Complex Manifold Research Articles

Estimates of Picard modular cusp forms

Abstract In this article, for n ≥ 2 {n\geq 2} , we compute asymptotic, qualitative, and quantitative estimates of the Bergman kernel of Picard modular cusp forms associated to torsion-free, cocompact subgroups of SU ⁢ ( ( n , 1 ) , ℂ ) {\mathrm{SU}((n,1),\mathbb{C})} . The main result of the article is the following result. Let Γ ⊂ SU ⁢ ( ( 2 , 1 ) , 𝒪 K ) {\Gamma\subset\mathrm{SU}((2,1),\mathcal{O}_{K})} be a torsion-free subgroup of finite index, where K is a totally imaginary field. Let ℬ Γ k {{{\mathcal{B}_{\Gamma}^{k}}}} denote the Bergman kernel associated to the 𝒮 k ⁢ ( Γ ) {\mathcal{S}_{k}(\Gamma)} , complex vector space of weight-k cusp forms with respect to Γ. Let 𝔹 2 {\mathbb{B}^{2}} denote the 2-dimensional complex ball endowed with the hyperbolic metric, and let X Γ := Γ \ 𝔹 2 {X_{\Gamma}:=\Gamma\backslash\mathbb{B}^{2}} denote the quotient space, which is a noncompact complex manifold of dimension 2. Let | ⋅ | pet {|\cdot|_{\mathrm{pet}}} denote the point-wise Petersson norm on 𝒮 k ⁢ ( Γ ) {\mathcal{S}_{k}(\Gamma)} . Then, for k ≫ 1 {k\gg 1} , we have the following estimate: sup z ∈ X Γ ⁡ | ℬ Γ k ⁢ ( z ) | pet = O Γ ⁢ ( k 5 2 ) , \sup_{z\in X_{\Gamma}}|{{\mathcal{B}_{\Gamma}^{k}}}(z)|_{\mathrm{pet}}=O_{% \Gamma}(k^{\frac{5}{2}}), where the implied constant depends only on Γ.

Phase-only beampattern synthesis for maximizing mainlobe gain via Riemannian Newton method

Beampattern synthesis with phase-only weights is extensively utilized in phased-array systems due to the virtue of low-cost and high power efficiency. It is always a non-convex optimization problem having constant modulus constraints (CMCs) and the performance significantly depends on the employed solvers. This paper discusses the phase-only beampattern synthesis for maximizing the mainlobe gain with the constraints on the mainlobe ripple and the relative sidelobe level. As CMC is equivalent to a complex circle manifold, the proposed beampattern synthesis is considered as a manifold optimization problem, and then decomposed into several subproblems under the framework of manifold alternating direction method of multipliers (ADMM). In particular, the subproblem for updating phase-only weights is efficiently tackled by a second-order Riemannian Newton method, which has faster convergence speed and no requirement on line search compared to widely-used first-order Riemannian manifold methods. The explicit convergence condition is derived and the computational complexity is also analyzed. Numerical results show that the proposed method has better performance on the mainlobe gain and sidelobe control accuracy, or less computational cost compared to several representative approaches, such as convex relaxation, standard ADMM, Riemannian gradient descent-based ADMM and Riemannian conjugate gradient methods.

Geometric pushforward in Hodge filtered complex cobordism and secondary invariants

We construct a functorial pushforward homomorphism in geometric Hodge filtered complex cobordism along proper holomorphic maps between arbitrary complex manifolds. This significantly improves previous results on such transfer maps and is a much stronger result than the ones known for differential cobordism of smooth manifolds. This enables us to define and provide a concrete geometric description of Hodge filtered fundamental classes for all proper holomorphic maps. Moreover, we give a geometric description of a cobordism analog of the Abel–Jacobi invariant for nullbordant maps which is mapped to the classical invariant under the Hodge filtered Thom morphism. For the latter we provide a new construction in terms of geometric cycles.

A note on p-Kähler structures on compact quotients of Lie groups

A p-Kähler structure on a complex manifold of complex dimension n is given by a d-closed transverse real (p, p)-form. In the paper, we study the existence of p-Kähler structures on compact quotients of simply connected Lie groups by discrete subgroups endowed with an invariant complex structure. In particular, we discuss the existence of p-Kähler structures on nilmanifolds, with a focus on the case p=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p =2$$\\end{document} and complex dimension n=4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n = 4$$\\end{document}. Moreover, we prove that a (n-2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(n-2)$$\\end{document}-Kähler almost abelian solvmanifold of complex dimension n≥3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n\\ge 3$$\\end{document} has to be Kähler.

A Structure Theorem for Neighborhoods of Compact Complex Manifolds

A Structure Theorem for Neighborhoods of Compact Complex Manifolds

Symplectic complexity of reductive group actions

Let a complex algebraic reductive group G act on a complex algebraic manifold X. For a G-invariant subvariety Ξ of the nilpotent cone N(g∗)⊂g∗ we define a notion of Ξ-symplectic complexity of X. This notion generalizes the notion of complexity defined in Vinberg (1986). We prove several properties of this notion, and relate it to the notion of Ξ-complexity defined in Aizenbud and Gourevitch (2024) motivated by its relation with representation theory.

A few remarks on the Poincaré metric on a singular holomorphic foliation

Let F be a Riemann surface foliation on M∖E, where M is a complex manifold and E⊂M is a closed set. Assume that F is hyperbolic, i.e. all the leaves of the foliation F are hyperbolic Riemann surfaces. Fix a Hermitian metric g on M. We consider the Verjovsky modulus of uniformization map η, which measures the largest possible derivative in the class of holomorphic maps from the unit disc into the leaves of F. Various results are known to ensure the continuity of the map η along the transverse directions, with suitable conditions on M, F and E. For a domain U⊂M, let FU be the holomorphic foliation given by the restriction of F to the domain U, i.e. F|U. We consider the modulus of uniformization map ηU corresponding to the foliation FU and study its variation when the corresponding domain U varies in the Caratheodory kernel sense, motivated by the work of Lins Neto and Canille Martins.

Algebraic cones of LCK manifolds with potential

A complex manifold X, dimX>2, is called “an LCK manifold with potential”, if it can be realized as a complex submanifold of a Hopf manifold. Let X˜ be its Z-covering, considered as a complex submanifold in Cn﹨0. We prove that X˜ is algebraic. We call the manifolds obtained this way the algebraic cones, and show that the affine algebraic structure on X˜ is independent from the choice of X. We give several intrinsic definitions of an algebraic cone, and prove that these definitions are equivalent.

Geometry of universal embedding spaces for almost complex manifolds

Geometry of universal embedding spaces for almost complex manifolds

Hermitian non-K<span class="BxUVEf ILfuVd" lang="en"><span class="hgKElc">ä</span></span>hler structures on products of principal $\mathrm{S}^1$-bundles over complex flag manifolds and applications in Hermitian geometry with torsion

Hermitian non-K<span class="BxUVEf ILfuVd" lang="en"><span class="hgKElc">ä</span></span>hler structures on products of principal $\mathrm{S}^1$-bundles over complex flag manifolds and applications in Hermitian geometry with torsion

Kodaira dimensions of almost complex manifolds II

Kodaira dimensions of almost complex manifolds II

Vaisman theorem for lcK spaces

Vaisman's theorem for locally conformally Kahler (lcK) compact manifolds states that any lcK metric on a compact complex manifold which admits a Kahler metric is, in fact, globally conformally Kahler (gcK). In this paper, we extend this theorem to compact complex spaces with singularities.

Estimates of Bergman kernels and Bergman metric on compact Picard surfaces

Let Γ⊂SU((2,1),C) be a torsion-free cocompact subgroup. Let B2 denote the 2-dimensional complex ball endowed with the hyperbolic metric μhyp, and let XΓ:=Γ﹨B2 denote the quotient space, which is a compact complex manifold of dimension 2. Let Λ:=ΩXΓ2 denote the line bundle on XΓ, whose sections are holomorphic (2,0)-forms. For any k≥1, the hyperbolic metric induces a point-wise metric on H0(XΓ,Λ⊗k), which we denote by |⋅|hyp. For any k≥1, let BΛk denote the Bergman kernel of the complex vector space H0(XΓ,Λ⊗k). For any k≥3, and z,w∈XΓ, the first main result of the article is the following off-diagonal estimate of the Bergman kernel BΛk|BΛk(z,w)|hyp=OXΓ(k2cosh3k−8⁡(dhyp(z,w)/2)), where dhyp(z,w) denotes the geodesic distance between the points z and w on XΓ, and the implied constant depends only on the Picard surface XΓ.For any k≥1, let μberk(z):=−i2π∂z∂z‾log⁡|BΛk(z,z)|hyp denote the Bergman metric associated the line bundle Λ⊗k, and let μberk,vol denote the associated volume form. For k≫1 sufficiently large, and ϵ>0, the second main result of the article is the following estimatesupz∈XΓ⁡|μberk,vol(z)μhypvol(z)|=OXΓ,ϵ(k4+ϵ), where μhypvol denotes the volume form associated to the hyperbolic metric μhyp, and the implied constant depends on the Picard surface XΓ, and on the choice of ϵ>0.

On a complex collar neighbourhood theorem

In this note we review and add an extra precision to the proof of our collar neighbourhood theorem for strictly pseudoconvex complex manifolds with boundary.

The continuity equation for Hermitian metrics: Calabi estimates, Chern scalar curvature, and Oeljeklaus–Toma manifolds

AbstractWe prove local Calabi and higher order estimates for solutions to the continuity equation introduced by La Nave–Tian and extended to Hermitian metrics by Sherman–Weinkove. We apply the estimates to show that on a compact complex manifold, the Chern scalar curvature of a solution must blow up at a finite‐time singularity. Additionally, starting from certain classes of initial data on Oeljeklaus–Toma manifolds, we prove Gromov–Hausdorff and smooth convergence of the metric to a particular nonnegative (1,1)‐form as .

Circle Actions on Four Dimensional Almost Complex Manifolds With Discrete Fixed Point Sets

Abstract We establish a necessary and sufficient condition for pairs of integers to arise as the weights at the fixed points of an effective circle action on a compact almost complex 4-manifold with a discrete fixed point set. As an application, we provide a necessary and sufficient condition for a pair of integers to arise as the Chern numbers of such an action, answering negatively a question by Sabatini whether $c_{1}^{2}[M] \leq 3 c_{2}[M]$ holds for any such manifold $M$. We achieve this by demonstrating that pairs of integers that arise as weights of a circle action also arise as weights of a restriction of a $\mathbb {T}^{2}$-action. Furthermore, we discuss applications to circle actions on complex/symplectic 4-manifolds and semi-free circle actions with discrete fixed point sets.

Mechanistic manifold in a hemoprotein-catalyzed cyclopropanation reaction with diazoketone

Hemoproteins have recently emerged as promising biocatalysts for new-to-nature carbene transfer reactions. However, mechanistic understanding of the interplay between productive and unproductive pathways in these processes is limited. Using spectroscopic, structural, and computational methods, we investigate the mechanism of a myoglobin-catalyzed cyclopropanation reaction with diazoketones. These studies shed light on the nature and kinetics of key catalytic steps in this reaction, including the formation of an early heme-bound diazo complex intermediate, the rate-determining nature of carbene formation, and the cyclopropanation mechanism. Our analyses further reveal the existence of a complex mechanistic manifold for this reaction that includes a competing pathway resulting in the formation of an N-bound carbene adduct of the heme cofactor, which was isolated and characterized by X-ray crystallography, UV-Vis, and Mössbauer spectroscopy. This species can regenerate the active biocatalyst, constituting a non-productive, yet non-destructive detour from the main catalytic cycle. These findings offer a valuable framework for both mechanistic analysis and design of hemoprotein-catalyzed carbene transfer reactions.

LMdist: Local Manifold distance accurately measures beta diversity in ecological gradients.

Differentiating ecosystems poses a complex, high-dimensional problem constrained by capturing relevant variation across species profiles. Researchers use pairwise distances and subsequent dimensionality reduction to highlight variation in a few dimensions. Despite popularity in analysis of ecological data, these low-dimensional visualizations can contain geometric abnormalities such as "arch" and "horseshoe" effects, potentially obscuring the impact of environmental gradients. These abnormalities appear in ordination but are in fact a product of oversaturated large pairwise distances. We present Local Manifold distance (LMdist), an unsupervised algorithm which adjusts pairwise beta diversity measures to better represent true ecological distances between samples. Beta diversity measures can have a bounded dynamic range in depicting long environmental gradients with high species turnover. Using a graph structure, LMdist projects pairwise distances onto a manifold and traverses the manifold surface to adjust pairwise distances at the upper end of the beta diversity measure's dynamic range. This allows for values beyond the range of the original measure. Not all datasets will have oversaturated pairwise distances, nor will capture variation that resembles a manifold, so LMdist adjusts only those pairwise values which may be undervalued in the presence of a sampled gradient. The adjusted distances serve as input for ordination and statistical testing. We demonstrate on real and simulated data that LMdist effectively recovers distances along known gradients and along complex manifolds such as the Swiss roll dataset. LMdist enables more powerful statistical tests for gradient effects and reveals variation orthogonal to the gradient. Available on GitHub at https://github.com/knights-lab/LMdist.

Berezin-type quantization on even-dimensional compact manifolds

In this article we show that a Berezin-type quantization can be achieved on a compact even dimensional manifold M 2d by removing a skeleton M 0 of lower dimension such that what remains is diffeomorphic to R 2d (cell decomposition) which we identify with Cd and embed in CPd . A local Poisson structure and Berezin-type quantization are induced from CPd . Thus we have a Hilbert space with a reproducing kernel. The symbols of bounded linear operators on the Hilbert space have a star product which satisfies the correspondence principle outside a set of measure zero. This construction depends on the diffeomorphism. One needs to keep track of the global holonomy and hence the cell decomposition of the manifold. As an example, we illustrate this type of quanitzation of the torus. We exhibit Berezin-Toeplitz quantization of a complex manifold in the same spirit as above.

Do products of compact complex manifolds admit LCK metrics?

AbstractAn locally conformally Kähler (LCK) manifold is a Hermitian manifold which admits a Kähler cover with deck group acting by holomorphic homotheties with respect to the Kähler metric. The product of two LCK manifolds does not have a natural product LCK structure. It is conjectured that a product of two compact complex manifolds is never LCK. We classify all known examples of compact LCK manifolds onto three not exclusive classes: LCK with potential, a class of manifolds we call of Inoue type, and those containing a rational curve. In the present paper, we prove that a product of an LCK manifold and an LCK manifold belonging to one of these three classes does not admit an LCK structure.