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We study deformed Hermitian Yang–Mills (dHYM) connections on ruled surfaces explicitly, using the momentum construction. As a main application, we provide many new examples of dHYM connections coupled to a variable background Kähler metric. These are solutions of the moment map partial differential equations given by the Hamiltonian action of the extended gauge group, coupling the dHYM equation to the scalar curvature of the background. The large radius limit of these coupled equations is the Kähler–Yang–Mills system of Álvarez-Cónsul, Garcia-Fernandez, and García-Prada, and in this limit, our solutions converge smoothly to those constructed by Keller and Tønnesen-Friedman. We also discuss other aspects of our examples including conical singularities, realization as B-branes, the small radius limit, and canonical representatives of complexified Kähler classes. [ABSTRACT FROM AUTHOR]

We extend classical results of Perego and Rapagnetta on moduli spaces of sheaves of type OG10 to moduli spaces of Bridgeland semistable objects on the Kuznetsov component of a cubic fourfold. In particular, we determine the period of this class of varieties and use it to understand when they become birational to moduli spaces of sheaves on a K3 surface. [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]

We prove a matrix Li–Yau–Hamilton inequality for the Green function on complete Kähler manifolds with nonnegative holomorphic bisectional curvature. This estimate is an elliptic analogue of the matrix estimate of Cao and Ni for the heat equation on Kähler manifolds. It is also the complex counterpart of the matrix estimate on Riemannian manifolds obtained previously by the author. [ABSTRACT FROM AUTHOR]

The workshop Geometrie, organized by Aaron Naber (Evanston), Andr'e Neves (Chicago), Eleonora Di Nezza (Paris), and Burkhard Wilking (Münster), was well attended with over 45 participants whose interests covered a considerable part of the current research in Differential Geometry. We had talks in such diverse subjects as geometric evolution equations, Kähler geometry, minimal surfaces, RCD spaces, hyperbolic metrics, or scalar curvature metrics. [ABSTRACT FROM AUTHOR]

In this paper, we show Liouville-type theorems for exponentially harmonic maps and p-harmonic maps between Kähler manifolds. We then study the quadratic form of an exponentially harmonic map of Kähler manifolds and its applications. We also prove that if f : M → N is an exponentially harmonic map from a compact Kähler manifold M into a Kähler manifold N with constant energy density such that the curvature of N is strongly semi-negative, then f is holomorphic or antiholomorphic. [ABSTRACT FROM AUTHOR]

Let (M 1 , g) and (M 2 , h) be two Hermitian manifolds. The twisted product Hermitian manifold (M 1 × M 2 f , G) is the product manifold M 1 × M 2 endowed with the Hermitian metric G = g + f 2 h , where f is a positive smooth function on M 1 × M 2 . In this paper, the Chern curvature, Chern Ricci curvature, Chern Ricci scalar curvature and holomorphic sectional curvature of the twisted product Hermitian manifold are derived. The necessary and sufficient conditions for the compact twisted product Hermitian manifold to have constant holomorphic sectional curvature are obtained. Under the condition that the logarithm of the twisted function is pluriharmonic, it is proved that the twisted product Hermitian manifold is Chern flat or Chern Ricci-flat, if and only if M 1 , g and M 2 , h are Chern flat or Chern Ricci-flat, respectively. [ABSTRACT FROM AUTHOR]

We first provide a stochastic formula for the Carathéodory distance in terms of general Markovian couplings and prove a comparison result between the Carathéodory distance and the complete Kähler metric with a negative lower curvature bound using the Kendall–Cranston coupling. This probabilistic approach gives a version of the Schwarz lemma on complete noncompact Kähler manifolds with a further decomposition Ricci curvature into the orthogonal Ricci curvature and the holomorphic sectional curvature, which cannot be obtained by using Yau–Royden's Schwarz lemma. We also prove coupling estimates on quaternionic Kähler manifolds. As a by‐product, we obtain an improved gradient estimate of positive harmonic functions on Kähler manifolds and quaternionic Kähler manifolds under lower curvature bounds. [ABSTRACT FROM AUTHOR]

Motivated by the recent studies of the harmonic maps on normal almost contact metric manifolds, in particular Sasakian, Kenmotsu, and cosymplectic manifolds, in this article we in-vestigate the harmonic maps on C12-manifolds which are non-normal almost contact metric manifolds. Finally, some ex-amples are presented. [ABSTRACT FROM AUTHOR]

Moreover, he studies the correspondence between soft HT ht -open in soft topological spaces and HT ht -open in topological spaces. By using this definition, they prove in a Riemannian setting that if an isoparametric hypersurface is Pythagorean, then it is totally umbilical with sectional curvature HT ht , where HT ht is the Golden Ratio. In addition to these, the characterizations of soft HT ht spaces and soft HT ht spaces are discussed. First, they find some necessary conditions for such a manifold to be HT ht -Einstein. [Extracted from the article]

In this paper, we study applications of some certain types of solitons such as conformal Ricci soliton, conformal η -Ricci–Yamabe soliton and η -Ricci soliton on Kählerian spacetime manifolds. Further, we have developed the characteristics of conformal Ricci soliton and conformal η -Ricci–Yamabe soliton on almost pseudo-symmetric Kählerian spacetime manifolds. Here, we have signalized the nature of solitons in terms of shrinking, steady or expanding and we have also presented the relationship between λ and μ in terms of conformal η -Ricci–Yamabe soliton. Finally, we have embellished the classification of the potential function with respect to gradient η -Ricci soliton on Kählerian spacetime manifolds. [ABSTRACT FROM AUTHOR]

Refining Yau's and Kołodziej's techniques, we establish very precise uniform a priori estimates for degenerate complex Monge--Ampère equations on compact Kähler manifolds, that allow us to control the blow up of the solutions as the cohomology class and the complex structure both vary. We apply these estimates to the study of various families of possibly singular Kähler varieties endowed with twisted Kähler--Einstein metrics, by analyzing the behavior of canonical densities, establishing uniform integrability properties, and developing the first steps of a pluripotential theory in families. This provides interesting information on the moduli space of stable varieties, extending works by Berman--Guenancia and Song, as well as on the behavior of singular Ricci-flat metrics on (log) Calabi--Yau varieties, generalizing works by Rong--Ruan--Zhang, Gross--Tosatti--Zhang, Collins--Tosatti and Tosatti--Weinkove--Yang. [ABSTRACT FROM AUTHOR]

In this paper, we shall study L ∞ estimate for complex quotient equations on compact Kähler manifolds. The method applies to more general equations satisfying a set of structure conditions. Moreover, this method also applies to the stability estimate for complex Hessian equations. [ABSTRACT FROM AUTHOR]

The unit exact Kahler manifolds are introduced as Kähler manifolds with complex distribution of codimension two, whose Kähler form Ω is exact. We study their essential examples as well as their fundamental properties and we give a concrete examples. Secondly, we investigate some symmetric properties on this kind of manifolds. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of generalized magnetic vector fields as magnetic maps from a Kählerian manifold to its tangent bundle endowed with a Berger type deformed Sasaki metric. Some properties of Killing magnetic vector fields are provided specially in the case of an Einstein manifold and a space form. In the last section, we consider the case of unit tangent bundle. [ABSTRACT FROM AUTHOR]

We survey on dimension estimate of holomorphic functions on Kähler manifolds with nonnegative curvature. [ABSTRACT FROM AUTHOR]

We prove lower bound estimates for the first nonzero eigenvalue of the Laplacian on a compact quaternion-Kähler manifold. For the closed or Neumann case, the lower bounds depend on dimension, diameter, and lower bound of scalar curvature, and they are derived as the large time implication of the modulus of continuity estimates for solutions of the heat equation. For the Dirichlet case, we establish lower bounds that depend on dimension, inradius, lower bound of scalar curvature, and lower bound of the second fundamental form of the boundary, via a Laplace comparison theorem for the distance to the boundary function. [ABSTRACT FROM AUTHOR]

In this paper, we make a correction to Theorem 1.2 of the aforementioned paper [J. Liu, Fano manifolds containing a negative divisor isomorphic to a rational homogeneous space of Picard number one, Int. J. Math. 31(9) (2020) 2050066]. [ABSTRACT FROM AUTHOR]

Given a Fano manifold (X , ω) , we develop a variational approach to characterize analytically the existence of Kähler–Einstein metrics with prescribed singularities, assuming that these singularities can be approximated algebraically. Moreover, we define a function α ω on the set of prescribed singularities which generalizes Tian's α-invariant, showing that its upper lever set { α ω ⁢ (⋅) > n n + 1 } produces a subset of the Kähler–Einstein locus, i.e. of the locus given by all prescribed singularities that admit Kähler–Einstein metrics. In particular, we prove that many K-stable manifolds admit all possible Kähler–Einstein metrics with prescribed singularities. Conversely, we show that enough positivity of the α-invariant function at nontrivial prescribed singularities (or other conditions) implies the existence of genuine Kähler–Einstein metrics. Finally, through a continuity method we also prove the strong continuity of Kähler–Einstein metrics on curves of totally ordered prescribed singularities when the relative automorphism groups are discrete. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new class of Kählerian manifolds and study their essential examples as well as their fundamental properties. Next, we investigate a particular type belonging to this class and we establish some basic results for Riemannian curvature tensor. Concrete examples are given. [ABSTRACT FROM AUTHOR]

As is well known, the holomorphic sectional curvature is just half of the sectional curvature in a holomorphic plane section on a Kähler manifold (Zheng, Complex differential geometry (2000)). In this article, we prove that if the holomorphic sectional curvature is half of the sectional curvature in a holomorphic plane section on a Hermitian manifold then the Hermitian metric is Kähler. [ABSTRACT FROM AUTHOR]

This work is intended as an attempt to extend some results of nearly Kählerian Finsler manifolds. We give a condition to generalized (a, b, J)−manifolds to be weakly Landsberg metric. Furthermore, we find the conditions under which a nearly Kähler Finsler manifold has relatively isotropic Landsberg curvature and relatively isotropic mean Landsberg curvature. [ABSTRACT FROM AUTHOR]

Given a compact Kähler manifold, we prove that all global isometries of the space of Kähler metrics are induced by biholomorphisms and anti-biholomorphisms of the manifold. In particular, there exist no global symmetries for Mabuchi's metric. Moreover, we show that the Mabuchi completion does not even admit local symmetries. Closely related to these findings, we provide a large class of metric geodesic segments that cannot be extended at one end, exhibiting the first such examples in the literature. [ABSTRACT FROM AUTHOR]

First we introduce the notion of parallel Ricci tensor ∇Ric = 0 for real hypersurfaces in the complex quadric Qm = SOm+2/SOmSO2 and show that the unit normal vector field N is singular. Next we give a new classification of real hypersurfaces in the complex quadric Qm = SOm+2/SOmSO2 with parallel Ricci tensor. [ABSTRACT FROM AUTHOR]

The global boundary regularity for the Ә-problem on a relatively compact domain with the C²-smooth boundary in a Kähler manifold that satisfies some "Hartogs-pseudoconvexity" condition is investigated in this paper. The applications to the Ә-problem and the Әb-problem, are thoroughly discussed. [ABSTRACT FROM AUTHOR]

We construct coherent states for each eigenspace of a magnetic Laplacian on the complex projective n-space in order to apply a quantization-dequantization method. Doing so allows to define the Berezin transform for these spaces. We then establish a formula for this transform as a function of the Fubini-Study Laplacian in a closed form involving of a terminating Kampé de Fériet function. For the lowest spherical Landau level on the Riemann sphere the obtained formula reduces to the one derived by Berezin himself. [ABSTRACT FROM AUTHOR]

Let P : = U (p + q) / U (p) × U (q) be the complex Grassmann manifold and F : T 1 , 0 P → [ 0 , + ∞) be an arbitrary U (p + q) -invariant strongly pseudoconvex complex Finsler metric. We prove that F is necessary a Kähler-Berwald metric which is not necessary Hermitian quadratic. We also prove that F is Hermitian quadratic if and only if F is a constant multiple of the canonical U (p + q) -invariant Kähler metric on P. In particular on the complex projective space CP n = U (n + 1) / U (n) × U (1) , there exists no U (n + 1) -invariant strongly pseudoconvex complex Finsler metric other than a constant multiple of the Fubini-Study metric. These invariant metrics are of particular interesting since they are the most important examples of strongly pseudoconvex complex Finsler metrics on P which are elliptic metrics in the sense that they enjoy very similar holomorphic sectional curvature and bisectional curvature properties as that of the U (p + q) -invariant Kähler metrics on P , nevertheless, these invariant metrics are not necessary Hermitian quadratic, hence provide nontrivial explicit examples for complex Finsler geometry in the compact cases. [ABSTRACT FROM AUTHOR]

A compact Kähler manifold with quasi-negative holomorphic sectional curvature must have ample canonical bundle. This was conjectured by Wu-Yau and is recently proved by the continuity method. In this note, we will give an alternative proof of this result by using the Kähler-Ricci flow. [ABSTRACT FROM AUTHOR]

We derive some consequences of the Liouville theorem for plurisubharmonic functions of L.‐F. Tam and the author. The first result provides a nonlinear version of the complex splitting theorem (which splits off a factor of ℂ isometrically from the simply connected Kähler manifold with nonnegative bisectional curvature and a linear growth holomorphic function) of L.‐F. Tam and the author. The second set of results concerns the so‐called k‐hyperbolicity and its connection with the negativity of the k‐scalar curvature (when k = 1 they are the negativity of holomorphic sectional curvature and Kobayashi hyperbolicity) introduced recently in [33] by F. Zheng and the author. We lastly prove a new Schwarz‐lemma‐type estimate in terms of only the holomorphic sectional curvatures of both domain and target manifolds. © 2020 Wiley Periodicals LLC. [ABSTRACT FROM AUTHOR]

We study solitons of almost pseudo symmetric Kählerian space-time manifold. It is considered that different curvature tensors like projective, conharmonic and conformal curvature tensors in almost pseudo symmetric Kählerian space-time manifolds are flat. It is shown that solitons are steady, expanding or shrinking under different relations of isotropic pressure, the cosmological constant, energy density and gravitational constant. [ABSTRACT FROM AUTHOR]

We give a new Tian–Todorov lemma on deformations of CR-structures and use it to reprove the deformation unobstructedness of normal compact strongly pseudoconvex CR-manifold under the assumption of d ′ d ′ ′ -lemma, more faithfully following Tian–Todorov's approach. [ABSTRACT FROM AUTHOR]

We prove the Kobayashi—Hitchin correspondence and the approximate Kobayashi—Hitchin correspondence for twisted holomorphic vector bundles on compact Kähler manifolds. More precisely, if X is a compact manifold and g is a Gauduchon metric on X, a twisted holomorphic vector bundle on X is g−polystable if and only if it is g−Hermite-Einstein, and if X is a compact Kähler manifold and g is a Kähler metric on X, then a twisted holomorphic vector bundle on X is g−semistable if and only if it is approximate g−Hermite-Einstein. [ABSTRACT FROM AUTHOR]

An Iwasawa manifold is a compact complex homogeneous manifold isomorphic to a quotient |$G/\Lambda $|⁠ , where |$G$| is the group of complex unipotent |$3 \times 3$| matrices and |$\Lambda \subset G$| is a cocompact lattice. In this work, we study holomorphic submanifolds in Iwasawa manifolds. We prove that any compact complex curve in an Iwasawa manifold is contained in a holomorphic subtorus. We also prove that any complex surface in an Iwasawa manifold is either an abelian surface or a Kähler non-projective isotrivial elliptic surface of Kodaira dimension one. In the Appendix, we show that any subtorus in Iwasawa manifold carries complex multiplication. [ABSTRACT FROM AUTHOR]

In this paper we prove classification results for gradient shrinking Ricci solitons under two invariant conditions, namely nonnegative orthogonal bisectional curvature and weakly PIC 1 , without any curvature upper bound. New results on ancient solutions for the Ricci and Kähler-Ricci flow are obtained. Applications to Kähler manifolds with almost nonnegative orthogonal bisectional curvature are derived as consequences. The main new feature is that no curvature upper bound is assumed. [ABSTRACT FROM AUTHOR]

In this paper we prove a gap theorem for Kähler manifolds with nonnegative orthogonal bisectional curvature and nonnegative Ricci curvature, which generalizes an earlier result of the first author [L. Ni, An optimal gap theorem, Invent. Math. 189 2012, 3, 737–761]. We also prove a Liouville theorem for plurisubharmonic functions on such a manifold, which generalizes a previous result of L.-F. Tam and the first author [L. Ni and L.-F. Tam, Plurisubharmonic functions and the structure of complete Kähler manifolds with nonnegative curvature, J. Differential Geom. 64 2003, 3, 457–524] and complements a recent result of Liu [G. Liu, Three-circle theorem and dimension estimate for holomorphic functions on Kähler manifolds, Duke Math. J. 165 2016, 15, 2899–2919]. [ABSTRACT FROM AUTHOR]

A CY bundle on a connected compact complex manifold X was a crucial ingredient in constructing differential systems for period integrals in an earlier paper by Lian and Yau, by lifting line bundles from the base X to the total space. A question was therefore raised as to whether there exists such a bundle that supports the liftings of all line bundles from X, simultaneously. This was a key step for giving a uniform construction of differential systems for arbitrary complete intersections in X. In this paper, we answer the existence question in the affirmative if X is assumed to be Kahler, and also in general if the Picard group ofX is assumed to be discrete. Furthermore, we prove a rigidity property of CY bundles if the principal group is an algebraic torus, showing that such a CY bundle is essentially determined by its character map. [ABSTRACT FROM AUTHOR]

Given a complex manifold X, any Kähler class defines an affine bundle over X, and any Kähler form in the given class defines a totally real embedding of X into this affine bundle. We formulate conditions under which the affine bundles arising this way are Stein and relate this question to other natural positivity conditions on the tangent bundle of X. For compact Kähler manifolds of non‐negative holomorphic bisectional curvature, we establish a close relation of this construction to adapted complex structures in the sense of Lempert–Szőke and to the existence question for good complexifications in the sense of Totaro. Moreover, we study projective manifolds for which the induced affine bundle is not just Stein but affine and prove that these must have big tangent bundle. In the course of our investigation, we also obtain a simpler proof of a result of Yang on manifolds having non‐negative holomorphic bisectional curvature and big tangent bundle. [ABSTRACT FROM AUTHOR]

In this paper, the author proves a generalized Donaldson-Uhlenbeck-Yau theorem for twisted holomorphic chain on a non-compact Kähler manifold. As an application, the author obtains a Bogomolov type Chern numbers inequality for semistable twisted holomorphic chain. [ABSTRACT FROM AUTHOR]

In this paper, we show that there are non-trivial complete rotationally symmetric conformal Kähler, Einstein metrics on Bd and ℂd, and there are non-trivial complete rotationally symmetric conformal Kähler, Einstein-Maxwell metrics on Bd* and ℂd*. [ABSTRACT FROM AUTHOR]

It is proved that Kirichenko-Uskorev structures induced on oriented hypersurfaces of a Kählerian manifold of dimension at least six are necessarily cosymplectic. [ABSTRACT FROM AUTHOR]

Given a semi-Riemannian 4-manifold (M, g) with two distinguished vector fields satisfying properties determined by their shear, twist and various Lie bracket relations, a family of Kähler metrics gK is constructed, defined on an open set in M, which coincides with M in many typical examples. Under certain conditions g and gK share various properties, such as a Killing vector field or a vector field with a geodesic flow. In some cases the Kähler metrics are complete. The Ricci and scalar curvatures of gK are computed under certain assumptions in terms of data associated to g. Many examples are described, including classical spacetimes inwarped products, for instance de Sitter spacetime, aswell as gravitational planewaves, metrics of Petrov type D such as Kerr and NUT metrics, and metrics for which gK is an SKR metric. For the latter an inverse ansatz is described, constructing g from the SKR metric. [ABSTRACT FROM AUTHOR]

We classify and investigate locally conformally Kähler structures on four-dimensional solvable Lie algebras up to linear equivalence. As an application we can produce many examples in higher dimension, here including lcK structures on Oeljeklaus-Toma manifolds, and we also give a geometric interpretation of some of the 4-dimensional structures in our classification. [ABSTRACT FROM AUTHOR]

We find a family of Kähler metrics invariantly defined on the radius r0 > 0 tangent disk bundle TM,r0 of any given real space-form M or any of its quotients by discrete groups of isometries. Such metrics are complete in the non-negative curvature case and non-complete in the negative curvature case. If dimM=2 and M has constant sectional curvature K ≠ 0, then the Kähler manifolds TM,r0 have holonomy SU(2); hence they are Ricci-flat. For M = S², just this dimension, the metric coincides with the Stenzel metric on the tangent manifold TS2, giving us a new most natural description of this well-know metric. [ABSTRACT FROM AUTHOR]

We describe explicitly the chamber structure of the movable cone for a general complete intersection Calabi–Yau threefold in a non-split (n + 4) -dimensional P n -ruled Fano manifold of index n + 1 and Picard number two. Moreover, all birational minimal models of such Calabi–Yau threefolds are found whose number is finite. [ABSTRACT FROM AUTHOR]

Ichijyō introduced (a; b; J)-manifolds as a special class of generalized Randers manifolds. We introduce generalized (a; b; J)-manifolds. A partial negative answer to Ichijyō’s open problem on nearly Kähler Finsler manifolds is given. The condition under which generalized (a; b; J)-manifolds are Berwaldian is obtained. Finally, we prove that under a mild assumption a nearly Kähler Finsler manifold is Landsbergian. [ABSTRACT FROM AUTHOR]

We prove a boundary regularity result for the complete Kähler-Einstein metrics of negative Ricci curvature near strictly pseudoconvex boundary points, and we deduce the asymptotic behavior of their holomorphic bisectional curvatures near such points. Finally, we prove that the same asymptotic behavior holds at boundary points at which the squeezing function tends to one. [ABSTRACT FROM AUTHOR]

The purpose of this paper is twofold. We first solve the Dirichlet problem for α -Hermitian–Einstein equations on I ± -holomorphic bundles over bi-Hermitian manifolds. Second, by using Uhlenbeck–Yau's continuity method, we prove that the existence of approximate α -Hermitian–Einstein structure and the α -semi-stability on I ± -holomorphic bundles over compact bi-Hermitian manifolds are equivalent. [ABSTRACT FROM AUTHOR]

Consider the holomorphic Hamiltonian action of a compact Lie group K on a compact Kähler manifold M with a moment map Φ : M → k ∗ . Assume that 0 is a regular value of the moment map. Weitsman raised the question of what we can say about the cohomology of the Kähler quotient M 0 ≔ Φ − 1 (0) ∕ K if all the ordinary cohomology of M is of type (p , p). In this paper, using the Cartan–Chern–Weil theory we show that in the above context there is a natural surjective Kirwan map from an equivariant version of the Dolbeault cohomology of M onto the Dolbeault cohomology of the Kähler quotient M 0. As an immediate consequence, this result provides an answer to the question posed by Weitsman. [ABSTRACT FROM AUTHOR]