Complex Manifold Research Articles

On the asymptotic geometry of finite-type $k$-surfaces in three-dimensional hyperbolic space

For 0<k<1 , a finite-type k - surface in 3 -dimensional hyperbolic space is a complete, immersed surface of finite area and of constant extrinsic curvature equal to k . In Smith (2021), we showed that such surfaces have finite genus and finitely many cusp-like ends. Each of these cusps is asymptotic to an immersed cylinder of exponentially decaying radius about a complete geodesic and terminates at an ideal point which we call the extremity of the cusp. We show that every cusp of any finite-type k -surface has a well-defined axis, which we will call the Steiner geodesic of the cusp. One of the end-points of this axis is the extremity, and we will call the other, which constitutes new geometric data, the Steiner point of the cusp. We prove a new identity involving extremities and Steiner points in terms of Möbius invariant vector fields over the Riemann sphere. We define two new functionals over the space of finite-type k -surfaces. The first, which will be called the generalized volume , is defined by the integral of a certain well-chosen form, and extends to the non-embedded case the concept of volume of the set bounded by the surface. The second, which will be called the renormalized energy , is related to the integral of the mean curvature of the surface, and is well-defined up to a choice of Busemann function. Upon describing natural parametrizations of the strata of the space of finite-type k -surfaces by open complex manifolds, we prove a new Schläfli-type formula relating the extremities and Steiner points to the first order variations of the generalized volume and the renormalized energy. In particular, Möbius invariance of this formula yields the aforementioned identity. We conclude by studying some applications of this identity and Schläfli-type formula.

STAR-RIS Assisted Wireless Powered IoT Networks

The paper proposes a novel design of simultaneously transmitting and reflecting reconfigurable intelligent surfaces (STAR-RIS) in a wireless powered Internet of Things (IoT) network, where two sensor node groups (SNGs) harvest energy from a power station (PS) and transmit their message to an access point (AP) with the harvested energy. The STAR-RIS, which is deployed in the middle of the SNGs and adopts the time splitting (TS) working mode, can help the energy transfer in the wireless energy transfer (WET) phase and the information transfer in the wireless information transfer (WIT) phase. The paper aims to maximize the sum throughput from the two SNGs to the AP by jointly designing the phase shifts of the STAR-RIS and the working time allocated to the two SNGs in the WET and WIT phases, respectively. To solve the formulated non-convex optimization problem, we propose a low-complexity algorithm where we first derive the optimal phase shifts of the STAR-RIS in the WIT phase. Then, we adopt the Lagrange dual method to simplify the optimization problem and optimize the phase shifts of the STAR-RIS in the WET phase by the Majorization-Minimization (MM) algorithm and the complex circle manifold (CCM) algorithm. Next, a two-layer iterative algorithm is used to obtain the optimal values of time allocated to the two SNGs. Finally, we evaluate the improvement of the proposed scheme by the simulation results compared with other benchmark schemes.

Kähler-Einstein metrics and obstruction flatness of circle bundles

Obstruction flatness of a strongly pseudoconvex hypersurface Σ in a complex manifold refers to the property that any (local) Kähler-Einstein metric on the pseudoconvex side of Σ, complete up to Σ, has a potential −log⁡u such that u is C∞-smooth up to Σ. In general, u has only a finite degree of smoothness up to Σ. In this paper, we study obstruction flatness of hypersurfaces Σ that arise as unit circle bundles S(L) of negative Hermitian line bundles (L,h) over Kähler manifolds (M,g). We prove that if (M,g) has constant Ricci eigenvalues, then S(L) is obstruction flat. If, in addition, all these eigenvalues are strictly less than one and (M,g) is complete, then we show that the corresponding disk bundle admits a complete Kähler-Einstein metric. Finally, we give a necessary and sufficient condition for obstruction flatness of S(L) when (M,g) is a Kähler surface (dim⁡M=2) with constant scalar curvature.

Poincaré metric of holomorphic foliations with non-degenerate singularities

Consider a Brody hyperbolic foliation [Formula: see text] with non-degenerate singularities on a compact complex manifold. We show that its leafwise Poincaré metric is transversally Hölder continuous with a logarithmic slope towards the singular set of [Formula: see text].

Rank of the Nijenhuis tensor on parallelizable almost complex manifolds

We study almost complex structures on parallelizable manifolds via the rank of their Nijenhuis tensor. First, we show how the computations of such rank can be reduced to finding smooth functions on the underlying manifold solving a system of first order PDEs. On specific manifolds, we find an explicit solution. Then we compute the Nijenhuis tensor on curves of almost complex structures, showing that there is no constraint (except for lower semi-continuity) to the possible jumps of its rank. Finally, we focus on [Formula: see text]-nilmanifolds and the associated Lie algebras. We classify which [Formula: see text]-dimensional, nilpotent, real Lie algebras admit almost complex structures whose Nijenhuis tensor has a given rank, deducing the corresponding classification for invariant structures on [Formula: see text]-nilmanifolds. We also find a topological upper-bound for the rank of the Nijenhuis tensor for invariant almost complex structures on solvmanifolds of any dimension, obtained as a quotient of a completely solvable Lie group. Our results are complemented by a large number of examples.

Integration of generalized complex structures

We solve the integration problem for generalized complex manifolds, obtaining as the natural integrating object a holomorphic stack with a shifted symplectic structure; in other words, a real symplectic groupoid with a compatible complex structure is defined only up to Morita equivalence. We explain how such objects differentiate to give generalized complex manifolds, and we show that a generalized complex manifold is integrable in this sense if and only if its underlying real Poisson structure is integrable. We describe several concrete examples of these integrations. Crucial to our solution are new technical tools, which are of independent interest, namely, a reduction procedure for Lie groupoid actions on Courant algebroids, as well as certain local-to-global extension results for multiplicative forms on local Lie groupoids.

Complex curves in hypercomplex nilmanifolds with [formula omitted]-solvable Lie algebras

An operator I on a real Lie algebra g is called a complex structure operator if I2=−Id and the −1-eigenspace g1,0 is a Lie subalgebra in the complexification of g. A hypercomplex structure on a Lie algebra g is a triple of complex structures I,J and K on g satisfying the quaternionic relations. We call a hypercomplex nilpotent Lie algebra H-solvable if there exists a sequence of H-invariant subalgebrasg1H⊃g2H⊃⋯⊃gk−1H⊃gkH=0, such that [giH,giH]⊂gi+1H. We give examples of H-solvable hypercomplex structures on a nilpotent Lie algebra and conjecture that all hypercomplex structures on nilpotent Lie algebras are H-solvable. Let (N,I,J,K) be a compact hypercomplex nilmanifold associated to an H-solvable hypercomplex Lie algebra. We prove that, for a general complex structure L induced by quaternions, there are no complex curves in a complex manifold (N,L).

Relative connections on principal bundles and relative equivariant structures

We investigate relative holomorphic connections on a principal bundle over a family of compact complex manifolds. A sufficient condition is given for the existence of a relative holomorphic connection on a holomorphic principal bundle over a complex analytic family. We also introduce the notion of relative equivariant bundles and establish its relation with relative holomorphic connections on principal bundles.

Quasi-Projective Manifolds Uniformized by Carathéodory Hyperbolic Manifolds and Hyperbolicity of Their Subvarieties

Abstract Let $M$ be a Carathéodory hyperbolic complex manifold. We show that $M$ supports a real-analytic bounded strictly plurisubharmonic function. If $M$ is also complete Kähler, we show that $M$ admits the Bergman metric. When $M$ is strongly Carathéodory hyperbolic and is the universal covering of a quasi-projective manifold $X$, the Bergman metric can be estimated in terms of a Poincaré-type metric on $X$. It is also proved that any quasi-projective (resp. projective) subvariety of $X$ is of log-general type (resp. general type), a result consistent with a conjecture of Lang.

Pairs of Associated Yamabe Almost Solitons with Vertical Potential on Almost Contact Complex Riemannian Manifolds

Almost contact complex Riemannian manifolds, also known as almost contact B-metric manifolds, are, in principle, equipped with a pair of mutually associated pseudo-Riemannian metrics. Each of these metrics is specialized as a Yamabe almost soliton with a potential collinear to the Reeb vector field. The resulting manifolds are then investigated in two important cases with geometric significance. The first is when the manifold is of Sasaki-like type, i.e., its complex cone is a holomorphic complex Riemannian manifold (also called a Kähler–Norden manifold). The second case is when the soliton potential is torse-forming, i.e., it satisfies a certain recurrence condition for its covariant derivative with respect to the Levi–Civita connection of the corresponding metric. The studied solitons are characterized. In the three-dimensional case, an explicit example is constructed, and the properties obtained in the theoretical part are confirmed.

Relative regular Riemann–Hilbert correspondence II

We develop the theory of relative regular holonomic $\mathcal {D}$-modules with a smooth complex manifold $S$ of arbitrary dimension as parameter space, together with their main functorial properties. In particular, we establish in this general setting the relative Riemann–Hilbert correspondence proved in a previous work in the one-dimensional case.

On the algebra generated by μ ¯ , ∂ ¯ , ∂ , μ \overline{\mu },\overline{\partial },\partial ,\mu

Abstract In this note, we determine the structure of the associative algebra generated by the differential operators μ ¯ , ∂ ¯ , ∂ \overline{\mu },\overline{\partial },\partial , and μ \mu that act on complex-valued differential forms of almost complex manifolds. This is done by showing that it is the universal enveloping algebra of the graded Lie algebra generated by these operators and determining the structure of the corresponding graded Lie algebra. We then determine the cohomology of this graded Lie algebra with respect to its canonical inner differential [ d , − ] \left[d,-] , as well as its cohomology with respect to all its inner differentials.

On singular generalizations of the Singer–Hopf conjecture

AbstractThe Singer–Hopf conjecture predicts the sign of the topological Euler characteristic of a closed aspherical manifold. In this note, we propose singular generalizations of the Singer–Hopf conjecture, formulated in terms of the Euler–Mather characteristic, intersection homology Euler characteristic and, resp., virtual Euler characteristic of a closed irreducible subvariety of an aspherical complex projective manifold. We prove these new conjectures under the assumption that the cotangent bundle of the ambient variety is numerically effective (nef), or, more generally, when the ambient manifold admits a finite morphism to a complex projective manifold with a nef cotangent bundle. The main ingredients in the proof are the semi‐positivity properties of nef vector bundles together with a topological version of the Riemann–Roch theorem, proved by Kashiwara.

Quasi-plurisubharmonic envelopes 3: Solving Monge–Ampère equations on hermitian manifolds

Abstract We develop a new approach to L ∞ L^{\infty} -a priori estimates for degenerate complex Monge–Ampère equations on complex manifolds. It only relies on compactness and envelopes properties of quasi-plurisubharmonic functions. In a prequel [Quasi-plurisubharmonic envelopes 1: Uniform estimates on Kähler manifolds, preprint (2021), https://arxiv.org/abs/2106.04273], we have shown how this method allows one to obtain new and efficient proofs of several fundamental results in Kähler geometry. In [Quasi-plurisubharmonic envelopes 2: Bounds on Monge–Ampère volumes, Algebr. Geom. 9 (2022), 6, 688–713], we have studied the behavior of Monge–Ampère volumes on hermitian manifolds. We extend here the techniques of the former to the hermitian setting and use the bounds established in the latter, producing new relative a priori estimates, as well as several existence results for degenerate complex Monge–Ampère equations on compact hermitian manifolds.

Q-stability conditions on Calabi–Yau-𝕏 categories

We introduce $q$ -stability conditions $(\sigma,s)$ on Calabi–Yau- $\mathbb {X}$ categories $\mathcal {D}_\mathbb {X}$ , where $\sigma$ is a stability condition on $\mathcal {D}_\mathbb {X}$ and $s$ a complex number. We prove the corresponding deformation theorem, that $\operatorname {QStab}_s\mathcal {D}_\mathbb {X}$ is a complex manifold of dimension $n$ for fixed $s$ , where $n$ is the rank of the Grothendieck group of $\mathcal {D}_\mathbb {X}$ over $\mathbb {Z}[q^{\pm 1}]$ . When $s=N$ is an integer, we show that $q$ -stability conditions can be identified with the stability conditions on $\mathcal {D}_N$ , provided the orbit category $\mathcal {D}_N=\mathcal {D}_\mathbb {X}/[\mathbb {X}-N]$ is well defined. To attack the questions on existence and deformation along the $s$ direction, we introduce the inducing method. Sufficient and necessary conditions are given, for a stability condition on an $\mathbb {X}$ -baric heart (that is, a usual triangulated category) of $\mathcal {D}_\mathbb {X}$ to induce $q$ -stability conditions on $\mathcal {D}_\mathbb {X}$ . As a consequence, we show that the space $\operatorname {QStab}^\oplus \mathcal {D}_\mathbb {X}$ of (induced) open $q$ -stability conditions is a complex manifold of dimension $n+1$ . Our motivating examples for $\mathcal {D}_\mathbb {X}$ are coming from (Keller's) Calabi–Yau- $\mathbb {X}$ completions of dg algebras. In the case of smooth projective varieties, the $\mathbb {C}^*$ -equivariant coherent sheaves on canonical bundles provide the Calabi–Yau- $\mathbb {X}$ categories. Another application is that we show perfect derived categories can be realized as cluster- $\mathbb {X}$ categories for acyclic quivers.

Approximation and accumulation results of holomorphic mappings with dense image

We present four approximation theorems for manifold–valued mappings. The first one approximates holomorphic embeddings on pseudoconvex domains in $\mathbb{C}^n$ with holomorphic embeddings with dense images. The second theorem approximates holomorphic mappings on complex manifolds with bounded images with holomorphic mappings with dense images. The last two theorems work the other way around, constructing (in different settings) sequences of holomorphic mappings (embeddings in the first one) converging to a mapping with dense image defined on a given compact minus certain points (thus in general not holomorphic).

D3-brane supergravity solutions from Ricci-flat metrics on canonical bundles of Kähler–Einstein surfaces

D3 brane solutions of type IIB supergravity can be obtained by means of a classical Ansatz involving a harmonic warp factor, H(textbf{y},bar{textbf{y}}) multiplying at power -1/2 the first summand, i.e., the Minkowski metric of the D3 brane world-sheet, and at power 1/2 the second summand, i.e., the Ricci-flat metric on a six-dimensional transverse space mathcal {M}_6, whose complex coordinates y are the arguments of the warp factor. Of particular interest is the case where mathcal {M}_6={text {tot}}[ Kleft[ left( mathcal {M}_Bright) right] is the total space of the canonical bundle over a complex Kähler surface mathcal {M}_B. This situation emerges in many cases while considering the resolution à la Kronheimer of singular manifolds of type mathcal {M}_6=mathbb {C}^3/Gamma , where Gamma subset mathrm {SU(3)} is a discrete subgroup. When Gamma = mathbb {Z}_4, the surface mathcal {M}_B is the second Hirzebruch surface endowed with a Kähler metric having mathrm {SU(2)times U(1)} isometry. There is an entire class {text {Met}}(mathcal{F}mathcal{V}) of such cohomogeneity one Kähler metrics parameterized by a single function mathcal{F}mathcal{K}(mathfrak {v}) that are best described in the Abreu–Martelli–Sparks–Yau (AMSY) symplectic formalism. We study in detail a two-parameter subclass {text {Met}}(mathcal{F}mathcal{V})_{textrm{KE}}subset {text {Met}}(mathcal{F}mathcal{V}) of Kähler–Einstein metrics of the aforementioned class, defined on manifolds that are homeomorphic to S^2times S^2, but are singular as complex manifolds. Actually, {text {Met}}(mathcal{F}mathcal{V})_{textrm{KE}}subset {text {Met}}(mathcal{F}mathcal{V})_{textrm{ext}}subset {text {Met}}(mathcal{F}mathcal{V}) is a subset of a four parameter subclass {text {Met}}(mathcal{F}mathcal{V})_{textrm{ext}} of cohomogeneity one extremal Kähler metrics originally introduced by Calabi in 1983 and translated by Abreu into the AMSY action-angle formalism.{text {Met}}(mathcal{F}mathcal{V})_{textrm{ext}} contains also a two-parameter subclass {text {Met}}(mathcal{F}mathcal{V})_{textrm{ext}mathbb {F}_2} disjoint from {text {Met}}(mathcal{F}mathcal{V})_{textrm{KE}} of extremal smooth metrics on the second Hirzebruch surface that we rederive using constraints on period integrals of the Ricci 2-form. The Kähler–Einstein nature of the metrics in {text {Met}}(mathcal{F}mathcal{V})_{textrm{KE}} allows the construction of the Ricci-flat metric on their canonical bundle via the Calabi Ansatz, which we recast in the AMSY formalism deriving some new elegant formulae. The metrics in {text {Met}}(mathcal{F}mathcal{V})_{textrm{KE}} are defined on the base manifolds of U(1) fibrations supporting the family of Sasaki–Einstein metrics textrm{SEmet}_5 introduced by Gauntlett et al. (Adv Theor Math Phys 8:711–734, 2004), and already appeared in Gibbons and Pope (Commun Math Phys 66:267–290, 1979). However, as we show in detail using Weyl tensor polynomial invariants, the six-dimensional Ricci-flat metric on the metric cone of {mathcal M}_5 in {text {Met}}(textrm{SE})_5 is different from the Ricci-flat metric on {text {tot}}[ Kleft[ left( mathcal {M}_{textrm{KE}}right) right] constructed via Calabi Ansatz. This opens new research perspectives. We also show the full integrability of the differential system of geodesics equations on mathcal {M}_B thanks to a certain conserved quantity which is similar to the Carter constant in the case of the Kerr metric.

Quadratic relations between periods of connections

We prove the existence of quadratic relations between periods of meromorphic flat bundles on complex manifolds with poles along a divisor with normal crossings under the assumption of "goodness". In dimension one, for which goodness is always satisfied, we provide methods to compute the various pairings involved. In an appendix, we give details on the classical results needed for the proofs. V3: Revised version, various proofs simplified in Section 3, exposition improved.

CYT and SKT Metrics on Compact Semi-Simple Lie Groups

A Hermitian metric on a complex manifold $(M, I)$ of complex dimension $n$ is called Calabi-Yau with torsion (CYT) or Bismut-Ricci flat, if the restricted holonomy of the associated Bismut connection is contained in ${\rm SU}(n)$ and it is called strong K\"ahler with torsion (SKT) or pluriclosed if the associated fundamental form $F$ is $\partial \overline \partial$-closed. In the paper we study the existence of left-invariant SKT and CYT metrics on compact semi-simple Lie groups endowed with a Samelson complex structure $I$. In particular, we show that if $I$ is determined by some maximal torus $T$ and $g$ is a left-invariant Hermitian metric, which is also invariant under the right action of the torus $T$, and is both CYT and SKT, then $g$ has to be Bismut flat.

The probabilistic vs the quantization approach to Kähler–Einstein geometry

In the probabilistic construction of Kähler–Einstein metrics on a complex projective algebraic manifold X—involving random point processes on X—a key role is played by the partition function. In this work a new quantitative bound on the partition function is obtained. It yields, in particular, a new direct analytic proof that X admits a Kähler–Einstein metrics if it is uniformly Gibbs stable. The proof makes contact with the quantization approach to Kähler–Einstein geometry.