Showing papers on "Complex dimension published in 2019"

Complex structure degenerations and collapsing of Calabi-Yau metrics

Abstract: In this paper, we make progress on understanding the collapsing behavior of Calabi-Yau metrics on a degenerating family of polarized Calabi-Yau manifolds. In the case of a family of smooth Calabi-Yau hypersurfaces in projective space degenerating into the transversal union of two smooth Fano hypersurfaces in a generic way, we obtain a complete result in all dimensions establishing explicit and precise relationships between the metric collapsing and complex structure degenerations. This result is new even in complex dimension two. This is achieved via gluing and singular perturbation techniques, and a key geometric ingredient involving the construction of certain (not necessarily smooth) Kahler metrics with torus symmetry. We also discuss possible extensions of this result to more general settings.

Anomaly cancellation in the topological string

Abstract: We describe the coupling of holomorphic Chern-Simons theory at large N with Kodaira-Spencer gravity. We explain a new anomaly cancellation mechanism at all loops in perturbation theory for open-closed topological B-model. At one loop this anomaly cancellation is analogous to the Green-Schwarz mechanism. As an application, we introduce a type I version of Kodaira-Spencer theory in complex dimensions 3 and 5. In complex dimension 5, we show that it can only be coupled consistently at the quantum level to holomorphic Chern-Simons theory with gauge group SO(32). This is analogous to the Green-Schwarz mechanism for the physical type I string. This coupled system is conjectured to be a supersymmetric localization of type I string theory. In complex dimension 3, the required gauge group is SO(8).

Special Lagrangian submanifolds of log Calabi-Yau manifolds

Abstract: We study the existence of special Lagrangian submanifolds of log Calabi-Yau manifolds equipped with the complete Ricci-flat Kahler metric constructed by Tian-Yau. We prove that if $X$ is a Tian-Yau manifold, and if the compact Calabi-Yau manifold at infinty admits a single special Lagrangian, then $X$ admits infinitely many disjoint special Lagrangians. In complex dimension $2$, we prove that if $Y$ is a del Pezzo surface, or a rational elliptic surface, and $D\in |-K_{Y}|$ is a smooth divisor with $D^2=d$, then $X= Y\backslash D$ admits a special Lagrangian torus fibration, as conjectured by Strominger-Yau-Zaslow and Auroux. In fact, we show that $X$ admits twin special Lagrangian fibrations, confirming a prediction of Leung-Yau. In the special case that $Y$ is a rational elliptic surface, or $Y= \mathbb{P}^2$ we identify the singular fibers for generic data, thereby confirming two conjectures of Auroux. Finally, we prove that after a hyper-Kahler rotation, $X$ can be compactified to the complement of a Kodaira type $I_{d}$ fiber appearing as a singular fiber in a rational elliptic surface $\check{\pi}: \check{Y}\rightarrow \mathbb{P}^1$.

Dolbeault cohomology of complex nilmanifolds foliated in toroidal groups

Abstract: It is conjectured that the Dolbeault cohomology of a complex nilmanifold $X$ is computed by left-invariant forms. We prove this under the assumption that $X$ is suitably foliated in toroidal groups and deduce that the conjecture holds in real dimension up to six. Our approach generalises previous methods, where the existence of a holomorphic fibration was a crucial ingredient.

The algebraic dimension of compact complex threefolds with vanishing second Betti numbers

Abstract: We investigate compact complex manifolds of dimension three and second Betti number $b_2(X) = 0$. We are interested in the algebraic dimension $a(X)$, which is by definition the transcendence degree of the field of meromorphic functions over the field of complex numbers. The topological Euler characteristic $\chi_{\mathrm{ top}}(X) $ equals the third Chern class $c_3(X)$ by a theorem of Hopf. Our main result is that, if $X$ is a compact 3-dimensional complex manifold with $b_2(X) = 0$ and $a(X) > 0$, then $c_3(X) = \chi_{\rm top}(X) = 0$, that is, we either have $b_1(X) = 0, \ b_3(X) = 2$ or $b_1(X) = 1, \ b_3(X) = 0.$

Volterra type integration operators from Bergman spaces to Hardy spaces

Abstract: We completely characterize the boundedness of the Volterra type integration operators $J_b$ acting from the weighted Bergman spaces $A^p_\alpha$ to the Hardy spaces $H^q$ of the unit ball of $\mathbb{C}^n$ for all $0

Three Generalizations Regarding Limit Sets for Complex Kleinian Groups

Abstract: We study three problems related to the limit sets of discrete subgroups of PSL(n+1,C). In Chapter 2, we study the dynamics of solvable discrete subgroups of PSL(n+1,C). We prove that solvable groups are virtually triangularizable and we provide a description of the all the possible Kulkarni limit sets of solvable subgroups of PSL(n+1,C). Finally, we give the representations of these groups. With this description, the full description of the dynamics of general discrete subgroups of PSL(n+1,C) will be almost complete. In Chapter 3, we propose a new definition for the concept of limit set for the action of a discrete subgroup of PSL(n+1,C), we call it the Frances limit set. In complex dimension n=2, the Kulkarni limit set seems to be the right notion of limit set. However, in dimension n>2, the Kulkarni limit set is difficult to compute and it is bigger than it needs to be. This new limit set is, in general, smaller than the Kulkarni limit set. Also, it is made up of projective subspaces of the same dimension. The action of a discrete subgroup of PSL(n+1,C) on the complement of its Frances limit set is proper and discontinuous, also, this limit set is purely dimensional and unstable under deformations. In Chapter 4, we propose a way to generalize Patterson-Sullivan measures to the complex setting. We consider the Kobayashi metric on the complement of the Kulkarni limit set of an irreducible subgroup of PSL(3,C). These domains are the complement of arrays of complex lines in general position. We parametrize the space of such arrays of lines and we prove that, if for some subgroup of PSL(3,C) the entropy volume is finite, then we can construct these measures. We also give some concrete ideas on how to guarantee that the entropy volume of the Kobayashi metric is finite for certain groups.

Biholomorphic equivalence to totally nondegenerate model CR manifolds

Abstract: Applying Elie Cartan’s classical method, we show that the biholomorphic equivalence problem to a totally nondegenerate Beloshapka’s model of CR dimension one and codimension $$k> 1$$ , whence of real dimension $$2+k$$ , is reducible to some absolute parallelism, namely to an $$\{e\}$$ -structure on a certain prolonged manifold of real dimension either $$3+k$$ or $$4+k$$ . The proof relies upon a careful weight analysis on the structure equations associated with the mentioned problem of equivalence. As one of the applications of the achieved results, we also reconfirm in CR dimension one Beloshapka’s maximum conjecture on the holomorphic rigidity of his models in certain lengths equal or greater than three.

A divergent horocycle in the horofunction compactification of the Teichmüller metric

Abstract: We give an example of a horocycle in the Teichmuller space of the five-times-punctured sphere that does not converge in the Gardiner--Masur compactification, or equivalently in the horofunction compactification of the Teichmuller metric. As an intermediate step, we exhibit a simple closed curve whose extremal length is periodic but not constant along the horocycle. The example lifts to any Teichmuller space of complex dimension greater than one via covering constructions.

Bott–Chern harmonic forms on complete Hermitian manifolds

Abstract: Let (M,J,g,ω) be a Hermitian manifold of complex dimension n. Assume that the torsion of the Chern connection ∇ is bounded, and that there exists a 𝒞∞exhausting function ρ : M → ℝ such that ∇ρ,∇2ρ ...

Holomorphicity of real Kaehler submanifolds

Abstract: Let $f\colon M^{2n}\to\mathbb{R}^{2n+p}$ denote an isometric immersion of a Kaehler manifold of complex dimension $n\geq 2$ into Euclidean space with codimension $p$. If $2p\leq 2n-1$, we show that generic rank conditions on the second fundamental form of the submanifold imply that $f$ has to be a minimal submanifold. In fact, for codimension $p\leq 11$ we prove that $f$ must be holomorphic with respect to some complex structure in the ambient space.

Method and apparatus for marking and displaying spatial size in virtual three-dimensional house model

Abstract: The present disclosure provides a method and an apparatus for automatically generating and displaying a relevant dimension in a virtual three-dimensional house model. The method includes: acquiring a plane layout and a three-dimensional house model of a single house; aligning a top view or a cross-sectional view of the three-dimensional house model with the plane layout to obtain a correspondence relationship between a unit length of the three-dimensional house model and a unit pixel of the plane layout; calculating a real length corresponding to the unit length of the three-dimensional house model according to the correspondence relationship; and calculating a real dimension of a room and/or an object in the house according to the real length corresponding to the unit length of the three-dimensional house model for the purpose of presentation.

A Poisson transform adapted to the Rumin complex

Abstract: Let $G$ be a semisimple Lie group with finite center, $K\subset G$ a maximal compact subgroup, and $P\subset G$ a parabolic subgroup. Following ideas of P.Y.\ Gaillard, one may use $G$-invariant differential forms on $G/K\times G/P$ to construct $G$-equivariant Poisson transforms mapping differential forms on $G/P$ to differential forms on $G/K$. Such invariant forms can be constructed using finite dimensional representation theory. In this general setting, we first prove that the transforms that always produce harmonic forms are exactly those that descend from the de Rham complex on $G/P$ to the associated Bernstein-Gelfand-Gelfand (or BGG) complex in a well defined sense. The main part of the article is devoted to an explicit construction of such transforms with additional favorable properties in the case that $G=SU(n+1,1)$. Thus $G/P$ is $S^{2n+1}$ with its natural CR structure and the relevant BGG complex is the Rumin complex, while $G/K$ is complex hyperbolic space of complex dimension $n+1$. The construction is carried out both for complex and for real differential forms and the compatibility of the transforms with the natural operators that are available on their sources and targets are analyzed in detail.

Existence criteria for special locally conformally Kähler metrics

Abstract: We investigate the relation between holomorphic torus actions on complex manifolds of locally conformally Kahler (LCK) type and the existence of special LCK metrics. We show that if the group of biholomorphisms of such a manifold (M, J) contains a compact torus which is not totally real, then there exists a Vaisman metric on the manifold, generalising a result of Kamishima–Ornea. Also, we obtain a new obstruction to the existence of LCK structures on a given complex manifold in terms of its automorphism group. As an application, we obtain a classification of manifolds of LCK type among all the manifolds having the structure of a holomorphic principal torus bundle. Moreover, we show that if the group of biholomorphisms contains a compact torus whose dimension is half the real dimension of M, then (M, J) admits an LCK metric with positive potential. Finally, we obtain new non-existence results for LCK metrics on certain products of complex manifolds.

On generalized Inoue manifolds

Abstract: This paper is about a generalization of famous Inoue's surfaces. Let $M$ be a matrix in $SL(2n+1,\mathbb{Z})$ having only one real eigenvalue which is simple. We associate to $M$ a complex manifold $T_M$ of complex dimension $n+1$. This manifold fibers over $S^1$ with the fiber $\mathbb{T}^{2n+1}$ and monodromy $M^\top$. Our construction is elementary and does not use algebraic number theory. We show that some of the Oeljeklaus-Toma manifolds are biholomorphic to the manifolds of type $T_M$. We prove that if $M$ is not diagonalizable, then $T_M$ does not admit a K\"ahler structure and is not homeomorphic to any of Oeljeklaus-Toma manifolds.

Mass, Kähler manifolds, and symplectic geometry

Abstract: In the author’s previous joint work with Hein (Commun Math Phys 347:183–221, 2016), a mass formula for asymptotically locally Euclidean Kahler manifolds was proved, assuming only relatively weak fall-off conditions on the metric. However, the case of real dimension 4 presented technical difficulties that led us to require fall-off conditions in this special dimension that are stronger than the Chruściel fall-off conditions that sufficed in higher dimensions. Nevertheless, the present article shows that techniques of four-dimensional symplectic geometry can be used to obtain all the major results of Hein-LeBrun (2016), assuming only Chruściel-type fall-off. In particular, the present article presents a new proof of our Penrose-type inequality for the mass of an asymptotically Euclidean Kahler manifold that only requires this very weak metric fall-off.

Convergence of the CR Yamabe flow

Abstract: We consider the CR Yamabe flow on a compact strictly pseudoconvex CR manifold M of real dimension $$2n+1$$ . We prove convergence of the CR Yamabe flow when $$n=1$$ or M is spherical.

Regular multi-types and the Bloom conjecture

Abstract: We prove equality of the vector field (iterated commutator) type and the regular contact type, which together with the Bloom theorem on equality of the Levi-form type and the regular contact type provides a complete solution of a long standing open problem of Bloom in the case of complex dimension three. For general dimensions, we verify the Bloom conjecture when $s=n-2$, which provides the first positive result in the pseudoconvexity sensitive case for a real hypersurface in ${\mathbb{C}}^n$ after his important work in 1981.

On the Laplace–Beltrami operator on compact complex spaces

Abstract: Let $(X,h)$ be a compact and irreducible Hermitian complex space of complex dimension $v>1$. In this paper we show that the Friedrichs extension of both the Laplace-Beltrami operator and the Hodge-Kodaira Laplacian acting on functions has discrete spectrum. Moreover we provide some estimates for the growth of the corresponding eigenvalues and we use these estimates to deduce that the associated heat operators are trace-class. Finally we give various applications to the Hodge-Dolbeault operator and to the Hodge-Kodaira Laplacian in the setting of Hermitian complex spaces of complex dimension $2$.

Bochner-Kähler and Bach flat manifolds

Abstract: We have classified Bochner-Kahler manifolds of real dimension $$> 4$$ , which are also Bach flat. In the 4-dimensional case, we have shown that if the scalar curvature is harmonic, then it is constant. Finally, we show that the gradient of scalar curvature of any Bochner-Kahler manifold is an infinitesimal harmonic transformation, and if it is conformal, then the scalar curvature is constant.

Three Topological Results on the Twistor Discriminant Locus in the 4-Sphere

Abstract: We exploit techniques from classical (real and complex) algebraic geometry for the study of the standard twistor fibration $${\pi : \mathbb{CP}^3 \rightarrow S^4}$$ . We prove three results about the topology of the twistor discriminant locus of an algebraic surface in $${\mathbb{CP}^3}$$ . First of all we prove that, with the exception of two special cases, the real dimension of the twistor discriminant locus of an algebraic surface is always equal to 2. Secondly we describe the possible intersections of a general surface with the family of twistor lines: we find that only 4 configurations are possible and for each of them we compute the dimension. Lastly we give a decomposition of the twistor discriminant locus of a given cone in terms of its singular locus and its dual variety.

Gluing and deformation of asymptotically cylindrical Calabi–Yau manifolds in complex dimension three

Abstract: We develop some consequences of the connection between Calabi-Yau structures and torsion-free G(2) structures on compact and asymptotically cylindrical six- and seven-dimensional manifolds. Firstly, we improve the known proof that matching asymptotically cylindrical Calabi-Yau threefolds can be glued. Secondly, we give an alternative proof that the moduli space of Calabi-Yau structures on a six dimensional real manifold is smooth, and extend it to the asymptotically cylindrical case. Finally, we prove that the gluing map of Calabi-Yau threefolds, extended between these moduli spaces, is a local diffeomorphism: that is, that every deformation of a glued Calabi-Yau threefold arises from an essentially unique deformation of the asymptotically cylindrical pieces. (C) 2018 Published by Elsevier B.V.

Stability of Random-Projection Based Classifiers. The Bayes Error Perspective

Abstract: In this paper we investigate the Bayes error and the stability of Bayes’ error when the dimension of the classification problem is reduced using random projections. We restrict our attention to the two-class problem. Furthermore, we assume that distributions in classes come from multivariate normal distributions with the same covariance matrices, i.e., differing only in the means. This is one of the few situations when the Bayes error expression can be written in a simple form of a compact final formula. The bias and the variance of the classification error introduced by random projections are determined. Both full-dimensional normal distributions and singular distributions were considered with a real dimension smaller than the ambient dimension. These results allow for the separation of the impact of random dimension reduction from the impact of the learning sample and provide lower bounds on classification errors. Relatively low variance of the Bayes error introduced by random projections confirms the stability of the random-projection based classifiers, at least under the proposed assumptions.

On a class of Kato manifolds

Abstract: We revisit Brunella's proof of the fact that Kato surfaces admit locally conformally K\" ahler metrics, and we show that it holds for a large class of higher dimensional complex manifolds containing a global spherical shell. On the other hand, we construct manifolds containing a global spherical shell which admit no locally conformally Kahler metric. We consider a specific class of these manifolds, which can be seen as a higher dimensional analogue of Inoue-Hirzebruch surfaces, and study several of their analytical properties. In particular, we give new examples, in any complex dimension $n \geq 3$, of compact non-exact locally conformally K\" ahler manifolds with algebraic dimension $n-2$, algebraic reduction bimeromorphic to $\mathbb{C}\mathbb{P}^{n-2}$ and admitting non-trivial holomorhic vector fields.

Mayer-Vietoris sequences and equivariant K-theory rings of toric varieties

Abstract: We apply a Mayer-Vietoris sequence argument to identify the Atiyah-Segal equivariant complex K-theory rings of certain toric varieties with rings of integral piecewise Laurent polynomials on the associated fans. We provide necessary and sufficient conditions for this identification to hold for toric varieties of complex dimension 2, including smooth and singular cases. We prove that it always holds for smooth toric varieties, regardless of whether or not the fan is polytopal or complete. Finally, we introduce the notion of fans with \distant singular cones," and prove that the identification holds for them. The identification has already been made by Hararda, Holm, Ray and Williams in the case of divisive weighted projective spaces; in addition to enlarging the class of toric varieties for which the identification holds, this work provides an example in which the identification fails. We make every effort to ensure that our work is rich in examples.

A rigidity result for Kählerian manifolds endowed with closed conformal vector fields

Abstract: We show that if a connected compact K¨ahlerian surface M with nonpositive Gaussian curvature is endowed with a closed conformal vector field ξ whose singular points are isolated, then M is isometric to a flat torus and ξ is parallel. We also consider the case of a connected complete K¨ahlerian manifod M of complex dimension n > 1 and endowed with a nontrivial closed conformal vector field ξ. In this case, it is well known that the singularities of ξ are automatically isolated and the nontrivial leaves of the distribution generated by ξ and Jξ are totally geodesic in M. Assuming that one such leaf is compact, has torsion normal holonomy group and that the holomorphic sectional curvature of M along it is nonpositive, we show that ξ is parallel and M is foliated by a family of totally geodesic isometric tori and also by a family of totally geodesic isometric complete K¨ahlerian manifolds of complex dimension n − 1. In particular, the universal covering of M is isometric to a Riemannian product having R2 as a factor. We also comment on a generic class of compact complex symmetric spaces not possessing nontrivial closed conformal vector fields, thus showing that we cannot get rid of the hypothesis of nonpositivity of the holomorphic sectional curvature in the direction of ξ.

Holomorphic Foliations Tangent to Levi-Flat Subsets

Abstract: An irreducible real analytic subvariety H of real dimension $$2n +1$$ in a complex manifold M is a Levi-flat subset if its regular part carries a complex foliation of dimension n. Locally, a germ of real analytic Levi-flat subset is contained in a germ of irreducible complex variety $$H^{\imath }$$ of dimension $$n+1$$ , called intrinsic complexification, which can be globalized to a neighborhood of H in M provided H is a coherent analytic subvariety. In this case, a singular holomorphic foliation $$\mathcal {F}$$ of dimension n in M that is tangent to H is also tangent to $$H^{\imath }$$ . In this paper, we prove integration results of local and global nature for the restriction to $$H^{\imath }$$ of a singular holomorphic foliation $$\mathcal {F}$$ tangent to a real analytic Levi-flat subset H. From a local viewpoint, if $$n=1$$ and $$H^{\imath }$$ has an isolated singularity, then $$\mathcal {F}|_{H^{\imath }}$$ has a meromorphic first integral. From a global perspective, when $$M = \mathbb {P}^N$$ and H is coherent and of low codimension, $$H^{\imath }$$ extends to an algebraic variety. In this case, $$\mathcal {F}|_{H^{\imath }}$$ has a rational first integral provided infinitely many leaves of $$\mathcal {F}$$ in H are algebraic.

Holomorphic Approximation via Dolbeault Cohomology

Abstract: The purpose of this paper is to study holomorphic approximation and approximation of $\bar\partial$-closed forms in complex manifolds of complex dimension $n\geq 1$. We consider extensions of the classical Runge theorem and the Mergelyan property to domains in complex manifolds for the smooth and the $L^2$ topology. We characterize the Runge or Mergelyan property in terms of certain Dolbeault cohomology groups and some geometric sufficient conditions are given.

On compact affine quaternionic curves and surfaces

Abstract: This paper is devoted to the study of affine quaternionic manifolds and to a possible classification of all compact affine quaternionic curves and surfaces. It is established that on an affine quaternionic manifold there is one and only one affine quaternionic structure. A direct result, based on the celebrated Kodaira Theorem that studies compact complex manifolds in complex dimension 2, states that the only compact affine quaternionic curves are the quaternionic tori and the primary Hopf surface S^3 x S^1. As for compact affine quaternionic surfaces, we restrict to the complete ones: the study of their fundamental groups, together with the inspection of all nilpotent hypercomplex simply connected 8-dimensional Lie Groups, identifies a path towards their classification.

Comments on the Newlander-Nirenberg theorem

Abstract: The Newlander-Nirenberg theorem says that a necessary and sufficient condition for the complex coordinates associated with a given almost complex structure tensor $I_M{}^N$ to exist is the vanishing of the Nijenhuis tensor ${\cal N}_{MN}{}^K$. In the first part of the paper, we give a simple explicit proof of this fact. In the second part, we discuss a supersymmetric interpretation of this theorem. ${\it (i)}$ The condition ${\cal N}_{MN}{}^K = 0$ is necessary for a certain $N=1$ supersymmetric mechanical sigma models to enjoy $N=2$ supersymmetry. ${\it (ii)}$ The sufficiency of this condition for the existence of complex coordinates implies that the representation of the supersymmetry algebra realized by the superfields associated with all the real coordinates and their superpartners can be presented as a direct sum of d irreducible representations (d is the complex dimension of the manifold).