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Journal articles on the topic 'Non-Kähler geometry'

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Published: 7 September 2024

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1

Dai, Song. "Lower order tensors in non-Kähler geometry and non-Kähler geometric flow." Annals of Global Analysis and Geometry 50, no. 4 (June 6, 2016): 395–418. http://dx.doi.org/10.1007/s10455-016-9518-0.

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2

Broder, Kyle. "The Schwarz lemma in Kähler and non-Kähler geometry." Asian Journal of Mathematics 27, no. 1 (2023): 121–34. http://dx.doi.org/10.4310/ajm.2023.v27.n1.a5.

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3

Fino, Anna, and Adriano Tomassini. "Non-Kähler solvmanifolds with generalized Kähler structure." Journal of Symplectic Geometry 7, no. 2 (2009): 1–14. http://dx.doi.org/10.4310/jsg.2009.v7.n2.a1.

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4

Verbitsky, M. S., V. Vuletescu, and L. Ornea. "Classification of non-Kähler surfaces and locally conformally Kähler geometry." Russian Mathematical Surveys 76, no. 2 (April 1, 2021): 261–89. http://dx.doi.org/10.1070/rm9858.

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5

Zheng, Fangyang. "Some recent progress in non-Kähler geometry." Science China Mathematics 62, no. 11 (May 22, 2019): 2423–34. http://dx.doi.org/10.1007/s11425-019-9528-1.

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6

Alessandrini, Lucia, and Giovanni Bassanelli. "Positive $$\partial \bar \partial - closed$$ currents and non-Kähler geometrycurrents and non-Kähler geometry." Journal of Geometric Analysis 2, no. 4 (July 1992): 291–316. http://dx.doi.org/10.1007/bf02934583.

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7

Cortés, Vicente, and Liana David. "Twist, elementary deformation and K/K correspondence in generalized geometry." International Journal of Mathematics 31, no. 10 (September 2020): 2050078. http://dx.doi.org/10.1142/s0129167x20500780.

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We define the conformal change and elementary deformation in generalized complex geometry. We apply Swann’s twist construction to generalized (almost) complex and Hermitian structures obtained by these operations and establish conditions for the Courant integrability of the resulting twisted structures. We associate to any appropriate generalized Kähler manifold [Formula: see text] with a Hamiltonian Killing vector field a new generalized Kähler manifold, depending on the choice of a pair of non-vanishing functions and compatible twist data. We study this construction when [Formula: see text] is toric, with emphasis on the four-dimensional case, and we apply it to deformations of the standard flat Kähler metric on [Formula: see text], the Fubini–Study metric on [Formula: see text] and the admissible Kähler metrics on Hirzebruch surfaces. As a further application, we recover the K/K (Kähler/Kähler) correspondence, by specializing to ordinary Kähler manifolds.

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8

Dunajski, Maciej. "Null Kähler Geometry and Isomonodromic Deformations." Communications in Mathematical Physics 391, no. 1 (December 8, 2021): 77–105. http://dx.doi.org/10.1007/s00220-021-04270-0.

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AbstractWe construct the normal forms of null-Kähler metrics: pseudo-Riemannian metrics admitting a compatible parallel nilpotent endomorphism of the tangent bundle. Such metrics are examples of non-Riemannian holonomy reduction, and (in the complexified setting) appear on the space of Bridgeland stability conditions on a Calabi–Yau threefold. Using twistor methods we show that, in dimension four—where there is a connection with dispersionless integrability—the cohomogeneity-one anti-self-dual null-Kähler metrics are generically characterised by solutions to Painlevé I or Painlevé II ODEs.

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9

YANG, BO. "A CHARACTERIZATION OF NONCOMPACT KOISO-TYPE SOLITONS." International Journal of Mathematics 23, no. 05 (May 2012): 1250054. http://dx.doi.org/10.1142/s0129167x12500541.

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We construct complete gradient Kähler–Ricci solitons of various types on the total spaces of certain holomorphic line bundles over compact Kähler–Einstein manifolds with positive scalar curvature. Those are noncompact analogues of the compact examples found by Koiso [On rotationally symmetric Hamilton's equations for Kähler–Einstein metrics, in Recent Topics in Differential and Analytic Geometry, Advanced Studies in Pure Mathematics, Vol. 18-I (Academic Press, Boston, MA, 1990), pp. 327–337]. Our examples can be viewed a generalization of previous examples by Cao [Existense of gradient Kähler–Ricci solitons, in Elliptic and Parabolic Methods in Geometry (Minneapolis, MN, 1994), pp. 1–16], Chave and Valent [On a class of compact and non-compact quasi-Einstein metrics and their renormalizability properties, Nuclear Phys. B 478 (1996) 758–778], Pedersen, Tønnesen-Friedman, and Valent [Quasi-Einstein Kähler metrics, Lett. Math. Phys. 50(3) (1999) 229–241], and Feldman, Ilmanen and Knopf [Rotationally symmetric shrinking and expanding gradient Kähler–Ricci solitons, J. Differential Geom. 65 (2003) 169–209]. We also prove a uniformization result on complete steady gradient Kähler–Ricci solitons with non-negative Ricci curvature under additional assumptions.

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10

Yau, Shing-Tung. "Existence of canonical metrics in non-Kähler geometry." Notices of the International Congress of Chinese Mathematicians 9, no. 1 (2021): 1–10. http://dx.doi.org/10.4310/iccm.2021.v9.n1.a1.

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11

ABREU, MIGUEL. "KÄHLER GEOMETRY OF TORIC VARIETIES AND EXTREMAL METRICS." International Journal of Mathematics 09, no. 06 (September 1998): 641–51. http://dx.doi.org/10.1142/s0129167x98000282.

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A (symplectic) toric variety X, of real dimension 2n, is completely determined by its moment polytope Δ ⊂ ℝn. Recently Guillemin gave an explicit combinatorial way of constructing "toric" Kähler metrics on X, using only data on Δ. In this paper, differential geometric properties of these metrics are investigated using Guillemin's construction. In particular, a nice combinatorial formula for the scalar curvature R is given, and the Euler–Lagrange condition for such "toric" metrics being extremal (in the sense of Calabi) is proven to be R being an affine function on Δ ⊂ ℝn. A construction, due to Calabi, of a 1-parameter family of extremal Kähler metrics of non-constant scalar curvature on [Formula: see text] is recast very simply and explicitly using Guillemin's approach. Finally, we present a curious combinatorial identity for convex polytopes Δ ⊂ ℝn that follows from the well-known relation between the total integral of the scalar curvature of a Kähler metric and the wedge product of the first Chern class of the underlying complex manifold with a suitable power of the Kähler class.

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12

LEBRUN, CLAUDE. "FANO MANIFOLDS, CONTACT STRUCTURES, AND QUATERNIONIC GEOMETRY." International Journal of Mathematics 06, no. 03 (June 1995): 419–37. http://dx.doi.org/10.1142/s0129167x95000146.

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Let Z be a compact complex (2n+1)-manifold which carries a complex contact structure, meaning a codimension-1 holomorphic sub-bundle D⊂TZ which is maximally non-integrable. If Z admits a Kähler-Einstein metric of positive scalar curvature, we show that it is the Salamon twistor space of a quaternion-Kähler manifold (M4n, g). If Z also admits a second complex contact structure [Formula: see text], then Z=CP2n+1. As an application, we give several new characterizations of the Riemannian manifold HPn= Sp(n+1)/(Sp(n)×Sp(1)).

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13

Angella, Daniele, Adriano Tomassini, and Misha Verbitsky. "On non-Kähler degrees of complex manifolds." Advances in Geometry 19, no. 1 (January 28, 2019): 65–69. http://dx.doi.org/10.1515/advgeom-2018-0026.

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Abstract We study cohomological properties of complex manifolds. In particular, under suitable metric conditions, we extend to higher dimensions a result by A. Teleman, which provides an upper bound for the Bott– Chern cohomology in terms of Betti numbers for compact complex surfaces according to the dichotomy b1 even or odd.

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14

Castrillón López, M., P. M. Gadea, and J. A. Oubiña. "Homogeneous Quaternionic Kähler Structures on Eight-Dimensional Non-Compact Quaternion-Kähler Symmetric Spaces." Mathematical Physics, Analysis and Geometry 12, no. 1 (December 13, 2008): 47–74. http://dx.doi.org/10.1007/s11040-008-9051-x.

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15

VACARU, SERGIU I. "FINSLER AND LAGRANGE GEOMETRIES IN EINSTEIN AND STRING GRAVITY." International Journal of Geometric Methods in Modern Physics 05, no. 04 (June 2008): 473–511. http://dx.doi.org/10.1142/s0219887808002898.

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We review the current status of Finsler–Lagrange geometry and generalizations. The goal is to aid non-experts on Finsler spaces, but physicists and geometers skilled in general relativity and particle theories, to understand the crucial importance of such geometric methods for applications in modern physics. We also would like to orient mathematicians working in generalized Finsler and Kähler geometry and geometric mechanics how they could perform their results in order to be accepted by the community of "orthodox" physicists. Although the bulk of former models of Finsler–Lagrange spaces where elaborated on tangent bundles, the surprising result advocated in our works is that such locally anisotropic structures can be modeled equivalently on Riemann–Cartan spaces, even as exact solutions in Einstein and/or string gravity, if nonholonomic distributions and moving frames of references are introduced into consideration. We also propose a canonical scheme when geometrical objects on a (pseudo) Riemannian space are nonholonomically deformed into generalized Lagrange, or Finsler, configurations on the same manifold. Such canonical transforms are defined by the coefficients of a prime metric and generate target spaces as Lagrange structures, their models of almost Hermitian/Kähler, or nonholonomic Riemann spaces. Finally, we consider some classes of exact solutions in string and Einstein gravity modeling Lagrange–Finsler structures with solitonic pp-waves and speculate on their physical meaning.

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16

Fine, Joel, and Dmitri Panov. "Hyperbolic geometry and non-Kähler manifolds with trivial canonical bundle." Geometry & Topology 14, no. 3 (July 13, 2010): 1723–63. http://dx.doi.org/10.2140/gt.2010.14.1723.

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17

Parton, Maurizio, and Victor Vuletescu. "Examples of non-trivial rank in locally conformal Kähler geometry." Mathematische Zeitschrift 270, no. 1-2 (October 28, 2010): 179–87. http://dx.doi.org/10.1007/s00209-010-0791-5.

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18

Boucetta, Mohamed. "On the Hermitian structures of the sequence of tangent bundles of an affine manifold endowed with a Riemannian metric." Complex Manifolds 9, no. 1 (January 1, 2022): 18–51. http://dx.doi.org/10.1515/coma-2021-0128.

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Abstract Let (M, ∇, 〈, 〉) be a manifold endowed with a flat torsionless connection r and a Riemannian metric 〈, 〉 and (TkM) k ≥1 the sequence of tangent bundles given by TkM = T(Tk −1 M) and T 1 M = TM. We show that, for any k ≥ 1, TkM carries a Hermitian structure (Jk , gk ) and a flat torsionless connection ∇k and when M is a Lie group and (∇, 〈, 〉) are left invariant there is a Lie group structure on each TkM such that (Jk , gk , ∇k ) are left invariant. It is well-known that (TM, J 1, g 1) is Kähler if and only if 〈, 〉 is Hessian, i.e, in each system of affine coordinates (x 1, . . ., xn ), 〈 ∂ x i , ∂ x j 〉 = ∂ 2 φ ∂ x i ∂ x j \left\langle {{\partial _x}_{_i},{\partial _{{x_j}}}} \right\rangle = {{{\partial ^2}\phi } \over {{\partial _x}_{_i}{\partial _x}_j}} . Having in mind many generalizations of the Kähler condition introduced recently, we give the conditions on (∇, 〈, 〉) so that (TM, J 1, g 1) is balanced, locally conformally balanced, locally conformally Kähler, pluriclosed, Gauduchon, Vaisman or Calabi-Yau with torsion. Moreover, we can control at the level of (∇, 〈, 〉) the conditions insuring that some (TkM, Jk , gk ) or all of them satisfy a generalized Kähler condition. For instance, we show that there are some classes of (M, ∇, 〈, 〉) such that, for any k ≥ 1, (TkM, Jk , gk ) is balanced non-Kähler and Calabi-Yau with torsion. By carefully studying the geometry of (M, ∇, 〈, 〉), we develop a powerful machinery to build a large classes of generalized Kähler manifolds.

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19

Yu, Tony Yue. "Gromov compactness in non-archimedean analytic geometry." Journal für die reine und angewandte Mathematik (Crelles Journal) 2018, no. 741 (August 1, 2018): 179–210. http://dx.doi.org/10.1515/crelle-2015-0077.

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Abstract Gromov’s compactness theorem for pseudo-holomorphic curves is a foundational result in symplectic geometry. It controls the compactness of the moduli space of pseudo-holomorphic curves with bounded area in a symplectic manifold. In this paper, we prove the analog of Gromov’s compactness theorem in non-archimedean analytic geometry. We work in the framework of Berkovich spaces. First, we introduce a notion of Kähler structure in non-archimedean analytic geometry using metrizations of virtual line bundles. Second, we introduce formal stacks and non-archimedean analytic stacks. Then we construct the moduli stack of non-archimedean analytic stable maps using formal models, Artin’s representability criterion and the geometry of stable curves. Finally, we reduce the non-archimedean problem to the known compactness results in algebraic geometry. The motivation of this paper is to provide the foundations for non-archimedean enumerative geometry.

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20

DI SCALA, ANTONIO J., ANDREA LOI, and FABIO ZUDDAS. "RIEMANNIAN GEOMETRY OF HARTOGS DOMAINS." International Journal of Mathematics 20, no. 02 (February 2009): 139–48. http://dx.doi.org/10.1142/s0129167x09005236.

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Let DF = {(z0, z) ∈ ℂn | |z0|2 < b, ||z||2 < F(|z0|2)} be a strongly pseudoconvex Hartogs domain endowed with the Kähler metric gF associated to the Kähler form [Formula: see text]. This paper contains several results on the Riemannian geometry of these domains. These are summarized in Theorems 1.1–1.3. In the first one we prove that if DF admits a non-special geodesic (see definition below) through the origin whose trace is a straight line then DF is holomorphically isometric to an open subset of the complex hyperbolic space. In the second theorem we prove that all the geodesics through the origin of DF do not self-intersect, we find necessary and sufficient conditions on F for DF to be geodesically complete and we prove that DF is locally irreducible as a Riemannian manifold. Finally, in Theorem 1.3, we compare the Bergman metric gB and the metric gF in a bounded Hartogs domain and we prove that if gB is a multiple of gF, namely gB = λ gF, for some λ ∈ ℝ+, then DF is holomorphically isometric to an open subset of the complex hyperbolic space.

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21

Di Scala, Antonio J., Naohiko Kasuya, and Daniele Zuddas. "Non-Kähler complex structures on $\mathbb{R}^4$, II." Journal of Symplectic Geometry 16, no. 3 (2018): 631–44. http://dx.doi.org/10.4310/jsg.2018.v16.n3.a2.

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22

Popovici, Dan. "Non-Kähler Mirror Symmetry of the Iwasawa Manifold." International Mathematics Research Notices 2020, no. 23 (November 7, 2018): 9471–538. http://dx.doi.org/10.1093/imrn/rny256.

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Abstract We propose a new approach to the mirror symmetry conjecture in a form suitable to possibly non-Kähler compact complex manifolds whose canonical bundle is trivial. We apply our methods by proving that the Iwasawa manifold $X$, a well-known non-Kähler compact complex manifold of dimension $3$, is its own mirror dual to the extent that its Gauduchon cone, replacing the classical Kähler cone that is empty in this case, corresponds to what we call the local universal family of essential deformations of $X$. These are obtained by removing from the Kuranishi family the two “superfluous” dimensions of complex parallelisable deformations that have a similar geometry to that of the Iwasawa manifold. The remaining four dimensions are shown to have a clear geometric meaning including in terms of the degeneration at $E_2$ of the Frölicher spectral sequence. On the local moduli space of “essential” complex structures, we obtain a canonical Hodge decomposition of weight $3$ and a variation of Hodge structures, construct coordinates and Yukawa couplings while implicitly proving a local Torelli theorem. On the metric side of the mirror, we construct a variation of Hodge structures parametrised by a subset of the complexified Gauduchon cone of the Iwasawa manifold using the sGG property (which means that all the Gauduchon metrics are strongly Gauduchon) of all the small deformations of this manifold proved in earlier joint work of the author with L. Ugarte. Finally, we define a mirror map linking the two variations of Hodge structures and we highlight its properties.

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23

Kruglikov, Boris, Vladimir Matveev, and Dennis The. "Submaximally symmetric c-projective structures." International Journal of Mathematics 27, no. 03 (March 2016): 1650022. http://dx.doi.org/10.1142/s0129167x16500221.

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[Formula: see text]-projective structures are analogues of projective structures in the almost complex setting. The maximal dimension of the Lie algebra of [Formula: see text]-projective symmetries of a complex connection on an almost complex manifold of [Formula: see text]-dimension [Formula: see text] is classically known to be [Formula: see text]. We prove that the submaximal dimension is equal to [Formula: see text]. If the complex connection is minimal (encoded as a normal parabolic geometry), the harmonic curvature of the [Formula: see text]-projective structure has three components and we specify the submaximal symmetry dimensions and the corresponding geometric models for each of these three pure curvature types. If the connection is non-minimal, we introduce a modified normalization condition on the parabolic geometry and use this to resolve the symmetry gap problem. We prove that the submaximal symmetry dimension in the class of Levi-Civita connections for pseudo-Kähler metrics is [Formula: see text], and specializing to the Kähler case, we obtain [Formula: see text]. This resolves the symmetry gap problem for metrizable [Formula: see text]-projective structures.

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24

Lin, Hsueh-Yung. "Compact Kähler threefolds with non-nef canonical bundle and symplectic geometry." Mathematical Research Letters 21, no. 6 (2014): 1341–52. http://dx.doi.org/10.4310/mrl.2014.v21.n6.a7.

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25

Rogov, Vasily. "Complex Geometry of Iwasawa Manifolds." International Mathematics Research Notices 2020, no. 23 (November 7, 2018): 9420–39. http://dx.doi.org/10.1093/imrn/rny230.

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Abstract 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.

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26

Bagaglini, Leonardo. "Non-orientable three-submanifolds of G2-manifolds." Advances in Geometry 19, no. 3 (July 26, 2019): 401–14. http://dx.doi.org/10.1515/advgeom-2018-0023.

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Abstract By analogy with associative and co-associative cases, we study a class of three-dimensional non-orientable submanifolds of manifolds with a G2-structure, modelled on planes lying in aspecial G2-orbit. An application of the Cartan–Kähler theory shows that some three-manifolds can be presented in this way. We also classify all the homogeneous ones in ℝℙ7.

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27

Sano, Taro. "Examples of non‐Kähler Calabi–Yau manifolds with arbitrarily large b2." Journal of Topology 14, no. 4 (November 24, 2021): 1448–60. http://dx.doi.org/10.1112/topo.12212.

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28

Chang, Yu-Lin. "Some results on compact Kähler surfaces with non-positive bisectional curvature." Geometriae Dedicata 145, no. 1 (July 29, 2009): 65–70. http://dx.doi.org/10.1007/s10711-009-9403-0.

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29

Zheng, Fangyang. "Examples of non-positively curved Kähler manifolds." Communications in Analysis and Geometry 4, no. 1 (1996): 129–60. http://dx.doi.org/10.4310/cag.1996.v4.n1.a3.

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30

Martelli, Dario, and James Sparks. "Resolutions of non-regular Ricci-flat Kähler cones." Journal of Geometry and Physics 59, no. 8 (August 2009): 1175–95. http://dx.doi.org/10.1016/j.geomphys.2009.06.005.

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31

Kasuya, Hisashi. "Hodge symmetry and decomposition on non-Kähler solvmanifolds." Journal of Geometry and Physics 76 (February 2014): 61–65. http://dx.doi.org/10.1016/j.geomphys.2013.10.012.

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32

Chrysikos, Ioannis, and Yusuke Sakane. "Homogeneous Einstein metrics on non-Kähler C-spaces." Journal of Geometry and Physics 160 (February 2021): 103996. http://dx.doi.org/10.1016/j.geomphys.2020.103996.

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33

Hashimoto, Kenji, and Taro Sano. "Examples of non-Kähler Calabi–Yau 3–folds with arbitrarily large b2." Geometry & Topology 27, no. 1 (May 1, 2023): 131–52. http://dx.doi.org/10.2140/gt.2023.27.131.

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34

Alonso, Izar, and Francesca Salvatore. "On the existence of balanced metrics on six-manifolds of cohomogeneity one." Annals of Global Analysis and Geometry 61, no. 2 (November 22, 2021): 309–31. http://dx.doi.org/10.1007/s10455-021-09807-z.

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AbstractWe consider balanced metrics on complex manifolds with holomorphically trivial canonical bundle, most commonly known as balanced SU(n)-structures. Such structures are of interest for both Hermitian geometry and string theory, since they provide the ideal setting for the Hull–Strominger system. In this paper, we provide a non-existence result for balanced non-Kähler $$\text {SU}(3)$$ SU ( 3 ) -structures which are invariant under a cohomogeneity one action on simply connected six-manifolds.

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35

MOLITOR, MATHIEU. "REMARKS ON THE STATISTICAL ORIGIN OF THE GEOMETRICAL FORMULATION OF QUANTUM MECHANICS." International Journal of Geometric Methods in Modern Physics 09, no. 03 (May 2012): 1220001. http://dx.doi.org/10.1142/s0219887812200010.

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A quantum system can be entirely described by the Kähler structure of the projective space [Formula: see text] associated to the Hilbert space [Formula: see text] of possible states; this is the so-called geometrical formulation of quantum mechanics. In this paper, we give an explicit link between the geometrical formulation (of finite dimensional quantum systems) and statistics through the natural geometry of the space [Formula: see text] of non-vanishing probabilities [Formula: see text] defined on a finite set En: = {x1, …, xn}. More precisely, we use the Fisher metric gF and the exponential connection ∇(1) (both being natural statistical objects living on [Formula: see text]) to construct, via the Dombrowski splitting theorem, a Kähler structure on [Formula: see text] which has the property that it induces the natural Kähler structure of a suitably chosen open dense subset of ℙ(ℂn). As a direct physical consequence, a significant part of the quantum mechanical formalism (in finite dimension) is encoded in the triple [Formula: see text].

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36

Qin, Lizhen, and Botong Wang. "A family of compact complex and symplectic Calabi–Yau manifolds that are non-Kähler." Geometry & Topology 22, no. 4 (April 5, 2018): 2115–44. http://dx.doi.org/10.2140/gt.2018.22.2115.

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37

GRIBACHEVA, DOBRINKA. "A NATURAL CONNECTION ON A BASIC CLASS OF RIEMANNIAN PRODUCT MANIFOLDS." International Journal of Geometric Methods in Modern Physics 09, no. 07 (September 7, 2012): 1250057. http://dx.doi.org/10.1142/s0219887812500570.

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A Riemannian manifold M with an integrable almost product structure P is called a Riemannian product manifold. Our investigations are on the manifolds (M, P, g) of the largest class of Riemannian product manifolds, which is closed with respect to the group of conformal transformations of the metric g. This class is an analogue of the class of locally conformal Kähler manifolds in almost Hermitian geometry. In the present paper we study a natural connection D on (M, P, g) (i.e. DP = Dg = 0). We find necessary and sufficient conditions, the curvature tensor of D to have properties similar to the Kähler tensor in Hermitian geometry. We pay attention to the case when D has a parallel torsion. We establish that the Weyl tensors for the connection D and the Levi-Civita connection coincide as well as the invariance of the curvature tensor of D with respect to the usual conformal transformation. We consider the case when D is a flat connection. We construct an example of the considered manifold by a Lie group where D is a flat connection with non-parallel torsion.

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38

Vu, Duc-Viet. "Relative non-pluripolar product of currents." Annals of Global Analysis and Geometry 60, no. 2 (May 26, 2021): 269–311. http://dx.doi.org/10.1007/s10455-021-09780-7.

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AbstractLet X be a compact Kähler manifold. Let $$T_1, \ldots , T_m$$ T 1 , … , T m be closed positive currents of bi-degree (1, 1) on X and T an arbitrary closed positive current on X. We introduce the non-pluripolar product relative to T of $$T_1, \ldots , T_m$$ T 1 , … , T m . We recover the well-known non-pluripolar product of $$T_1, \ldots , T_m$$ T 1 , … , T m when T is the current of integration along X. Our main results are a monotonicity property of relative non-pluripolar products, a necessary condition for currents to be of relative full mass intersection in terms of Lelong numbers, and the convexity of weighted classes of currents of relative full mass intersection. The former two results are new even when T is the current of integration along X.

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39

Winkelmann, Jörg. "On Manifolds with Trivial Logarithmic Tangent Bundle: The Non-Kähler Case." Transformation Groups 13, no. 1 (March 2008): 195–209. http://dx.doi.org/10.1007/s00031-008-9003-3.

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40

BISWAS, INDRANIL, MAHAN MJ, and HARISH SESHADRI. "3-MANIFOLD GROUPS, KÄHLER GROUPS AND COMPLEX SURFACES." Communications in Contemporary Mathematics 14, no. 06 (October 8, 2012): 1250038. http://dx.doi.org/10.1142/s0219199712500381.

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Let G be a Kähler group admitting a short exact sequence [Formula: see text] where N is finitely generated. (i) Then Q cannot be non-nilpotent solvable. (ii) Suppose in addition that Q satisfies one of the following: (a) Q admits a discrete faithful non-elementary action on ℍn for some n ≥ 2. (b) Q admits a discrete faithful non-elementary minimal action on a simplicial tree with more than two ends. (c) Q admits a (strong-stable) cut R such that the intersection of all conjugates of R is trivial. Then G is virtually a surface group. It follows that if Q is infinite, not virtually cyclic, and is the fundamental group of some closed 3-manifold, then Q contains as a finite index subgroup either a finite index subgroup of the three-dimensional Heisenberg group or the fundamental group of the Cartesian product of a closed oriented surface of positive genus and the circle. As a corollary, we obtain a new proof of a theorem of Dimca and Suciu in [Which 3-manifold groups are Kähler groups? J. Eur. Math. Soc.11 (2009) 521–528] by taking N to be the trivial group. If instead, G is the fundamental group of a compact complex surface, and N is finitely presented, then we show that Q must contain the fundamental group of a Seifert-fibered 3-manifold as a finite index subgroup, and G contains as a finite index subgroup the fundamental group of an elliptic fibration. We also give an example showing that the relation of quasi-isometry does not preserve Kähler groups. This gives a negative answer to a question of Gromov which asks whether Kähler groups can be characterized by their asymptotic geometry.

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41

Biswas, Indranil, and Sorin Dumitrescu. "Branched Holomorphic Cartan Geometries and Calabi–Yau Manifolds." International Mathematics Research Notices 2019, no. 23 (February 7, 2018): 7428–58. http://dx.doi.org/10.1093/imrn/rny003.

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Abstract We introduce the concept of a branched holomorphic Cartan geometry. It generalizes to higher dimension the definition of branched (flat) complex projective structure on a Riemann surface introduced by Mandelbaum [25]. This new framework is much more flexible than that of the usual holomorphic Cartan geometries. We show that all compact complex projective manifolds admit a branched flat holomorphic projective structure. We also give an example of a non-flat branched holomorphic normal projective structure on a compact complex surface. It is known that no compact complex surface admits such a structure with empty branching locus. We prove that non-projective compact simply connected Kähler Calabi–Yau manifolds do not admit any branched holomorphic projective structure. The key ingredient of its proof is the following result of independent interest: if E is a holomorphic vector bundle over a compact simply connected Kähler Calabi–Yau manifold and E admits a holomorphic connection, then E is a trivial holomorphic vector bundle and any holomorphic connection on E is trivial.

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Agricola, Ilka, Giulia Dileo, and Leander Stecker. "Homogeneous non-degenerate 3-(α,δ)-Sasaki manifolds and submersions over quaternionic Kähler spaces." Annals of Global Analysis and Geometry 60, no. 1 (April 26, 2021): 111–41. http://dx.doi.org/10.1007/s10455-021-09762-9.

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AbstractWe show that every 3-$$(\alpha ,\delta )$$ ( α , δ ) -Sasaki manifold of dimension $$4n + 3$$ 4 n + 3 admits a locally defined Riemannian submersion over a quaternionic Kähler manifold of scalar curvature $$16n(n+2)\alpha \delta$$ 16 n ( n + 2 ) α δ . In the non-degenerate case we describe all homogeneous 3-$$(\alpha ,\delta )$$ ( α , δ ) -Sasaki manifolds fibering over symmetric Wolf spaces and over their non-compact dual symmetric spaces. If $$\alpha \delta > 0$$ α δ > 0 , this yields a complete classification of homogeneous 3-$$(\alpha ,\delta )$$ ( α , δ ) -Sasaki manifolds. For $$\alpha \delta < 0$$ α δ < 0 , we provide a general construction of homogeneous 3-$$(\alpha , \delta )$$ ( α , δ ) -Sasaki manifolds fibering over non-symmetric Alekseevsky spaces, the lowest possible dimension of such a manifold being 19.

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Cheng, Xiaoliang, and Yihong Hao. "On the non-existence of common submanifolds of Kähler manifolds and complex space forms." Annals of Global Analysis and Geometry 60, no. 1 (May 10, 2021): 167–80. http://dx.doi.org/10.1007/s10455-021-09776-3.

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Chau, Albert, and Luen-Fai Tam. "Non-negatively curved Kähler manifolds with average quadratic curvature decay." Communications in Analysis and Geometry 15, no. 1 (2007): 121–46. http://dx.doi.org/10.4310/cag.2007.v15.n1.a4.

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Yang, Bo, and Fangyang Zheng. "$U(n)$-invariant Kähler–Ricci flow with non-negative curvature." Communications in Analysis and Geometry 21, no. 2 (2013): 251–94. http://dx.doi.org/10.4310/cag.2013.v21.n2.a1.

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Abreu, Miguel, and Rosa Sena-Dias. "Scalar-flat Kähler metrics on non-compact symplectic toric 4-manifolds." Annals of Global Analysis and Geometry 41, no. 2 (July 3, 2011): 209–39. http://dx.doi.org/10.1007/s10455-011-9280-2.

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Yur'ev, D. V. "Non-Euclidean geometry of mirrors and prequantization on the homogeneous Kähler manifoldM= Diff+(S1)/Rot(S1)." Russian Mathematical Surveys 43, no. 2 (April 30, 1988): 187–88. http://dx.doi.org/10.1070/rm1988v043n02abeh001724.

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Nill, Benjamin, and Andreas Paffenholz. "Examples of Kähler–Einstein toric Fano manifolds associated to non-symmetric reflexive polytopes." Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry 52, no. 2 (May 1, 2011): 297–304. http://dx.doi.org/10.1007/s13366-011-0041-y.

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BELLUCCI, STEFANO, SERGIO FERRARA, MURAT GÜNAYDIN, and ALESSIO MARRANI. "CHARGE ORBITS OF SYMMETRIC SPECIAL GEOMETRIES AND ATTRACTORS." International Journal of Modern Physics A 21, no. 25 (October 10, 2006): 5043–97. http://dx.doi.org/10.1142/s0217751x06034355.

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We study the critical points of the black hole scalar potential V BH in N = 2, d = 4 supergravity coupled to nV vector multiplets, in an asymptotically flat extremal black hole background described by a 2(nV+1)-dimensional dyonic charge vector and (complex) scalar fields which are coordinates of a special Kähler manifold. For the case of homogeneous symmetric spaces, we find three general classes of regular attractor solutions with nonvanishing Bekenstein–Hawking entropy. They correspond to three (inequivalent) classes of orbits of the charge vector, which is in a 2(nV+1)-dimensional representation RV of the U-duality group. Such orbits are nondegenerate, namely they have nonvanishing quartic invariant (for rank-3 spaces). Other than the ½-BPS one, there are two other distinct non-BPS classes of charge orbits, one of which has vanishing central charge. The three species of solutions to the N = 2 extremal black hole attractor equations give rise to different mass spectra of the scalar fluctuations, whose pattern can be inferred by using invariance properties of the critical points of V BH and some group theoretical considerations on homogeneous symmetric special Kähler geometry.

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SAKAGUCHI, MAKOTO. "FOUR-DIMENSIONAL N=2 SUPERSTRING BACKGROUNDS AND THE REAL HEAVENS." International Journal of Modern Physics A 11, no. 07 (March 20, 1996): 1279–97. http://dx.doi.org/10.1142/s0217751x96000572.

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We study N=2 superstring backgrounds which are four-dimensional, non-Kählerian and contain nontrivial dilaton and torsion fields. In particular we consider the case where the backgrounds possess at least one U(1) isometry and are characterized by the continual Toda equation and the Laplace equation. We obtain a string background associated with a nontrivial solution of the continual Toda equation, which is mapped, under the T duality transformation, to the Taub-NUT instanton background. It is shown that the integrable property of the non-Kählerian spaces has a direct origin in the real heavens: real, self-dual, Euclidean, Einstein spaces. The Laplace equation and the continual Toda equation imposed on quasi-Kähler geometry for consistent string propagation are related to the self-duality conditions of the real heavens with “translational” and “rotational” Killing symmetry respectively.

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