and let ℙn be an n-dimensional complex projective space with the metric Eℙn. Moreover, let φ: [0, 1) → (0,∞) be a smoothly increasing function. A holomorphic mapping f : 핌m → ℙn is called φ-normal if (φ||z||))−1(Eℙn(f(z), df (z))(ξ)) is bounded above for z ∈ 핌m and ξ ∈ ℂm such that ||ξ|| = 1, where df (z) is a map from Tz(핌m) into Tf(z) (ℙn) induced by f. For n = 1, f is called a φ-normal function. We present an extension of Lappan's five-valued theorem to the class of φ-normal functions. [ABSTRACT FROM AUTHOR]

In this paper, the authors discuss a generalization of Lappan's theorem to higher dimensional complex projective space and get the following result: Let f be a holomorphic mapping of Δ into ℙn(ℂ), and let H1, ⋯, Hq be hyperplanes in general position in ℙn(ℂ). Assume that sup { (1 − | z | 2) f ♯ (z) : z ∈ ∪ j = 1 q f − 1 (H j) } < ∞ , if q ≥ 2n2 + 3, then f is normal. [ABSTRACT FROM AUTHOR]

We prove a uniqueness theorem for linearly nondegenerate holomorphic mappings from an annulus to a complex projective space ℙn(ℂ) with different multiple values and a general condition on the intersections of the inverse images of the hyperplanes. [ABSTRACT FROM AUTHOR]

In this paper, we extend Montel's normal criterion to a family F of holomorphic mappings from a domain D of the complex plane C into a complex projective space P n (C) of dimension n. Furthermore, Lappan's five-valued theorem is extended to holomorphic curves. [ABSTRACT FROM AUTHOR]

Due to fundamental results of Igusa and Mumford the N = 2g-1(2g + 1) theta constants of first kind ∑ n integral exp π i (Z[n + a/2] + 2b'(n + a/2)), a,b integral. define for each genus g an injective holomorphic map of the Satake compactification Xg (4,8) = Hg/Γg [4,8] into the projective space PN-1. Moreover, this map is biholomorphic onto the image outside the Satake boundary. It is not biholomorphic on the whole in the cases g ≥≥ 6. Igusa also proved that in the cases g ≤ 2 this map is biholomorphic onto the image. In this paper we extend this result to the case g = 3. So we show that the theta map X3(4, 8) → ℙ35 is biholomorphic onto the image. This is equivalent to the statement that the image is a normal subvariety of ℙ35. [ABSTRACT FROM AUTHOR]

We present a new list of irreducible components for the space of k-dimensional holomorphic foliations on Pn, n ≥ 3, k ≥ 2. They are associated to pull-back of dimension one foliations on Pn-k+1 by non-linear rational maps. [ABSTRACT FROM AUTHOR]

In this paper we consider the order of holomorphic curves with maximal deficiency sum in the complex plane. The purpose of this paper is to weaken the condition treated in the paper. [ABSTRACT FROM AUTHOR]

In this paper we study harmonic maps from super Riemann surfaces in complex projective spaces and projective spaces associated with the super skew-field D . In both cases, we develop the theory of Gauß transforms and study the notion of isotropy, in particular its relation to holomorphic differentials on the super Riemann surface. Moreover, we give a definition of finite type harmonic maps for a special class of maps into C P n | n + 1 and thus obtain a classification for certain harmonic super tori. Furthermore, we investigate the equations satisfied by the underlying objects and give an example of a harmonic super torus in D P 2 whose underlying map is not harmonic. [ABSTRACT FROM AUTHOR]

We study compact invariant sets for holomorphic foliations on Stein manifold. As application, we show some dynamical properties concerning minimal sets (with singularities) of foliations and real analytic Levi-flat hypersurfaces in projective spaces. [ABSTRACT FROM AUTHOR]

We consider the set K( n, c, xdy − ydx) of codimension one holomorphic foliations on P, n ≥ 3, with normal Chern class c, and with a compact, connected Kupka set of radial transversal type. We will prove that foliations in this set, have a rational first integral and define an irreducible component of the space of foliations. [ABSTRACT FROM AUTHOR]

We show that the efficiency of a natural pairing between certain projectively invariant Hardy spaces on dual strongly ℂ-convex real hypersurfaces in complex projective space is measured by the norm of the corresponding Leray transform. [ABSTRACT FROM AUTHOR]

We describe the possible noncommutative deformations of complex projective three-space by exhibiting the Calabi–Yau algebras that serve as their homogeneous coordinate rings. We prove that the space parametrizing such deformations has exactly six irreducible components, and we give explicit presentations for the generic members of each family in terms of generators and relations. The proof uses deformation quantization to reduce the problem to a similar classification of unimodular quadratic Poisson structures in four dimensions, which we extract from Cerveau and Lins Neto's classification of degree-two foliations on projective space. Corresponding to the “exceptional” component in their classification is a quantization of the third symmetric power of the projective line that supports bimodule quantizations of the classical Schwarzenberger bundles. [ABSTRACT FROM AUTHOR]

In the paper, we study the existence of holomorphic isometric immersions from nonhomogeneous Kähler–Einstein manifolds into infinite dimensional complex projective space. It can also be regarded as an application of explicit solutions of complex Monge–Ampère equations on some pseudoconvex domains. [ABSTRACT FROM AUTHOR]

The possibility of reversion of the inequality in the Second Main Theorem of Cartan in the theory of holomorphic curves in projective space is discussed. A new version of this theorem is proved that becomes an asymptotic equality for a class of holomorphic curves defined by solutions of linear differential equations. [ABSTRACT FROM AUTHOR]

We show that under mild boundary conditions the moduli space of non-compact curves on a complex surface is (locally) an analytic subset of a ball in a Banach manifold, defined byfinitelymany holomorphic functions. [ABSTRACT FROM PUBLISHER]

A complex spaceXis in class 𝒬Gif it is a semistable quotient of the complement to an analytic subset of a Stein manifold by a holomorphic action of a reductive complex Lie groupG. It is shown that every pseudoconvex unramified domain overXis also in 𝒬G. [ABSTRACT FROM PUBLISHER]

Abstract: A holomorphy potential is a complex valued function whose complex gradient, with respect to some Kähler metric, is a holomorphic vector field. Given holomorphic vector fields on a compact complex manifold, form, for a given Kähler metric, a product of the following type: a function of the scalar curvature multiplied by functions of the holomorphy potentials of each of the vector fields. It is shown that the stipulation that such a product be itself a holomorphy potential for yet another vector field singles out critical metrics for a particular functional. This may be regarded as a generalization of the extremal metric variation of Calabi, where and the functional is the square of the -norm of the scalar curvature. The existence question for such metrics is examined in a number of special cases. Examples are constructed in the case of certain multifactored product manifolds. For the SKR metrics investigated by Derdzinski and Maschler and residing in the complex projective space, it is shown that only one type of nontrivial criticality holds in dimension three and above. [Copyright &y& Elsevier]

Clunie and Hayman proved that if the spherical derivative ∥f'∥ of an entire function satisfies ∥f'∥(z) = O(∣z∣σ), then T(r, f) = O(rσ+1). We generalize this to holomorphic curves in projective space of dimension n omitting n hyperplanes in general position. [ABSTRACT FROM AUTHOR]

For the most general off-shell N = 2 supersymmetric σ-model in projective superspace, we elaborate on its formulation in terms of N = 1 chiral superfields. A universal (model-independent) expression is obtained for the holomorphic symplectic two-form, which determines the second supersymmetry transformation. This two-form is associated with the two complex structureähe hyperkähler target space, which are complimentary to the one used to realize the target space as a Kähler manifold [ABSTRACT FROM AUTHOR]

Abstract: We define holomorphic structures on canonical line bundles of the quantum projective space and identify their space of holomorphic sections. This determines the quantum homogeneous coordinate ring of the quantum projective space. We show that the fundamental class of is naturally presented by a twisted positive Hochschild cocycle. Finally, we verify the main statements of the Riemann–Roch formula and the Serre duality for and . [Copyright &y& Elsevier]

We consider the inclusion of the space of algebraic (regular) maps between real algebraic varieties in the space of all continuous maps. For a certain class of real algebraic varieties, which include real projective spaces, it is well known that the space of real algebraic maps is a dense subset of the space of all continuous maps. Our first result shows that, for this class of varieties, the inclusion is also a homotopy equivalence. After proving this, we restrict the class of varieties to real projective spaces. In this case, the space of algebraic maps has a ‘minimum degree’ filtration by finite-dimensional subspaces and it is natural to expect that the homotopy types of the terms of the filtration approximate closer and closer the homotopy type of the space of continuous mappings as the degree increases. We prove this and compute the lower bounds of this approximation of these spaces. This result can be seen as a generalization of the results of Mostovoy, Vassiliev and others on the topology of the space of real rational maps and the space of real polynomials without n-fold roots. It can also be viewed as a real analogue of Mostovoy's work on the topology of the space of holomorphic maps between complex projective spaces, which generalizes Segal's work on the space of complex rational maps. [ABSTRACT FROM PUBLISHER]

We find new examples of compact Spin(7)-manifolds using a construction of Joyce (J. Differ. Geom., 53:89-130, ; Compact manifolds with special holonomy. Oxford University Press, Oxford, ). The essential ingredient in Joyce's construction is a Calabi-Yau 4-orbifold with particular singularities admitting an antiholomorphic involution, which fixes the singularities. We search the class of well-formed quasismooth hypersurfaces in weighted projective spaces for suitable Calabi-Yau 4-orbifolds. [ABSTRACT FROM AUTHOR]

By utilizing Jacobian sections introduced by Stoll, we prove a second main theorem of holomorphic curves into projective spaces for hypersurfaces under certain conditions on the jets of the curves. The precise conditions are listed in three equations in the introduction. This solves a weaker form of Griffiths' conjecture and only for a special class of holomorphic curves. [ABSTRACT FROM AUTHOR]

We give a characterization of a minimal real hypersurface with respect to the condition for the sectional curvature. [ABSTRACT FROM AUTHOR]

The first goal of the article is to solve several fundamental problems in the theory of holomorphic bundles over non-algebraic manifolds. For instance, we prove that stability and semi-stability are Zariski open properties in families when the Gauduchon degree map is a topological invariant, or when the parameter manifold is compact. Second, we show that, for a generically stable family of bundles over a Kähler manifold, the Petersson–Weil form extends as a closed positive current on the whole parameter space of the family. This extension theorem uses classical tools from Yang–Mills theory (e.g., the Donaldson functional on the space of Hermitian metrics and its properties). We apply these results to study families of bundles over a Kählerian manifold Y parametrized by a non-Kählerian surface X, proving that such families must satisfy very restrictive conditions. These results play an important role in our program to prove existence of curves on class VII surfaces [22–24]. [ABSTRACT FROM AUTHOR]

In this article, we prove a uniqueness result for two algebraically non-degenerate holomorphic maps into a projective space sharing the same hypersurfaces. [ABSTRACT FROM AUTHOR]

We consider complex manifolds with a class of holomorphic coordinate functions satisfying the condition that each transition function is given by the standard action on CP2n−1 of some element in Sp(2n,C)/Z2. We show that such a manifold has a natural contact structure. Given any contact manifold, one can associate with it a symplectic manifold. It is shown that the symplectic manifolds arising from complex manifolds with special coordinate functions of the above type admit a canonical quantization. [Copyright &y& Elsevier]

By introducing a Second Main Theorem of Nevanlinna theory with a strong defect relation for holomorphic curves f in P n (C) and hyperplanes H 1 , ... , H q satisfying f ⁎ , z = 0 on ∪ j = 1 q f − 1 (H j) , we give a normality criterion for families of holomorphic mappings of a domain D in C m into the projective space P n (C) under a condition of uniform boundedness of their tangent mappings on intersections of compact subsets of D and inverse image of a finite union of hyperplanes in P n (C) (rather than on entire compact subsets). [ABSTRACT FROM AUTHOR]