Complex Manifold Research Articles

A $dd^c$ -TYPE CONDITION BEYOND THE KÄHLER REALM

AbstractThis paper introduces a generalization of the $dd^c$ -condition for complex manifolds. Like the $dd^c$ -condition, it admits a diverse collection of characterizations, and is hereditary under various geometric constructions. Most notably, it is an open property with respect to small deformations. The condition is satisfied by a wide range of complex manifolds, including all compact complex surfaces, and all compact Vaisman manifolds. We show there are computable invariants of a real homotopy type which in many cases prohibit it from containing any complex manifold satisfying such $dd^c$ -type conditions in low degrees. This gives rise to numerous examples of almost complex manifolds which cannot be homotopy equivalent to any of these complex manifolds.

A lower bound in the problem of realization of cycles

AbstractWe consider the classical Steenrod problem on realization of integral homology classes by continuous images of smooth oriented manifolds. Let be the smallest positive integer such that any integral ‐dimensional homology class becomes realizable in the sense of Steenrod after multiplication by . The best known upper bound for was obtained independently by Brumfiel and Buchstaber in 1969. All known lower bounds for were very far from this upper bound. The main result of this paper is a new lower bound for that is asymptotically equivalent to the Brumfiel–Buchstaber upper bound (in the logarithmic scale). For , we prove that our lower bound is exact. Also, we obtain analogous results for the case of realization of integral homology classes by continuous images of smooth stably complex manifolds.

Deformations of astheno-Kähler metrics

AbstractThe property of admitting an astheno-Kähler metric is not stable under the action of small deformations of the complex structure of a compact complex manifold. In this study, we prove necessary cohomological conditions for the existence of curves of astheno-Kähler metrics along curves of deformations starting from an initial compact complex manifold endowed with an astheno-Kähler metric. Furthermore, we apply our results providing obstructions to the existence of curves of astheno-Kähler metrics on two different families of real eight-dimensional nilmanifolds endowed with invariant nilpotent complex structures.

Characterization of Lipschitz normally embedded complex curves

The main result of the paper states that a connected complex affine algebraic curve is LNE in Cn if and only if its germ at any singular point is a finite union of non-singular complex curve germs which are pairwise transverse, and its projective closure is in general position with the hyperplane at infinity. To this aim, we completely characterize complex analytic curves of a compact complex manifold which are LNE (regardless of the given Riemannian structure), and therefore we can relate when a projective algebraic curve is LNE in CPn with the property of its general affine traces being LNE in Cn. This allows us to deduce that Lipschitz classification of LNE curves is topological. We describe a complete invariant for the (outer and inner) Lipschitz equivalence of affine LNE curves and show that most such curves cannot be bi-Lipschitz homeomorphic to a plane one.

Angle-of-Arrival Estimation With Practical Phone Antenna Configurations

With the advances of the Internet of Things and mobile connectivity, location-based services are becoming increasingly popular and continue to enhance our experience. Multiple antennas have been pivotal in providing reliable wireless communications and high-resolution localization. If the antennas of the array are isotropic, then the simplified array manifold determined by the array geometry can be used to estimate the angle-of-arrival (AOA). However, in the real world, mobile handsets tend to have very limited space, where the practical antennas are equipped on the same ground plane, and the array geometry hardly obeys the rule of half-wavelength spacing. Therefore, a practical antenna couples signals from other antennas, causing a mutual coupling effect. Complex array manifolds are produced on an antenna even if the received signal is propagated through a single path channel. In addition, the irregular radiation pattern of each antenna further impairs the AOA estimation capability. Given the above effects, the simplified array manifold determined by the array geometry can no longer provide precise localization. In this paper, we propose a generic array manifold model for both isotropic and practical antennas. We also present an efficient algorithm to enable AOA estimation on practical antennas on the basis of the proposed model and implement it on a 5G phone at a mid-band spectrum with a 100MHz channel bandwidth. Results reveal the promising performance of the proposed model, with the AOA estimation errors lower than 10∘ in over 90% of the scenarios.

On Cohomology Vanishing with Polynomial Growth on Complex Manifolds with Pseudoconvex Boundary

The \bar{\partial} cohomology groups with polynomial growth H^{r,s}_{\mathrm{p.g.}} will be studied. It will be shown that, given a complex manifold M , a locally pseudoconvex bounded domain \Omega\Subset M satisfying certain geometric boundary conditions and a holomorphic vector bundle E\to M , H^{r,s}_{\mathrm{p.g.}}(\Omega,E)=0 holds for all s\geq1 if E is Nakano positive and r=\dim{M} . It will also be shown that H^{r,s}_{\mathrm{p.g.}}(\Omega,E)=0 for all r and s with r+s>\dim{M} if, moreover, \operatorname{rank}E=1 . By the comparison theorem due to Deligne, Maltsiniotis (Astérisque 17 (1974), 141–160) and Sasakura (Inst. Math. Sci. 17 (1981), 371–552), it follows in particular that, for any smooth projective variety X , for any ample line bundle L\to X and for any effective divisor D on X such that [D]|_{| D|}\geq0 , the algebraic cohomology H^{s}_{\mathrm{alg}}(X\setminus | D | , \Omega_X^r(L)) vanishes if r+s>\dim{X} .

Submanifolds with ample normal bundle

AbstractWe construct germs of complex manifolds of dimension along projective submanifolds of dimension with ample normal bundle and without nonconstant meromorphic functions whenever . We also show that our methods do not allow the construction of similar examples when by establishing an algebraicity criterion for foliations on projective spaces that generalizes a classical result by Van den Ven characterizing linear subspaces of projective spaces as the only submanifolds with split tangent sequence.

Entire Curves Producing Distinct Nevanlinna Currents

Abstract First, inspired by a question of Sibony, we show that in every compact complex manifold $Y$ with certain Oka property, there exists some entire curve $f: \mathbb {C}\rightarrow Y$ generating all Nevanlinna/Ahlfors currents on $Y$, by holomorphic disks $\{f\restriction _{\mathbb {D}(c, r)}\}_{c\in \mathbb {C}, r>0}$. Next, we answer positively a question of Yau, by constructing some entire curve $g: \mathbb {C}\rightarrow X$ in the product $X:=E_{1}\times E_{2}$ of two elliptic curves $E_{1}$ and $E_{2}$, such that by using concentric holomorphic disks $\{g\restriction _{\mathbb {D}_{ r}}\}_{r>0}$ we can obtain infinitely many distinct Nevanlinna/Ahlfors currents proportional to the extremal currents of integration along curves $[\{e_{1}\}\times E_{2}]$, $[E_{1}\times \{e_{2}\}]$ for all $e_{1}\in E_{1}, e_{2}\in E_{2}$ simultaneously. This phenomenon is new, and it shows tremendous holomorphic flexibility of entire curves in large scale geometry. Dedicated to Julien Duval with admiration

L∞ liftings of semiregularity maps via Chern–Simons classes

We introduce the notion of Chern–Simons classes for curved DG-pairs and we prove that a particular case of this general construction provides canonical L∞ liftings of Buchweitz–Flenner semiregularity maps for coherent sheaves on complex manifolds.

Bounding nonminimality and a conjecture of Borovik–Cherlin

Motivated by the search for methods to establish strong minimality of certain low order algebraic differential equations, a measure of how far a finite rank stationary type is from being minimal is introduced and studied: The degree of nonminimality is the minimum number of realisations of the type required to witness a nonalgebraic forking extension. Conditional on the truth of a conjecture of Borovik and Cherlin on the generic multiple-transitivity of homogeneous spaces definable in the stable theory being considered, it is shown that the nonminimality degree is bounded by the U -rank plus 2. The Borovik–Cherlin conjecture itself is verified for algebraic and meromorphic group actions, and a bound of U -rank plus 1 is then deduced unconditionally for differentially closed fields and compact complex manifolds. An application is given regarding transcendence of solutions to algebraic differential equations.

Solution to the Thomson Problem for Clifford Tori with an Application to Wigner Crystals.

In its original version, the Thomson problem consists of the search for the minimum-energy configuration of a set of point-like electrons that are confined to the surface of a two-dimensional sphere () that repel each other according to Coulomb's law, in which the distance is the Euclidean distance in the embedding space of the sphere, i.e., . In this work, we consider the analogous problem where the electrons are confined to an n-dimensional flat Clifford torus with n = 1, 2, 3. Since the torus can be embedded in the complex manifold , we define the distance in the Coulomb law as the Euclidean distance in , in analogy to what is done for the Thomson problem on the sphere. The Thomson problem on a Clifford torus is of interest because supercells with the topology of a Clifford torus can be used to describe periodic systems such as Wigner crystals. In this work, we numerically solve the Thomson problem on a square Clifford torus. To illustrate the usefulness of our approach, we apply it to Wigner crystals. We demonstrate that the equilibrium configurations we obtain for large numbers of electrons are consistent with the predicted structures of Wigner crystals. Finally, in the one-dimensional case, we analytically obtain the energy spectrum and the phonon dispersion law.

Benchmarking variational AutoEncoders on cancer transcriptomics data.

Deep generative models, such as variational autoencoders (VAE), have gained increasing attention in computational biology due to their ability to capture complex data manifolds which subsequently can be used to achieve better performance in downstream tasks, such as cancer type prediction or subtyping of cancer. However, these models are difficult to train due to the large number of hyperparameters that need to be tuned. To get a better understanding of the importance of the different hyperparameters, we examined six different VAE models when trained on TCGA transcriptomics data and evaluated on the downstream tasks of cluster agreement with cancer subtypes and survival analysis. We studied the effect of the latent space dimensionality, learning rate, optimizer, initialization and activation function on the quality of subsequent downstream tasks on the TCGA samples. We found β-TCVAE and DIP-VAE to have a good performance, on average, despite being more sensitive to hyperparameters selection. Based on these experiments, we derived recommendations for selecting the different hyperparameters settings. To ensure generalization, we tested all hyperparameter configurations on the GTEx dataset. We found a significant correlation (ρ = 0.7) between the hyperparameter effects on clustering performance in the TCGA and GTEx datasets. This highlights the robustness and generalizability of our recommendations. In addition, we examined whether the learned latent spaces capture biologically relevant information. Hereto, we measured the correlation and mutual information of the different representations with various data characteristics such as gender, age, days to metastasis, immune infiltration, and mutation signatures. We found that for all models the latent factors, in general, do not uniquely correlate with one of the data characteristics nor capture separable information in the latent factors even for models specifically designed for disentanglement.

Quasi-monotone convergence of plurisubharmonic functions

The complex Monge-Ampère operator has been defined for locally bounded plurisubharmonic functions by Bedford-Taylor in the 80's. This definition has been extended to compact complex manifolds, and to various classes of mildly unbounded quasi-plurisubharmonic functions by various authors. As this operator is not continuous for the L1-topology, several stronger topologies have been introduced over the last decades to remedy this, while maintaining efficient compactness criteria. The purpose of this note is to show that these stronger topologies are essentially equivalent to the natural quasi-monotone topology that we introduce and study here.

On Bott–Chern and Aeppli cohomologies of almost complex manifolds and related spaces of harmonic forms

In this paper we introduce several new cohomologies of almost complex manifolds, among which stands a generalization of Bott–Chern and Aeppli cohomologies defined using the operators d, dc. We explain how they are connected to already existing cohomologies of almost complex manifolds and we study the spaces of harmonic forms associated to d, dc, showing their relation with Bott–Chern and Aeppli cohomologies and to other well-studied spaces of harmonic forms. Notably, Bott–Chern cohomology of 1-forms is finite-dimensional on compact manifolds and provides an almost complex invariant hd+dc1 that distinguishes between almost complex structures. On almost Kähler 4-manifolds, the spaces of harmonic forms we consider are particularly well-behaved and are linked to harmonic forms considered by Tseng and Yau in the study of symplectic cohomology.

Almost complex manifolds with total Betti number three

In this paper, we show the minimal total Betti number of a closed almost complex manifold of dimension [Formula: see text] is four, thus confirming a conjecture of Sullivan except for dimension [Formula: see text]. Along the way, we prove the only simply connected closed complex manifold having total Betti number three is the complex projective plane.

Descent of dg cohesive modules for open covers on complex manifolds

Descent of dg cohesive modules for open covers on complex manifolds

Geometric Hodge filtered complex cobordism

We construct a geometric cycle model for a Hodge filtered extension of complex cobordism for every smooth manifold with a filtration on its de Rham complex with complex coefficients. Using a refinement of the Pontryagin–Thom construction, we construct an explicit isomorphism between our geometric model and the abstract model of Hodge filtered complex cobordism of Hopkins–Quick for every complex manifold with the Hodge filtration.

Functionals for the study of lcK metrics on compact complex manifolds

We propose an approach to the existence problem for locally conformally Kähler metrics on compact complex manifolds by introducing and studying a functional that is different according to whether the complex dimension of the manifold is 2 or higher.

Deletion-contraction triangles for Hausel–Proudfoot varieties

To a graph, Hausel and Proudfoot associate two complex manifolds, \mathfrak{B} and \mathfrak{D} , which behave, respectively, like moduli of local systems on a Riemann surface and moduli of Higgs bundles. For instance, \mathfrak{B} is a moduli space of microlocal sheaves, which generalize local systems, and \mathfrak{D} carries the structure of a complex integrable system. We show the Euler characteristics of these varieties count spanning subtrees of the graph, and the point-count over a finite field for \mathfrak{B} is a generating polynomial for spanning subgraphs. This polynomial satisfies a deletion-contraction relation, which we lift to a deletion-contraction exact triangle for the cohomology of \mathfrak{B} . There is a corresponding triangle for \mathfrak{D} . Finally, we prove that \mathfrak{B} and \mathfrak{D} are diffeomorphic, the diffeomorphism carries the weight filtration on the cohomology of \mathfrak{B} to the perverse Leray filtration on the cohomology of \mathfrak{D} , and all these structures are compatible with the deletion-contraction triangles.

Non-Kähler Calabi-Yau geometry and pluriclosed flow

Hermitian, pluriclosed metrics with vanishing Bismut-Ricci form give a natural extension of Calabi-Yau metrics to the setting of complex, non-Kähler manifolds, and arise independently in mathematical physics. We reinterpret this condition in terms of the Hermitian-Einstein equation on an associated holomorphic Courant algebroid, and thus refer to solutions as Bismut Hermitian-Einstein. This implies Mumford-Takemoto slope stability obstructions, and using these we exhibit infinitely many topologically distinct complex manifolds in every dimension with vanishing first Chern class which do not admit Bismut Hermitian-Einstein metrics. This reformulation also leads to a new description of pluriclosed flow in terms of Hermitian metrics on holomorphic Courant algebroids, implying new global existence results, in particular on all complex non-Kähler surfaces of Kodaira dimension κ≥0. On complex manifolds which admit Bismut-flat metrics we show global existence and convergence of pluriclosed flow to a Bismut-flat metric, which in turn gives a classification of generalized Kähler structures on these spaces.