Showing papers on "Complex dimension published in 2022"

Special Hermitian metrics on Oeljeklaus–Toma manifolds

Abstract: Oeljeklaus–Toma (OT) manifolds are higher dimensional analogues of Inoue-Bombieri surfaces and their construction is associated to a finite extension K $K$ of Q $\mathbb {Q}$ and a subgroup of units U $U$ . We characterize the existence of pluriclosed metrics (also known as strongly Kähler with torsion (SKT) metrics) on any OT manifold X ( K , U ) $X(K, U)$ purely in terms of number-theoretical conditions, yielding restrictions on the third Betti number b 3 $b_3$ and the Dolbeault cohomology group H ∂ ¯ 2 , 1 $H^{2,1}_{\overline{\partial }}$ . Combined with the main result in (Dubickas, Results Math. 76 (2021), 78), these numerical conditions render explicit examples of pluriclosed OT manifolds in arbitrary complex dimension. We prove that in complex dimension 4 and type ( 2 , 2 ) $(2, 2)$ , the existence of a pluriclosed metric on X ( K , U ) $X(K, U)$ is entirely topological, namely, it is equivalent to b 3 = 2 $b_3=2$ . Moreover, we provide an explicit example of an OT manifold of complex dimension 4 carrying a pluriclosed metric. Finally, we show that no OT manifold admits balanced metrics, but all of them carry instead locally conformally balanced metrics.

On the dimension of Dolbeault harmonic $(1,1)$-forms on almost Hermitian $4$-manifolds

Abstract: We prove that the dimension $h^{1,1}_{\overline\partial}$ of the space of Dolbeault harmonic $(1,1)$-forms is not necessarily always equal to $b^-$ on a compact almost complex 4-manifold endowed with an almost Hermitian metric which is not locally conformally almost K\"ahler. Indeed, we provide examples of non integrable, non locally conformally almost K\"ahler, almost Hermitian structures on compact 4-manifolds with $h^{1,1}_{\overline\partial}=b^-+1$. This answers to a question by Holt.

Compatibility between non-kähler structures on complex (nil)manifolds

Abstract: We study the interplay between the following types of special non-Kähler Hermitian metrics on compact complex manifolds (locally conformally Kähler, k-Gauduchon, balanced, and locally conformally balanced) and prove that a locally conformally Kähler compact nilmanifold carrying a balanced or a left-invariant k-Gauduchon metric is necessarily a torus. Combined with the main result in [FV16], this leads to the fact that a compact complex 2-step nilmanifold endowed with whichever two of the following types of metrics—balanced, pluriclosed and locally conformally Kähler—is a torus. Moreover, we construct a family of compact nilmanifolds in any dimension carrying both balanced and locally conformally balanced metrics and finally we show a compact complex nilmanifold does not support a left-invariant locally conformally hyper-Kähler structure.

Compactness for $$\Omega $$-Yang–Mills connections

Abstract: On a Riemannian manifold of dimension n we extend the known analytic results on Yang–Mills connections to the class of connections called $$\Omega $$ -Yang–Mills connections, where $$\Omega $$ is a smooth, not necessarily closed, $$(n-4)$$ -form on M. Special cases include $$\Omega $$ -anti-self-dual connections and Hermitian–Yang–Mills connections over general complex manifolds. By a key observation, a weak compactness result is obtained for moduli space of smooth $$\Omega $$ -Yang–Mills connections with uniformly $$L^2$$ bounded curvature, and it can be improved in the case of Hermitian–Yang–Mills connections over general complex manifolds. A removable singularity theorem for singular $$\Omega $$ -Yang–Mills connections on a trivial bundle with small energy concentration is also proven. As an application, it is shown how to compactify the moduli space of smooth Hermitian–Yang–Mills connections on unitary bundles over a class of balanced manifolds of Hodge-Riemann type. This class includes the metrics coming from multipolarizations, and in particular, the Kähler metrics. In the case of multipolarizations on a projective algebraic manifold, the compactification of smooth irreducible Hermitian–Yang–Mills connections with fixed determinant modulo gauge transformations inherits a complex structure from algebro-geometric considerations.

Toric Kato manifolds

Abstract: We introduce and study a special class of Kato manifolds, which we call toric Kato manifolds. Their construction stems from toric geometry, as their universal covers are open subsets of toric algebraic varieties of non-finite type. This generalizes previous constructions of Tsuchihashi and Oda, and in complex dimension 2, retrieves the properly blown-up Inoue surfaces. We study the topological and analytical properties of toric Kato manifolds and link certain invariants to natural combinatorial data coming from the toric construction. Moreover, we produce families of flat degenerations of any toric Kato manifold, which serve as an essential tool in computing their Hodge numbers. In the last part, we study the Hermitian geometry of Kato manifolds. We give a characterization result for the existence of locally conformally Kähler metrics on any Kato manifold. Finally, we prove that no Kato manifold carries balanced metrics and that a large class of toric Kato manifolds of complex dimension ≥3 do not support pluriclosed metrics.

Stable Submanifolds in the Product of Projective Spaces

Abstract: We provide a classification theorem for compact stable minimal immersions (CSMI) of codimension 1 or dimension 1 (codimension 1 and 2 or dimension 1 and 2) in the product of a complex (quaternionic) projective space with any other Riemannian manifold. We characterize the complex minimal immersions of codimension 2 or dimension 2 as the only CSMI in the product of two complex projective spaces. As an application, we characterize the CSMI of codimension 1 or dimension 1 (codimension 1 and 2 or dimension 1 and 2) in the product of a complex (quaternionic) projective space with any compact rank one symmetric space.

A reasonable notion of dimension for singular intersection homology

Abstract: M. Goresky and R. MacPherson intersection homology is also defined from the singular chain complex of a filtered space by H. King, with a key formula to make selections among singular simplexes. This formula needs a notion of dimension for subspaces $S$ of a Euclidean simplex, which is usually taken as the smallest dimension of the skeleta containing~$S$. Later, P. Gajer employed another dimension based on the dimension of polyhedra containing $S$. This last one allows traces of pullbacks of singular strata in the interior of the domain of a singular simplex. In this work, we prove that the two corresponding intersection homologies are isomorphic for Siebenmann's CS sets. In terms of King's paper, this means that polyhedral dimension is a ``reasonable'' dimension. The proof uses a Mayer-Vietoris argument which needs an adaptated subdivision. With the polyhedral dimension, that is a subtle issue. General position arguments are not sufficient and we introduce strong general position. With it, a stability is added to the generic character and we can do an inductive cutting of each singular simplex. This decomposition is realised with pseudo-barycentric subdivisions where the new vertices are not barycentres but close points of them.

From Dimension-Free Manifolds to Dimension-varying Control Systems

Abstract: Starting from the vector multipliers, the inner product, norm, distance, as well as addition of two vectors of different dimensions are proposed, which makes the spaces into a topological vector space, called the Euclidean space of different dimension (ESDD). An equivalence is obtained via distance. As a quotient space of ESDDs w.r.t. equivalence, the dimension-free Euclidean spaces (DFESs) and dimension-free manifolds (DFMs) are obtained, which have bundled vector spaces as its tangent space at each point. Using the natural projection from a ESDD to a DFES, a fiber bundle structure is obtained, which has ESDD as its total space and DFES as its base space. Classical objects in differential geometry, such as smooth functions, (co-)vector fields, tensor fields, etc., have been extended to the case of DFMs with the help of projections among different dimensional Euclidean spaces. Then the dimension-varying dynamic systems (DVDSs) and dimension-varying control systems (DVCSs) are presented, which have DFM as their state space. The realization, which is a lifting of DVDSs or DVCSs from DFMs into ESDDs, and the projection of DVDSs or DVCSs from ESDDs onto DFMs are investigated.

Point-Dimension Theory (Part I): The Point-Extended Box Dimension

Abstract: This article is an introductory work to a larger research project devoted to pure, applied and philosophical aspects of dimension theory. It concerns a novel approach toward an alternate dimension theory foundation: the point-dimension theory. For this purpose, historical research on this notion and related concepts, combined with critical analysis and philosophical development proved necessary. Hence, our main objective is to challenge the conventional zero dimension assigned to the point. This reconsideration allows us to propose two new ways of conceiving the notion of dimension, which are the two sides of the same coin. First as an organization; accordingly, we suggest the existence of the Dimensionad, an elementary particle conferring dimension to objects and space-time. The idea of the existence of this particle could possibly adopted as a projection to create an alternative way to unify quantum mechanics and Einstein's general relativity. Secondly, in connection with Boltzmann and Shannon entropies, dimension appears essentially as a comparison between entropies of sets. Thus, we started from the point and succeeded in constructing a point-dimension notion allowing us to extend the principle of box dimension in many directions. More precisely, we introduce the notion of point-extended box dimension in the large framework of topological vector spaces, freeing it from the notion of metric. This general setting permits us to treat the case of finite, infinite and invisible dimensions. This first part of our research project focuses essentially on general properties and is particularly oriented towards establishing a well founded framework for infinite dimension. Among others, one prospect is to test the possibility of using other types of spaces as a setting for quantum mechanics, instead of limiting it to the exclusive Hilbertian framework.

The meaning of dimensional space

Abstract: Abstract The concept of 0 dimension,1 dimension ,2 dimension,and 3 dimension,the more high dimension,until infinite dimension ,namely the concept of gradually increasings of dimension is non-existence.Instead,there is only a concept of infinitely great that is an one quantitative continuum. The space we see is just the defference of 1 dimension finite quantities and infinitely great quantities that is open space and is implied by change of direction .The accurate description of this one quantitative continuum is that its parts are connected each other as a unity at the infinite distance（infinitely great）relative to any orientation (all orientations)of our existence.it is unity in which its random parts is this infinitely great quantities and thus we call this unity as infinite quantities of infinite dimensions.

Stable submanifolds in the product of projective spaces

Abstract: We provide a classification theorem for compact stable minimal immersions (CSMI) of codimension $1$ or dimension $1$ (codimension $1$ and $2$ or dimension $1$ and $2$) in the product of a complex (quaternionic) projective space with any other Riemannian manifold. We characterize the complex minimal immersions of codimension $2$ or dimension $2$ as the only CSMI in the product of two complex projective spaces. As an application, we characterize the CSMI of codimension $1$ or dimension $1$ (codimension $1$ and $2$ or dimension $1$ and $2$) in the product of a complex (quaternionic) projective space with any compact rank one symmetric space.

The Dimension of the Moduli Space of Pointed Algebraic Curves of Low Genus

Abstract: We explicitly compute the moduli space pointed algebraic curves with a given numerical semigroup as Weierstrass semigroup for many cases of genus at most seven and determine the dimension for all semigroups of genus seven.

On the dimension of Dolbeault harmonic (1,1)-forms on almost Hermitian 4-manifolds

Abstract: We prove that the dimension $h^{1,1}_{\overline\partial}$ of the space of Dolbeault harmonic $(1,1)$-forms is not necessarily always equal to $b^-$ on a compact almost complex 4-manifold endowed with an almost Hermitian metric which is not locally conformally almost K\"ahler. Indeed, we provide examples of non integrable, non locally conformally almost K\"ahler, almost Hermitian structures on compact 4-manifolds with $h^{1,1}_{\overline\partial}=b^-+1$. This answers to a question by Holt.

The H/Q-correspondence and a generalization of the supergravity c-map

Abstract: Given a hypercomplex manifold with a rotating vector field (and additional data), we construct a conical hypercomplex manifold. As a consequence, we associate a quaternionic manifold to a hypercomplex manifold of the same dimension with a rotating vector field. This is a generalization of the HK/QK-correspondence. As an application, we show that a quaternionic manifold can be associated to a conical special complex manifold of half its dimension. Furthermore, a projective special complex manifold (with a canonical c-projective structure) associates with a quaternionic manifold. The latter is a generalization of the supergravity c-map. We do also show that the tangent bundle of any special complex manifold carries a canonical Ricci-flat hypercomplex structure, thereby generalizing the rigid c-map.

Generalized Poincaré Conjecture via Alexander trick over C-isomorphism extension to h-cobordism on inclusion maps with associated Kan-complex

Abstract: Abstract The generalized norm of the Poincaré conjecture was initiated with the extension of Henri Poincaré,s conjecture for n-dimensions. Steps have been taken thoroughly to prove the same from dimension-1 to dimension ≥5. Perelman closed the proof for dimension-3 thereby solving the millennium prize problem. Michael Freedman showed the validation in dimension-4 and Stephen Smale showed the validation in dimension ≥5. This paper would introduce a step-by-step proof taking Kan-fibration and Kan-complex being channeled from orders of Whitehead group, C-isomorphism, for any boundary manifold that implies a disjoint union among two n-dimensional manifolds for inclusion maps satisfying the vanishing torsion for a categorical correspondence on h-cobordism and Whitehead groups.

Jordan property for groups of bimeromorphic self-maps of complex manifolds with large Kodaira dimension

Abstract: We prove that the image of the pluricanonical representation of a group of bimeromorphic automorphisms of a complex manifold has bounded finite subgroups. As a consequence, we show that the group of bimeromorphic automorphisms of an $n$-dimensional complex manifold whose Kodaira dimension is at least $n-2$, satisfies the Jordan property.

Closed 3-forms in five dimensions and embedding problems

Abstract: We consider the question if a five dimensional manifold can be embedded into a Calabi-Yau manifold of complex dimension three such that the real part of the holomorphic volume form induces a given closed 3-form on the 5-manifold. We define an open set of 3-forms in dimension five which we call strongly pseudoconvex, and show that for closed strongly pseudoconvex 3-forms the perturbative version of this embedding problem can be solved if a finite dimensional vector space of obstructions vanishes.