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In this paper, we show the minimal total Betti number of a closed almost complex manifold of dimension 2 n ≥ 8 is four, thus confirming a conjecture of Sullivan except for dimension 6. Along the way, we prove the only simply connected closed complex manifold having total Betti number three is the complex projective plane. [ABSTRACT FROM AUTHOR]

Inspired by the classical Riemannian systolic inequality of Gromov, we present a combinatorial analogue providing a lower bound on the number of vertices of a simplicial complex in terms of its edge-path systole. Similarly to the Riemannian case, where the inequality holds under a topological assumption of "essentiality", our proofs rely on a combinatorial analogue of that assumption. Under a stronger assumption, expressed in terms of cohomology cup-length, we improve our results quantitatively. We also illustrate our methods in the continuous setting, generalizing and improving quantitatively the Minkowski principle of Balacheff and Karam; a corollary of this result is the extension of the Guth–Nakamura cup-length systolic bound from manifolds to complexes. [ABSTRACT FROM AUTHOR]

The sociolinguistic background of multilingual rural societies in West Africa and the prevailing conditions of language transmission are quite different from those found in most immigrant situations in the Global North. In the case focused upon here, the target language itself is under constant pressure from other, more dominant contact languages, and the usual repertoire of a fully competent speaker already involves a larger number of language sources and internal variations. This article explores the verbal behaviour of three speakers of the endangered language Pana (Gur/Niger-Congo; Mali/Burkina Faso) who experienced varying degrees of interrupted language transmission in earlier life times. They were all brought up in situations where only one of the parents spoke Pana as a first language and where it was not part of their general linguistic environment. The speakers find themselves now in a setting where local people prefer Pana and consider it the most appropriate code for village dwellers in community-internal communication. Accordingly, the speakers under scrutiny struggle with the communicative obligations and try to cope with their usually fully competent conversation partners' expectations. The presented analysis of discourse data shows the manifold and complex linguistic and social implications of such a situation. It will be argued that it is correspondingly difficult to disentangle general language contact phenomena from variation introduced through incomplete second-language acquisition. Furthermore, the data strongly suggests that the background of a diffuse linguistic system and a relatively unfocused society entails a greater liberty for the scrutinized speakers' communicative possibilities. Regarding norm adherence, the partners in discourse seem to stretch the acceptance of linguistic variation to the very limits of the already diffuse linguistic system as long as social conduct and behavioural norms of communication are respected. [ABSTRACT FROM AUTHOR]

In this paper, we study the Z / 2 action on complex Grassmann manifolds G n (C 2 n) given by taking orthogonal complement. We completely compute the associated Z / 2 Fadell–Husseini index. Our study is parallel to the study of the index of real Grassmann manifolds G n (R 2 n) by Baralić et al. [Forum Math., 30 (2018), pp. 1539–1572]. [ABSTRACT FROM AUTHOR]

Employing a variant of H¨ormander's approach to Andreotti--Grauert's theory, a comparison theorem is proved between some bundle-valued weighted L2 ¯∂- cohomology groups of a class of locally pseudoconvex bounded domains in complex manifolds. As a result, such domains will turn out to be domains of meromorphy if the manifold admits a Hermitian line bundle whose curvature form is positive on the boundary of the domain. Particularly, a domain in a compact complex algebraic surface admits a meromorphic function whose essential singularities are the points on the boundary if the complement of the domain is a complex curve of self-intersection zero. [ABSTRACT FROM AUTHOR]

We investigate the relative lie algebroid connections on a holomorphic vector bundle over a family of compact complex manifolds (or smooth projective varieties over ℂ). We provide a sufficient condition for the existence of a relative Lie algebroid connection on a holomorphic vector bundle over a complex analytic family of compact complex manifolds. We show that the relative Lie algebroid Chern classes of a holomorphic vector bundle admitting relative Lie algebroid connection vanish, if each of the fibers of the complex analytic family is compact and Kähler. Moreover, we consider the moduli space of relative Lie algebroid connections and we show that there exists a natural relative compactification of this moduli space. [ABSTRACT FROM AUTHOR]

We describe a rigidity phenomenon exhibited by the second Chern Ricci curvature of a Hermitian metric on a compact complex manifold. This yields a characterisation of second Chern Ricci‐flat Hermitian metrics on several types of manifolds as well as a range of non‐existence results for such metrics. [ABSTRACT FROM AUTHOR]

In this paper, we derive the second order estimate to the 2nd Hessian type equation on a compact almost Hermitian manifold. [ABSTRACT FROM AUTHOR]

Introduction: Treatment with chimeric antigen receptor T (CAR-T) cells involves a large number of interdisciplinary stakeholders and is associated with complex processes ranging from patient-specific production to follow-up care. Due to the complexity, maximum process optimization is required in order to avoid efficiency losses. This study aimed at systematically determining the preconditions for a frictionless flow of the CAR-T process by surveying the stakeholders involved. Methods: A Group Concept Mapping (GCM) analysis, a mixed-methods participatory research, was conducted. CAR-T experts from different professional backgrounds went through three steps: 1) Brainstorming relevant aspects (statements) for a frictionless process, 2) Sorting the collected statements based on their similarity, and 3) Rating the importance and feasibility of each statement. A cluster map reflecting the overarching topics was derived, and mean ratings per statement and cluster were calculated. Results: Overall, 20 CAR-T experts participated. A total of 80 statements were collected, resulting in a map of the following 10 clusters (mean importance/feasibility): Information for patients and physicians (4.16/3.77), Supportive network (4.03/3.53), Eligibility of patients (4.41/3.63), Evidence, transparency and communication (4.01/3.33), Paperwork (4.1/2.52), Interface with pharmaceutical manufacturer (4.03/2.85), Reimbursement (4.29/2.31), Quality Management (4.17/3.18), Infrastructure of CAR-T clinics (4.1/2.93), and Patientoriented processes (4.46/3.32). Discussion: The 80 statements underlined the complex and manifold nature of the CAR-T treatment process. Our results reflect the first step in overcoming hurdles: identifying potential hurdles and required preconditions. Decisionmakers and stakeholders can use the results to derive strategies and measures to further promote a frictionless process. [ABSTRACT FROM AUTHOR]

We show that the parabolic quaternionic Monge–Ampère equation on a compact hyperkähler manifold has always a long-time solution which, once normalized, converges smoothly to a solution of the quaternionic Monge–Ampère equation. This is the same setting in which Dinew and Sroka (2023) prove the conjecture of Alesker and Verbitsky (2010). We also introduce an analogue of the Chern–Ricci flow in hyperhermitian manifolds. [ABSTRACT FROM AUTHOR]

In this paper, we study the spaces of (d+dc)-harmonic forms and of (d+dΛ)-harmonic forms, a natural generalization of the spaces of Bott–Chern harmonic forms (respectively, symplectic harmonic forms) from complex (respectively, symplectic) manifolds to almost Hermitian manifolds. We apply the same techniques to compact complex surfaces, computing their Bott–Chern and Aeppli numbers and their spaces of (d+dΛ)-harmonic forms. We give several applications to compact quotients of Lie groups by a lattice. [ABSTRACT FROM AUTHOR]

Let X and S be complex analytic manifolds where S plays the role of a parameter space. Using the sheaf D∞X×S/S​ of relative differential operators of infinite order, we construct functorially the regular holonomic DX×S/S​-module Mreg​ associated to a relative holonomic DX×S/S​-module M, extending to the relative case classical theorems by Kashiwara–Kawai: denoting by M∞ the tensor product of M by D∞X×S/S​ we make M∞ explicit in terms of the sheaf of holomorphic solutions of M. As a consequence of the relative Riemann–Hilbert correspondence we conclude that M∞ and M∞​reg are isomorphic. [ABSTRACT FROM AUTHOR]

Mirror symmetry was first observed in worldsheet string constructions, and was shown to have profound implications in the Effective Field Theory (EFT) limit of string compactifications, and for the properties of Calabi–Yau manifolds. It opened up a new field in pure mathematics, and was utilised in the area of enumerative geometry. Spinor–Vector Duality (SVD) is an extension of mirror symmetry. This can be readily understood in terms of the moduli of toroidal compactification of the Heterotic String, which includes the metric the antisymmetric tensor field and the Wilson line moduli. In terms of the toroidal moduli, mirror symmetry corresponds to mappings of the internal space moduli, whereas Spinor–Vector Duality corresponds to maps of the Wilson line moduli. In the past few of years, we demonstrated the existence of Spinor–Vector Duality in the effective field theory compactifications of string theories. This was achieved by starting with a worldsheet orbifold construction that exhibited Spinor–Vector Duality and resolving the orbifold singularities, hence generating a smooth, effective field theory limit with an imprint of the Spinor–Vector Duality. Just like mirror symmetry, the Spinor–Vector Duality can be used to study the properties of complex manifolds with vector bundles. Spinor–Vector Duality offers a top-down approach to the "Swampland" program, by exploring the imprint of the symmetries of the ultra-violet complete worldsheet string constructions in the effective field theory limit. The SVD suggests a demarcation line between (2,0) EFTs that possess an ultra-violet complete embedding versus those that do not. [ABSTRACT FROM AUTHOR]

The existence or nonexistence of a complex structure on a differential manifold is a central problem in differential geometry. In particular, this problem on S 6 was a long-standing unsolved problem, and differential geometry is an important tool. Recently, G. Clemente found a necessary and sufficient condition for almost-complex structures on a general differential manifold to be complex structures by using a covariant exterior derivative in three articles. However, in two of them, G. Clemente used a stronger condition instead of the published one. From there, G. Clemente proved the nonexistence of the complex structure on S 6 . We study the related differential operators and give some examples of nilmanifolds. And we prove that the earlier condition is too strong for an almost complex structure to be integrable. In another word, we clarify the situation of this problem. [ABSTRACT FROM AUTHOR]

The Dirac-Dolbeault operator for a compact Kähler manifold is a special case of Dirac operator. The Green function for the Dirac Laplacian over a Riemannian manifold with boundary allows the expression of the values of the sections of the Dirac bundle in terms of the values on the boundary, extending the mean value theorem of harmonic analysis. Utilizing this representation and the Nash–Moser generalized inverse function theorem, we prove the existence of complex submanifolds of a complex projective manifold satisfying globally a certain partial differential equation under a certain injectivity assumption. Thereby, internal symmetries of Dolbeault and rational Hodge cohomologies play a crucial role. Next, we show the existence of complex submanifolds whose fundamental classes span the rational Hodge classes, proving the Hodge conjecture for complex projective manifolds. [ABSTRACT FROM AUTHOR]

Given a complex manifold X and a smooth positive function η thereon, we perturb the standard differential operator d = ∂ + ∂ ¯ acting on differential forms to a first-order differential operator D η whose principal part is η ∂ + ∂ ¯ . The role of the zero-th order part is to force the integrability property D η 2 = 0 that leads to a cohomology isomorphic to the de Rham cohomology of X, while the components of types (0 , 1) and (1 , 0) of D η induce cohomologies isomorphic to the Dolbeault and conjugate-Dolbeault cohomologies. We compute Bochner-Kodaira-Nakano-type formulae for the Laplacians induced by these operators and a given Hermitian metric on X. The computations throw up curvature-like operators of order one that can be made (semi-)positive under appropriate assumptions on the function η . As applications, we obtain vanishing results for certain harmonic spaces on complete, non-compact, manifolds and for the Dolbeault cohomology of compact complex manifolds that carry certain types of functions η . This study continues and generalises the one of the operators d h = h ∂ + ∂ ¯ that we introduced and investigated recently for a positive constant h that was then let to converge to 0 and, more generally, for constants h ∈ C . The operators d h had, in turn, been adapted to complex structures from the well-known adiabatic limit construction for Riemannian foliations. Allowing now for possibly non-constant functions η creates positivity in the curvature-like operator that stands one in good stead for various kinds of applications. [ABSTRACT FROM AUTHOR]

In this paper, we give sufficient conditions for Cauchy-completeness of Kobayashi hyperbolic domains in complex manifolds. The first result gives a sufficient condition for completeness for relatively compact domains in several large classes of manifolds. This follows from our second result, which may be of independent interest, in a much more general setting. This extends a result of Gaussier to the setting of manifolds. [ABSTRACT FROM AUTHOR]

In this paper, we consider the L2-existence of the Cauchy–Riemann equation on weakly pseudoconvex manifolds with strong Donnelly–Fefferman property. [ABSTRACT FROM AUTHOR]

This paper focuses on algorithms for the truncated Takagi factorization of complex symmetric matrices. The problem is formulated as a Riemannian optimization problem on a complex Stiefel manifold and then is converted into a real Riemannian optimization problem on the intersection of the real Stiefel manifold and the quasi-symplectic set. The steepest descent, the Riemannian nonmonotone conjugate gradient, Newton, and hybrid methods are used for solving the problem and they are compared in their performance for the optimization task. Numerical experiments are provided to illustrate the efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

We prove an L²loc-Hartogs-Bochner type extension theorem for unbounded domain D in a complex manifold X of complex dimension n ≥ 2. More precisely, we show that if Φ is a paracompactifying family of closed subsets of X not containing X, then the ...-cohomology group of (0, 1)-currents of class ... on X with supports in Φ is isomorphic to the ...-cohomology group of (0, 1)-forms with L²loc(X)-coefficients and with supports in Φ. Moreover, we prove that a sufficient condition for CR L²loc-functions, defined on the boundary ∂ D of D, being extended holomorphically to D is that the ...-cohomology groups must vanish. Similar results are given in the ... and Lp-categories. [ABSTRACT FROM AUTHOR]

The prescribed rational functions constitute a subset of rational functions satisfying certain symmetry and analyticity conditions. Over a smooth complex curve M, we construct explicitly a bundle 풲M with values in the prescribed rational functions. An intrinsic coordinate-independent formulation (the main result of the paper, Proposition 1) for such bundles is given. The construction presented in this paper is useful for studies of the canonical cosimplicial cohomology of infinite-dimensional Lie algebras on smooth manifolds, as well as for the purposes of conformal field theory and the theory of foliations. [ABSTRACT FROM AUTHOR]

Given a compact almost complex manifold (M2n,J), the almost complex invariant hJp,q is defined as the complex dimension of the cohomology space {[α] ∈ HdRp+q(M2n; ℂ)|α ∈ Ap,q(M2n),dα = 0}. Its properties have been studied mainly when 2n = 4. If we endow (M2n,J) with an almost Hermitian metric g, then the number hdp,q, i.e. the complex dimension of the space of Hodge–de Rham harmonic (p,q)-forms, does not depend on the choice of almost Kähler metrics when 2n = 4. In this paper, we study the relationship between hJp,q and hdp,q in dimension 2n ≥ 4. We prove hJn,0 = 0 if J is non-integrable and observe that hdp,0 = h Jp,0 if the metric is almost Kähler. If M2n is a compact quotient of a completely solvable Lie group and (J,g,ω) is a left-invariant almost Kähler structure on M, we prove hd1,1 = h J1,1. Finally, we study the 풞∞-pure and 풞∞-full properties of J on n-forms for the special dimension 2n = 4m. [ABSTRACT FROM AUTHOR]

We extend the Hirzebruch–Milnor class of a hypersurface |$X$| in an ambient complex algebraic manifold to the case where the normal bundle is nontrivial and |$X$| cannot be defined by a global function, using the associated line bundle and the graded quotients of the monodromy filtration. The earlier definition requiring a global defining function of |$X$| can be applied rarely to projective hypersurfaces with non-isolated singularities. Indeed, it is surprisingly difficult to get a one-parameter smoothing with total space smooth without destroying the singularities by blowing-ups (except certain quite special cases). As an application, assuming the singular locus is a projective variety, we show that the minimal exponent of a hypersurface can be captured by the spectral Hirzebruch–Milnor class, and higher du Bois and rational singularities of a hypersurface are detectable by the unnormalized Hirzebruch–Milnor class. Here the unnormalized class can be replaced by the normalized one in the higher du Bois case, but for the higher rational case, we must use also the decomposition of the Hirzebruch–Milnor class by the action of the semisimple part of the monodromy (which is equivalent to the spectral Hirzebruch–Milnor class). We cannot extend these arguments to the non-projective compact case by Hironaka's example. [ABSTRACT FROM AUTHOR]

The focus of this paper is on a volume form defined on a pseudoconvex hypersurface M in a complex Calabi–Yau manifold (that is, a complex n -manifold with a nowhere-vanishing holomorphic n -form). We begin by defining this volume form and observing that it can be viewed as a generalization of the affine-invariant volume form on a convex hypersurface in R n . We compute the first variation, which leads to a similar generalization of the affine mean curvature. In Sec. 2, we investigate the constrained variational problem, for pseudoconvex hypersurfaces M bounding compact domains Ω ⊂ Z. That is, we study critical points of the volume functional A (M) where the ordinary volume V (Ω) is fixed. The critical points are analogous to constant mean curvature submanifolds. We find that Sasaki–Einstein hypersurfaces satisfy the condition, and in particular the standard sphere S 2 n − 1 ⊂ C n does. The main work in the paper comes in Sec. 3 where we compute the second variation about the sphere. We find that it is negative in "most" directions but non-negative in directions corresponding to deformations of S 2 n − 1 by holomorphic diffeomorphisms. We are led to conjecture a "minimax" characterization of the sphere. We also discuss connections with the affine geometry case and with Kähler–Einstein geometry. Our original motivation for investigating these matters came from the case n = 3 and the embedding problem studied in our previous paper [S. Donaldson and F. Lehmann, Closed 3-forms in five dimensions and embedding problems, preprint (2022), arXiv:2210.16208]. There are some special features in this case. The volume functional can be defined without reference to the embedding in Z using only a closed "pseudoconvex" real 3 -form on M. In Sec. 4, we review this and develop some of the theory from the point of the symplectic structure on exact 3 -forms on M and the moment map for the action of the diffeomorphisms of M. [ABSTRACT FROM AUTHOR]

Generalized Calabi–Gray manifolds are non-Kähler complex manifolds with very explicit geometry yet not being homogeneous. In this note, we demonstrate how generalized Calabi–Gray manifolds can be used to answer some questions in non-Kähler geometry. [ABSTRACT FROM AUTHOR]

In 1954, Hirzebruch reported a problem posed by Kodaira and Spencer: on compact almost complex manifolds, is the dimension h∂̄p,q of the kernel of the Dolbeault Laplacian independent of the choice of almost Hermitian metric? In this paper, we review recent progresses on the original problem and we introduce a similar one: on compact almost complex manifolds, find a generalization of Bott–Chern and Aeppli numbers which is metric-independent. We find a solution to our problem valid on almost Kähler 4-manifolds. [ABSTRACT FROM AUTHOR]

The kinase pathway plays a crucial role in blood vessel function. Particular attention is paid to VEGFR type 2 angiogenesis and vascular morphogenesis as the tyrosine kinase pathway is preferentially activated. In silico studies were performed on several peptides that affect VEGFR2 in both stimulating and inhibitory ways. This investigation aims to examine the molecular properties of VEGFR2, a molecule primarily involved in the processes of vasculogenesis and angiogenesis. These relationships were defined by the interactions between Vascular Endothelial Growth Factor receptor 2 (VEGFR2) and the structural features of the systems. The chemical space of the inhibitory peptides and stimulators was described using topological and energetic properties. Furthermore, chimeric models of stimulating and inhibitory proteins (for VEGFR2) were computed using the protein system structures. The interaction between the chimeric proteins and VEGFR was computed. The chemical space was further characterized using complex manifolds and high-dimensional data visualization. The results show that a slightly similar chemical area is shared by VEGFR2 and stimulating and inhibitory proteins. On the other hand, the stimulator peptides and the inhibitors have distinct chemical spaces. [ABSTRACT FROM AUTHOR]

We generalize the notion of semi-universality in the classical deformation problems to the context of derived deformation theories. A criterion for a formal moduli problem to be semi-prorepresentable is produced. This can be seen as an analogue of Schlessinger's conditions for a functor of Artinian rings to have a semi-universal element. We also give a sufficient condition for a semi-prorepresentable formal moduli problem to admit a G-equivariant structure in a sense specified below, where G is a linearly reductive group. Finally, by making use of these criteria, we derive many classical results including the existence of (G-equivariant) formal semi-universal deformations of algebraic schemes and that of complex compact manifolds. [ABSTRACT FROM AUTHOR]

Let f:M2n→R2n+p$f: M^{2n}\rightarrow \mathbb {R}^{2n+p}$ denote an isometric immersion of a Kähler manifold with complex dimension n≥2$n\ge 2$ into Euclidean space with codimension p$p$. We show that generic rank conditions on the second fundamental form of a non‐minimal complete real Kähler submanifold f$f$ imply that f$f$ is a cylinder over a real Kähler submanifold g:N2p→R2p+p$g: N^{2p}\rightarrow \mathbb {R}^{2p+p}$. [ABSTRACT FROM AUTHOR]

Using tools from the theory of Lie groupoids, we study the category of logarithmic flat connections on principal G-bundles, where G is a complex reductive structure group. Flat connections on the affine line with a logarithmic singularity at the origin are equivalent to representations of a groupoid associated to the exponentiated action of C . We show that such representations admit a canonical Jordan–Chevalley decomposition and may be linearized by converting the C -action to a C ∗ -action. We then apply these results to give a functorial classification. Flat connections on a complex manifold with logarithmic singularities along a hypersurface are equivalent to representations of a twisted fundamental groupoid. Using a Morita equivalence, whose construction is inspired by Deligne's notion of paths with tangential basepoints, we prove a van Kampen type theorem for this groupoid. This allows us to show that the category of representations of the twisted fundamental groupoid can be localized to the normal bundle of the hypersurface. As a result, we obtain a functorial Riemann–Hilbert correspondence for logarithmic connections in terms of generalized monodromy data. [ABSTRACT FROM AUTHOR]

This article examines the manifold and complex responses of fifteenth-century elite politicians and writers to 'public' politics in Castile. Through analysis of a variety of sources including chronicles, allegorical poems, treatises, glosses and letters, it shows how the multiple conceptions of non-elite agency and attitudes to it can nuance our understanding of Castilian politics in the late Middle Ages. It argues that fifteenth-century chronicles, glosses and allegorical poems demonstrate a new attention from the elite to wider contexts beyond the confines of the traditional political society, which responded both to literary fashions and to real changes in the political reality of late medieval society. Moreover, their complex and even contradictory responses, which denigrated, appropriated and addressed these wider 'publics', ought to be considered integral to the development of 'public opinion', as part of a set of discursive and institutional struggles for the right to express political opinions. [ABSTRACT FROM AUTHOR]

The Petrie conjecture asserts that if a homotopy \mathbb {CP}^n admits a non-trivial circle action, its Pontryagin class agrees with that of \mathbb {CP}^n. Petrie proved this conjecture in the case where the manifold admits a T^n-action. An almost complex torus manifold is a 2n-dimensional compact connected almost complex manifold equipped with an effective T^n-action that has fixed points. For an almost complex torus manifold, there exists a graph that encodes information about the weights at the fixed points. We prove that if a 2n-dimensional almost complex torus manifold M only shares the Euler number with the complex projective space \mathbb {CP}^n, the graph of M agrees with the graph of a linear T^n-action on \mathbb {CP}^n. Consequently, M has the same weights at the fixed points, Chern numbers, cobordism class, Hirzebruch \chi _y-genus, Todd genus, and signature as \mathbb {CP}^n, endowed with the standard linear action. Furthermore, if M is equivariantly formal, the equivariant cohomology and the Chern classes of M and \mathbb {CP}^n also agree. [ABSTRACT FROM AUTHOR]

To a graph, Hausel and Proudfoot associate two complex manifolds, B and D, which behave, respectively, like moduli of local systems on a Riemann surface and moduli of Higgs bundles. For instance, B is a moduli space of microlocal sheaves, which generalize local systems, and 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 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 B. There is a corresponding triangle for D. Finally, we prove that B and D are diffeomorphic, the diffeomorphism carries the weight filtration on the cohomology of B to the perverse Leray filtration on the cohomology of D, and all these structures are compatible with the deletion-contraction triangles. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

A notable example due to Heier, Lu, Wong, and Zheng shows that there exist compact complex Kähler manifolds with ample canonical line bundle such that the holomorphic sectional curvature is negative semi‐definite and vanishes along high‐dimensional linear subspaces in every tangent space. The main result of this note is an upper bound for the dimensions of these subspaces. Due to the holomorphic sectional curvature being a real‐valued bihomogeneous polynomial of bidegree (2,2) on every tangent space, the proof is based on making a connection with the work of D'Angelo on complex subvarieties of real algebraic varieties and the decomposition of polynomials into differences of squares. Our bound involves an invariant that we call the holomorphic sectional curvature square decomposition length, and our arguments work as long as the holomorphic sectional curvature is semi‐definite, be it negative or positive. [ABSTRACT FROM AUTHOR]

In this paper, we apply a result of the classification of a compact cohomogeneity one Riemannian manifold with a compact Lie group G to obtain a classification of compact cohomogeneity one locally conformal Kähler manifolds. In particular, we prove that the compact complex manifold is a complex one-dimensional torus bundle over a projective rational homogeneous, or cohomogeneity one manifold except of a class of manifolds with a generalized Hopf surface bundle over a projective rational homogeneous space. Additionally, it is a homogeneous compact complex manifold under the complexification G C of the given compact Lie group G under an extra condition that the related closed one form is cohomologous to zero on the generic G orbit. Moreover, the semi-simple part S of the Lie group action has hypersurface orbits, i.e., it is of cohomogeneity one with respect to the semi-simple Lie group S in that special case. [ABSTRACT FROM AUTHOR]

We show that the knowledge of the Dirichlet–to–Neumann map for a nonlinear magnetic Schrödinger operator on the boundary of a compact complex manifold, equipped with a Kähler metric and admitting sufficiently many global holomorphic functions, determines the nonlinear magnetic and electric potentials uniquely. [ABSTRACT FROM AUTHOR]

Let |$(M,I,J,K,\Omega)$| be a compact HKT manifold, and let us denote with |$\partial$| the conjugate Dolbeault operator with respect to I , |$\partial_J:=J^{-1}\overline\partial J$|⁠ , |$\partial^\Lambda:=[\partial,\Lambda]$|⁠ , where Λ is the adjoint of |$L:=\Omega\wedge-$|⁠. Under suitable assumptions, we study Hodge theory for the complexes |$(A^{\bullet,0},\partial,\partial_J)$| and |$(A^{\bullet,0},\partial,\partial^\Lambda)$| showing a similar behavior to Kähler manifolds. In particular, several relations among the Laplacians, the spaces of harmonic forms and the associated cohomology groups, together with Hard Lefschetz properties, are proved. Moreover, we show that for a compact HKT |$\mathrm{SL}(n,\mathbb{H})$| -manifold, the differential graded algebra |$(A^{\bullet,0},\partial)$| is formal and this will lead to an obstruction for the existence of an HKT |$\mathrm{SL}(n,\mathbb{H})$| structure |$(I,J,K,\Omega)$| on a compact complex manifold (M ,  I). Finally, balanced HKT structures on solvmanifolds are studied. [ABSTRACT FROM AUTHOR]

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 [ABSTRACT FROM AUTHOR]

This paper proposes an optimal beamforming strategy for a downlink multi-user multi-input–multi-output (MIMO) dual-function radar communication (DFRC) system with dirty paper coding (DPC) adopted at the transmitter. We aim to achieve the maximum weighted sum rate of communicating users while adhering to a predetermined transmit covariance constraint for radar performance assurance. To make the intended problem trackable, we leverage the equivalence of the weighted sum rate and the weighted minimum mean squared error (MMSE) to reframe the issue and devise a block coordinate descent (BCD) approach to iteratively calculate transmit and receive beamforming solutions. Through this methodology, we demonstrate that the optimal receive beamforming aligns with the traditional MMSE approach, whereas the optimal transmit beamforming design can be cast into a quadratic optimization problem defined on a complex Stiefel manifold. Based on the majorization–minimization (MM) method, an iterative algorithm is then developed to compute the optimal transmit beamforming design by solving a series of orthogonal Procrustes problems (OPPs) that admit closed-form optimal solutions. Numerical findings serve to validate the efficacy of our scheme. It is demonstrated that our approach can achieve at least 73% higher spectral efficiency than the existing methods in a high signal-to-noise ratio (SNR) regime. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to investigate the relationship between continuation of pluriharmonic functions from the boundary of an unbounded domain and the vanishing of the Bott-Chern cohomology with supports in a paracompactifying family of closed subset of a complex manifold X. We moreover give a relation between distributional boundary values and extensible currents. [ABSTRACT FROM AUTHOR]

For a foliation ℱ of a smooth complex manifold, we introduce the category of V -structures associated to a vertex operator algebra V and the category of its modules. The main result consists of the construction of V -structures and canonicity proof of on ℱ. [ABSTRACT FROM AUTHOR]

Strominger–Yau–Zaslow (SYZ) proposed a way of constructing mirror pairs as pairs of torus fibrations. We apply this SYZ construction to toric Fano surfaces as complex manifolds, and discuss the homological mirror symmetry, where we consider Morse homotopy of the moment polytope instead of the Fukaya category. [ABSTRACT FROM AUTHOR]

We show that the same algebraic data that permit to construct the Lax pair and the r-matrix of an integrable non-linear σ-model in 1 + 1 dimensions can be also used for the construction of Lax pairs and of r-matrices of several other non-trivial integrable theories in 1 + 0 dimension. We call those new integrable theories the point particle E -models, we describe their structure and give their physical interpretation. We work out in detail the point particle E -modelsassociated to the bi-Yang–Baxter deformation of the SU(N) principal chiral model. In particular, for each complex flag manifold we thus obtain a two-parameter family of integrable models living on it. [ABSTRACT FROM AUTHOR]

We construct an injective map from the set of holomorphic equivalence classes of neighborhoods M of a compact complex manifold C into C m for some m < ∞ when (T M) | C is fixed and the normal bundle of C in M is either weakly negative or 2-positive. [ABSTRACT FROM AUTHOR]

Tucker decomposition is widely used for image representation, data reconstruction, and machine learning tasks, but the calculation cost for updating the Tucker core is high. Bilevel form of triple decomposition (TriD) overcomes this issue by decomposing the Tucker core into three low-dimensional third-order factor tensors and plays an important role in the dimension reduction of data representation. TriD, on the other hand, is incapable of precisely encoding similarity relationships for tensor data with a complex manifold structure. To address this shortcoming, we take advantage of hypergraph learning and propose a novel hypergraph regularized nonnegative triple decomposition for multiway data analysis that employs the hypergraph to model the complex relationships among the raw data. Furthermore, we develop a multiplicative update algorithm to solve our optimization problem and theoretically prove its convergence. Finally, we perform extensive numerical tests on six real-world datasets, and the results show that our proposed algorithm outperforms some state-of-the-art methods. [ABSTRACT FROM AUTHOR]