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Journal articles on the topic 'Kähler metric'

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Published: 10 March 2023

Last updated: 6 September 2023

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1

Kawamura, Masaya. "On Kähler-like and G-Kähler-like almost Hermitian manifolds." Complex Manifolds 7, no. 1 (April 3, 2020): 145–61. http://dx.doi.org/10.1515/coma-2020-0009.

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AbstractWe introduce Kähler-like, G-Kähler-like metrics on almost Hermitian manifolds. We prove that a compact Kähler-like and G-Kähler-like almost Hermitian manifold equipped with an almost balanced metric is Kähler. We also show that if a Kähler-like and G-Kähler-like almost Hermitian manifold satisfies B_{\bar i\bar j}^\lambda B_{\lambda j}^i \ge 0, then the metric is almost balanced and the almost complex structure is integrable, which means that the metric is balanced. We investigate a G-Kähler-like almost Hermitian manifold under some assumptions.

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2

Calderbank, David M. J., Vladimir S. Matveev, and Stefan Rosemann. "Curvature and the c-projective mobility of Kähler metrics with hamiltonian 2-forms." Compositio Mathematica 152, no. 8 (April 26, 2016): 1555–75. http://dx.doi.org/10.1112/s0010437x16007302.

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The mobility of a Kähler metric is the dimension of the space of metrics with which it is c-projectively equivalent. The mobility is at least two if and only if the Kähler metric admits a nontrivial hamiltonian 2-form. After summarizing this relationship, we present necessary conditions for a Kähler metric to have mobility at least three: its curvature must have nontrivial nullity at every point. Using the local classification of Kähler metrics with hamiltonian 2-forms, we describe explicitly the Kähler metrics with mobility at least three and hence show that the nullity condition on the curvature is also sufficient, up to some degenerate exceptions. In an appendix, we explain how the classification may be related, generically, to the holonomy of a complex cone metric.

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3

Hall, Stuart James, and Thomas Murphy. "Numerical Approximations to Extremal Toric Kähler Metrics with Arbitrary Kähler Class." Proceedings of the Edinburgh Mathematical Society 60, no. 4 (January 10, 2017): 893–910. http://dx.doi.org/10.1017/s0013091516000444.

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AbstractWe develop new algorithms for approximating extremal toric Kähler metrics. We focus on an extremal metric on , which is conformal to an Einstein metric (the Chen–LeBrun–Weber metric). We compare our approximation to one given by Bunch and Donaldson and compute various geometric quantities. In particular, we demonstrate a small eigenvalue of the scalar Laplacian of the Einstein metric that gives numerical evidence that the Einstein metric is conformally unstable under Ricci flow.

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4

Fino, Anna, Gueo Grantcharov, and Luigi Vezzoni. "Astheno–Kähler and Balanced Structures on Fibrations." International Mathematics Research Notices 2019, no. 22 (February 5, 2017): 7093–117. http://dx.doi.org/10.1093/imrn/rnx337.

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Abstract We study the existence of three classes of Hermitian metrics on certain types of compact complex manifolds. More precisely, we consider balanced, strong Kähler with torsion (SKT), and astheno-Kähler metrics. We prove that the twistor spaces of compact hyperkähler and negative quaternionic-Kähler manifolds do not admit astheno-Kähler metrics. Then we provide a construction of astheno-Kähler structures on torus bundles over Kähler manifolds leading to new examples. In particular, we find examples of compact complex non-Kähler manifolds which admit a balanced and an astheno-Kähler metric, thus answering to a question in [52] (see also [24]). One of these examples is simply connected. We also show that the Lie groups SU(3) and G2 admit SKT and astheno-Kähler metrics, which are different. Furthermore, we investigate the existence of balanced metrics on compact complex homogeneous spaces with an invariant volume form, showing in particular that if a compact complex homogeneous space M with invariant volume admits a balanced metric, then its first Chern class c1(M) does not vanish. Finally we characterize Wang C-spaces admitting SKT metrics.

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5

DUNAJSKI, MACIEJ, and PAUL TOD. "Four–dimensional metrics conformal to Kähler." Mathematical Proceedings of the Cambridge Philosophical Society 148, no. 3 (January 5, 2010): 485–503. http://dx.doi.org/10.1017/s030500410999048x.

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AbstractWe derive some necessary conditions on a Riemannian metric (M, g) in four dimensions for it to be locally conformal to Kähler. If the conformal curvature is non anti–self–dual, the self–dual Weyl spinor must be of algebraic type D and satisfy a simple first order conformally invariant condition which is necessary and sufficient for the existence of a Kähler metric in the conformal class. In the anti–self–dual case we establish a one to one correspondence between Kähler metrics in the conformal class and non–zero parallel sections of a certain connection on a natural rank ten vector bundle over M. We use this characterisation to provide examples of ASD metrics which are not conformal to Kähler.We establish a link between the ‘conformal to Kähler condition’ in dimension four and the metrisability of projective structures in dimension two. A projective structure on a surface U is metrisable if and only if the induced (2, 2) conformal structure on M = TU admits a Kähler metric or a para–Kähler metric.

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6

Fujiki, Akira. "Remarks on extremal Kähler metrics on ruled manifolds." Nagoya Mathematical Journal 126 (June 1992): 89–101. http://dx.doi.org/10.1017/s0027763000004001.

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Let X be a compact Kähler manifold and γ Kähler class. For a Kàhler metric g on X we denote by Rg the scalar curvature on X According to Calabi [3][4], consider the functional defined on the set of all the Kähler metrics g whose Kähler forms belong to γ, where dvg is the volume form associated to g. Such a Kähler metric is called extremal if it gives a critical point of Ф. In particular, if Rg is constant, g is extremal. The converse is also true if dim L(X) = 0, where L(X) is the maximal connected linear algebraic subgroup of AutoX (cf. [5]). Note also that any Kähler-Einstein metric is of constant scalar curvature.

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7

SIMANCA, SANTIAGO R. "PRECOMPACTNESS OF THE CALABI ENERGY." International Journal of Mathematics 07, no. 02 (April 1996): 245–54. http://dx.doi.org/10.1142/s0129167x96000141.

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For any complex manifold of Kähler type, the L2-norm of the scalar curvature of an extremal Kähler metric is a continuous function of the Kähler class. In particular, if a convergent sequence of Kähler classes are represented by extremal Kähler metrics, the corresponding sequence of L2-norms of the scalar curvatures is convergent. Similarly, the sequence of holomorphic vector fields associated with a sequence of extremal Kähler metrics with converging Kähler classes is convergent.

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8

Borówka, Aleksandra. "Quaternion-Kähler manifolds near maximal fixed point sets of $$S^1$$-symmetries." Annali di Matematica Pura ed Applicata (1923 -) 199, no. 3 (October 17, 2019): 1243–62. http://dx.doi.org/10.1007/s10231-019-00920-2.

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Abstract Using quaternionic Feix–Kaledin construction, we provide a local classification of quaternion-Kähler metrics with a rotating $$S^1$$S1-symmetry with the fixed point set submanifold S of maximal possible dimension. For any real-analytic Kähler manifold S equipped with a line bundle with a real-analytic unitary connection with curvature proportional to the Kähler form, we explicitly construct a holomorphic contact distribution on the twistor space obtained by the quaternionic Feix–Kaledin construction from these data. Conversely, we show that quaternion-Kähler metrics with a rotating $$S^1$$S1-symmetry induce on the fixed point set of maximal dimension a Kähler metric together with a unitary connection on a holomorphic line bundle with curvature proportional to the Kähler form and the two constructions are inverse to each other. Moreover, we study the case when S is compact, showing that in this case the quaternion-Kähler geometry is determined by the Kähler metric on the fixed point set (of maximal possible dimension) and by the contact line bundle along the corresponding submanifold on the twistor space. Finally, we relate the results to the c-map construction showing that the family of quaternion-Kähler manifolds obtained from a fixed Kähler metric on S by varying the line bundle and the hyperkähler manifold obtained by hyperkähler Feix–Kaledin construction from S are related by hyperkähler/quaternion-Kähler correspondence.

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9

Guenancia, Henri. "Kähler–Einstein metrics: From cones to cusps." Journal für die reine und angewandte Mathematik (Crelles Journal) 2020, no. 759 (February 1, 2020): 1–27. http://dx.doi.org/10.1515/crelle-2018-0001.

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AbstractIn this note, we prove that on a compact Kähler manifold \hskip-0.569055pt{X}\hskip-0.569055pt carrying a smooth divisor D such that {K_{X}+D} is ample, the Kähler–Einstein cusp metric is the limit (in a strong sense) of the Kähler–Einstein conic metrics when the cone angle goes to 0. We further investigate the boundary behavior of those and prove that the rescaled metrics converge to a cylindrical metric on {\mathbb{C}^{*}\times\mathbb{C}^{n-1}}.

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10

RUAN, WEI-DONG. "DEGENERATION OF KÄHLER–EINSTEIN MANIFOLDS I: THE NORMAL CROSSING CASE." Communications in Contemporary Mathematics 06, no. 02 (April 2004): 301–13. http://dx.doi.org/10.1142/s0219199704001331.

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In this paper we prove that the Kähler–Einstein metrics for a degeneration family of Kähler manifolds with ample canonical bundles converge in the sense of Cheeger–Gromov to the complete Kähler–Einstein metric on the smooth part of the central fiber when the central fiber has only normal crossing singularities inside smooth total space. We also prove the incompleteness of the Weil–Peterson metric in this case.

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11

Ida, Cristian, Alexandru Ionescu, and Adelina Manea. "A note on para-holomorphic Riemannian–Einstein manifolds." International Journal of Geometric Methods in Modern Physics 13, no. 09 (September 20, 2016): 1650107. http://dx.doi.org/10.1142/s0219887816501073.

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The aim of this note is the study of Einstein condition for para-holomorphic Riemannian metrics in the para-complex geometry framework. First, we make some general considerations about para-complex Riemannian manifolds (not necessarily para-holomorphic). Next, using a one-to-one correspondence between para-holomorphic Riemannian metrics and para-Kähler–Norden metrics, we study the Einstein condition for a para-holomorphic Riemannian metric and the associated real para-Kähler–Norden metric on a para-complex manifold. Finally, it is shown that every semi-simple para-complex Lie group inherits a natural para-Kählerian–Norden Einstein metric.

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12

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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13

Fernández, Marisa, Vicente Muñoz, and José A. Santisteban. "Cohomologically Kähler manifolds with no Kähler metrics." International Journal of Mathematics and Mathematical Sciences 2003, no. 52 (2003): 3315–25. http://dx.doi.org/10.1155/s0161171203211327.

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We show some examples of compact symplectic solvmanifolds, of dimension greater than four, which are cohomologically Kähler and do not admit Kähler metric since their fundamental groups cannot be the fundamental group of any compact Kähler manifold. Some of the examples that we study were considered by Benson and Gordon (1990). However, whether such manifolds have Kähler metrics was an open question. The formality and the hard Lefschetz property are studied for the symplectic submanifolds constructed by Auroux (1997) and some consequences are discussed.

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14

GUAN, DANIEL. "EXISTENCE OF EXTREMAL METRICS ON ALMOST HOMOGENEOUS MANIFOLDS OF COHOMOGENEITY ONE — III." International Journal of Mathematics 14, no. 03 (May 2003): 259–87. http://dx.doi.org/10.1142/s0129167x03001806.

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In this paper we prove that on certain manifolds Nn with nonnegative first Chern class the existence of extremal metric in a Kähler class is the same as the stability of the Kähler class. We also obtain many new Kähler classes with extremal metrics, in particular, the Kähler-Einstein metrics for Nn with n > 2. We also compare the problem of finding Calabi extremal metrics with the similar problem of finding Hermitian–Einstein metrics on the holomorphic vector bundles. We explain the geodesic stability and found that the stability for the manifold is much more complicated

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15

MASCHLER, GIDEON. "METRIC PAIRS AND THE FUTAKI CHARACTER." International Journal of Mathematics 13, no. 01 (February 2002): 1–9. http://dx.doi.org/10.1142/s0129167x02001125.

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The non-vanishing of the Futaki character gives an obstruction to the existence of Kähler metrics of constant scalar curvature, having a Kähler form belonging to a fixed Kähler class [4, 6]. It is shown that, in combination with the resolution of the Calabi conjecture [18], one has an analogous obstruction on pairs of metrics having Kähler forms belonging to a fixed pair of Kähler classes. If the difference of the Futaki characters on two classes of fixed total volume does not vanish identically, there cannot exist a pair of metrics, with Kähler forms in these classes, having the same Ricci form and the same harmonic Ricci form. When the obstruction vanishes, results in [8] are used to construct non-trivial examples of such pairs, which are also extremal in the sense of Calabi [3].

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16

Maschler, Gideon, and Christina W. Tønnesen-Friedman. "Generalizations of Kähler-Ricci solitons on projective bundles." MATHEMATICA SCANDINAVICA 108, no. 2 (June 1, 2011): 161. http://dx.doi.org/10.7146/math.scand.a-15165.

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We prove that an admissible manifold (as defined by Apostolov, Calderbank, Gauduchon and Tønnesen-Friedman), arising from a base with a local Kähler product of constant scalar curvature metrics, admits Generalized Quasi-Einstein Kähler metrics (as defined by D. Guan) in all "sufficiently small" admissible Kähler classes. We give an example where the existence of Generalized Quasi-Einstein metrics fails in some Kähler classes while not in others. We also prove an analogous existence theorem for an additional metric type, defined by the requirement that the scalar curvature is an affine combination of a Killing potential and its Laplacian.

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17

Di Nezza, Eleonora, and Vincent Guedj. "Geometry and topology of the space of Kähler metrics on singular varieties." Compositio Mathematica 154, no. 8 (July 19, 2018): 1593–632. http://dx.doi.org/10.1112/s0010437x18007170.

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Let $Y$ be a compact Kähler normal space and let $\unicode[STIX]{x1D6FC}\in H_{\mathit{BC}}^{1,1}(Y)$ be a Kähler class. We study metric properties of the space ${\mathcal{H}}_{\unicode[STIX]{x1D6FC}}$ of Kähler metrics in $\unicode[STIX]{x1D6FC}$ using Mabuchi geodesics. We extend several results of Calabi, Chen, and Darvas, previously established when the underlying space is smooth. As an application, we analytically characterize the existence of Kähler–Einstein metrics on $\mathbb{Q}$-Fano varieties, generalizing a result of Tian, and illustrate these concepts in the case of toric varieties.

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18

MABUCHI, TOSHIKI. "UNIQUENESS OF EXTREMAL KÄHLER METRICS FOR AN INTEGRAL KÄHLER CLASS." International Journal of Mathematics 15, no. 06 (August 2004): 531–46. http://dx.doi.org/10.1142/s0129167x04002429.

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For an integral Kähler class on a compact connected complex manifold, an extremal Kähler metric, if any, in the class is unique up to the action of Aut 0(M). This generalizes a recent result of Donaldson (see [4] for cases of metrics of constant scalar curvature) and that of Chen [3] for c1(M)≤0.

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19

SMOCZYK, KNUT, GUOFANG WANG, and YONGBING ZHANG. "THE SASAKI–RICCI FLOW." International Journal of Mathematics 21, no. 07 (July 2010): 951–69. http://dx.doi.org/10.1142/s0129167x10006331.

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In this paper, we introduce the Sasaki–Ricci flow to study the existence of η-Einstein metrics. In the positive case any η-Einstein metric can be homothetically transformed to a Sasaki–Einstein metric. Hence it is an odd-dimensional counterpart of the Kähler–Ricci flow. We prove its well-posedness and long-time existence. In the negative or null case the flow converges to the unique η-Einstein metric. In the positive case the convergence remains in general open. The paper can be viewed as an odd-dimensional counterpart of Cao's results on the Kähler–Ricci flow.

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20

Kawamura, Masaya. "On the Kähler-likeness on almost Hermitian manifolds." Complex Manifolds 6, no. 1 (January 1, 2019): 366–76. http://dx.doi.org/10.1515/coma-2019-0020.

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AbstractWe define a Kähler-like almost Hermitian metric. We will prove that on a compact Kähler-like almost Hermitian manifold (M2n, J, g), if it admits a positive ∂ ̄∂-closed (n − 2, n − 2)-form, then g is a quasi-Kähler metric.

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21

Zhang, Xi, and Xiangwen Zhang. "Generalized Kähler–Einstein Metrics and Energy Functionals." Canadian Journal of Mathematics 66, no. 6 (December 1, 2014): 1413–35. http://dx.doi.org/10.4153/cjm-2013-034-3.

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Abstract.In this paper, we consider a generalized Kähler–Einstein equation on a Kähler manifold M. Using the twisted 𝒦–energy introduced by Song and Tian, we show that the existence of generalized Kähler–Einstein metrics with semi–positive twisting (1, 1)–form θ is also closely related to the properness of the twisted 𝒦-energy functional. Under the condition that the twisting form θ is strictly positive at a point or M admits no nontrivial Hamiltonian holomorphic vector field, we prove that the existence of generalized Kähler–Einstein metric implies a Moser–Trudinger type inequality.

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22

Sun, Song. "Ricci curvature in Kähler geometry." International Journal of Mathematics 28, no. 09 (August 2017): 1740009. http://dx.doi.org/10.1142/s0129167x17400092.

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These are the notes for lectures given at the Sanya winter school in complex analysis and geometry in January 2016. In Sec. 1, we review the meaning of Ricci curvature of Kähler metrics and introduce the problem of finding Kähler–Einstein metrics. In Sec. 2, we describe the formal picture that leads to the notion of K-stability of Fano manifolds, which is an algebro-geometric criterion for the existence of a Kähler–Einstein metric, by the recent result of Chen–Donaldson–Sun. In Sec. 3, we discuss algebraic structure on Gromov–Hausdorff limits, which is a key ingredient in the proof of the Kähler–Einstein result. In Sec. 4, we give a brief survey of the more recent work on tangent cones of singular Kähler–Einstein metrics arising from Gromov–Hausdorff limits, and the connections with algebraic geometry.

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23

Chen, Xiuxiong, Song Sun, and Bing Wang. "Kähler–Ricci flow, Kähler–Einstein metric, and K–stability." Geometry & Topology 22, no. 6 (September 23, 2018): 3145–73. http://dx.doi.org/10.2140/gt.2018.22.3145.

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24

De, Uday Chand, Sudhakar K. Chaubey, and Young Jin Suh. "A note on almost co-Kähler manifolds." International Journal of Geometric Methods in Modern Physics 17, no. 10 (August 26, 2020): 2050153. http://dx.doi.org/10.1142/s0219887820501534.

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We characterize almost co-Kähler manifolds with gradient Yamabe, gradient Einstein and quasi-Yamabe solitons. It is proved that if the metric of a [Formula: see text]-almost co-Kähler manifold [Formula: see text] is a gradient Yamabe soliton, then [Formula: see text] is either [Formula: see text]-almost co-Kähler or [Formula: see text]-almost co-Kähler or the metric of [Formula: see text] is a trivial gradient Yamabe soliton. A [Formula: see text]-almost co-Kähler manifold with gradient Einstein soliton is [Formula: see text]-almost co-Kähler. Finally, it is shown that an almost co-Kähler manifold admitting a quasi-Yamabe soliton, whose soliton vector is pointwise collinear with the Reeb vector field of the manifold, is [Formula: see text]-almost co-Kähler. Consequently, some results of almost co-Kähler manifolds are deduced.

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25

Calamai, Simone. "The Calabi metric for the space of Kähler metrics." Mathematische Annalen 353, no. 2 (June 25, 2011): 373–402. http://dx.doi.org/10.1007/s00208-011-0690-z.

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26

Yang, Kichoon. "Invariant Kähler metrics and projective embeddings of the flag manifold." Bulletin of the Australian Mathematical Society 49, no. 2 (April 1994): 239–47. http://dx.doi.org/10.1017/s0004972700016300.

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We determine explicitly the space of invariant Hermitian and Kähler metrics on the flag manifold. In particular, we show that a Killing metric is not Kähler. The Chern forms are also computed in terms of the Maurer–Cartan form, and this calculation is used to prove that the flag manifold is projective algebraic. An explicit projective embedding of the flag manifold is also given.

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27

BIELAWSKI, ROGER. "Ricci-flat Kähler metrics on canonical bundles." Mathematical Proceedings of the Cambridge Philosophical Society 132, no. 3 (May 2002): 471–79. http://dx.doi.org/10.1017/s030500410100576x.

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We prove the existence of a (unique) S1-invariant Ricci-flat Kähler metric on a neighbourhood of the zero section in the canonical bundle of a real-analytic Kähler manifold X, extending the metric on X.

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28

Aazami, Amir Babak, and Gideon Maschler. "Kähler metrics via Lorentzian Geometry in dimension four." Complex Manifolds 7, no. 1 (November 15, 2019): 36–61. http://dx.doi.org/10.1515/coma-2020-0002.

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AbstractGiven 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 in warped products, for instance de Sitter spacetime, as well as gravitational plane waves, 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.

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29

Piovani, Riccardo, and Tommaso Sferruzza. "Deformations of Strong Kähler with torsion metrics." Complex Manifolds 8, no. 1 (January 1, 2021): 286–301. http://dx.doi.org/10.1515/coma-2020-0120.

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Abstract Existence of strong Kähler with torsion metrics, shortly SKT metrics, on complex manifolds has been shown to be unstable under small deformations. We find necessary conditions under which the property of being SKT is stable for a smooth curve of Hermitian metrics {ω t } t which equals a fixed SKT metric ω for t = 0, along a differentiable family of complex manifolds {M t } t .

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30

Biswas, Gour, Xiaomin Chen, and Uday De. "Riemann solitons on almost co-Kähler manifolds." Filomat 36, no. 4 (2022): 1403–13. http://dx.doi.org/10.2298/fil2204403b.

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The aim of the present paper is to characterize almost co-K?hler manifolds whose metrics are the Riemann solitons. At first we provide a necessary and sufficient condition for the metric of a 3-dimensional manifold to be Riemann soliton. Next it is proved that if the metric of an almost co-K?hler manifold is a Riemann soliton with the soliton vector field ?, then the manifold is flat. It is also shown that if the metric of a (?, ?)-almost co-K?hler manifold with ? < 0 is a Riemann soliton, then the soliton is expanding and ?, ?, ? satisfies a relation. We also prove that there does not exist gradient almost Riemann solitons on (?, ?)-almost co-K?hler manifolds with ? < 0. Finally, the existence of a Riemann soliton on a three dimensional almost co-K?hler manifold is ensured by a proper example.

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31

Dancer, Andrew S., and Ian A. B. Strachan. "Kähler–Einstein metrics with SU(2) action." Mathematical Proceedings of the Cambridge Philosophical Society 115, no. 3 (May 1994): 513–25. http://dx.doi.org/10.1017/s0305004100072273.

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The aim of this paper is to analyse Riemannian Kähler–Einstein metrics g in real dimension four admitting an isometric action of SU(2) with generically three-dimensional orbits. The Kähler condition means that there is a complex structure I, with respect to which the metric is hermitian, such that the two-form Ωdefined byis closed. It is well-known that if this condition holds then Ω is in fact covariant constant.

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32

Visinescu, Mihai. "Sasaki–Ricci flow equation on five-dimensional Sasaki–Einstein space Yp,q." Modern Physics Letters A 35, no. 14 (March 20, 2020): 2050114. http://dx.doi.org/10.1142/s021773232050114x.

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We analyze the transverse Kähler–Ricci flow equation on Sasaki-Einstein space [Formula: see text]. Explicit solutions are produced representing new five-dimensional Sasaki structures. Solutions which do not modify the transverse metric preserve the Sasaki–Einstein feature of the contact structure. If the transverse metric is altered, the deformed metrics remain Sasaki, but not Einstein.

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33

Feng, Zhiming. "The first two coefficients of the Bergman function expansions for Cartan–Hartogs domains." International Journal of Mathematics 29, no. 06 (June 2018): 1850043. http://dx.doi.org/10.1142/s0129167x1850043x.

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Let [Formula: see text] be a globally defined real Kähler potential on a domain [Formula: see text], and [Formula: see text] be a Kähler metric on the Hartogs domain [Formula: see text] associated with the Kähler potential [Formula: see text]. First, we obtain explicit formulas of the coefficients [Formula: see text] of the Bergman function expansion for the Hartogs domain [Formula: see text] in a momentum profile [Formula: see text]. Second, using explicit expressions of [Formula: see text], we obtain necessary and sufficient conditions for the coefficients [Formula: see text] to be constants. Finally, we obtain all the invariant complete Kähler metrics on Cartan–Hartogs domains such that their the coefficients [Formula: see text] of the Bergman function expansions are constants.

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34

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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35

Bejan, Cornelia-Livia, Galia Nakova, and Adara M. Blaga. "On Bochner Flat Kähler B-Manifolds." Axioms 12, no. 4 (March 30, 2023): 336. http://dx.doi.org/10.3390/axioms12040336.

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We obtain on a Kähler B-manifold (i.e., a Kähler manifold with a Norden metric) some corresponding results from the Kählerian and para-Kählerian context concerning the Bochner curvature. We prove that such a manifold is of constant totally real sectional curvatures if and only if it is a holomorphic Einstein, Bochner flat manifold. Moreover, we provide the necessary and sufficient conditions for a gradient Ricci soliton or a holomorphic η-Einstein Kähler manifold with a Norden metric to be Bochner flat. Finally, we show that a Kähler B-manifold is of quasi-constant totally real sectional curvatures if and only if it is a holomorphic η-Einstein, Bochner flat manifold.

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Alekseevsky, D. V., and S. Marchiafava. "Quaternionic Transformations of a Non-Positive Quaternionic Kähler Manifold." International Journal of Mathematics 08, no. 03 (May 1997): 301–16. http://dx.doi.org/10.1142/s0129167x97000147.

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Let (M,g,Q) be a simply connected, complete, quaternionic Kähler manifold without flat de Rham factor. Then any 1-parameter group of transformations of M which preserve the quaternionic structure Q preserves also the metric g. Moreover, if (M,g) is irreducible then the quaternionic Kähler metric g on (M,Q) is unique up to a homothety.

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37

Perego, Arvid. "Kobayashi—Hitchin correspondence for twisted vector bundles." Complex Manifolds 8, no. 1 (January 1, 2021): 1–95. http://dx.doi.org/10.1515/coma-2020-0107.

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Abstract 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.

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38

Nave, Gabriele La, Gang Tian, and Zhenlei Zhang. "Bounding Diameter Of Singular Kähler Metric." American Journal of Mathematics 139, no. 6 (2017): 1693–731. http://dx.doi.org/10.1353/ajm.2017.0042.

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39

Li, Yan. "Bounding Diameter of Conical Kähler Metric." Journal of Geometric Analysis 28, no. 2 (April 20, 2017): 950–82. http://dx.doi.org/10.1007/s12220-017-9850-z.

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40

Isaev, A. V. "Kähler-Einstein metric on Reinhardt domains." Journal of Geometric Analysis 5, no. 2 (June 1995): 237–54. http://dx.doi.org/10.1007/bf02921676.

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41

Gursky, Matthew J., and Jeffrey Streets. "A formal Riemannian structure on conformal classes and the inverse Gauss curvature flow." Geometric Flows 4, no. 1 (January 1, 2019): 30–50. http://dx.doi.org/10.1515/geofl-2019-0003.

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Abstract We define a formal Riemannian metric on a given conformal class of metrics with signed curvature on a closed Riemann surface. As it turns out this metric is the well-known Mabuchi-Semmes-Donaldson metric of Kähler geometry in a different guise. The metric has many interesting properties, and in particular we show that the classical Liouville energy is geodesically convex. This suggests a different approach to the uniformization theorem by studying the negative gradient flow of the normalized Liouville energy with respect to this metric, a new geometric flow whose principal term is the inverse of the Gauss curvature. We prove long time existence of solutions with arbitrary initial data and weak convergence to constant scalar curvature metrics by exploiting the metric space structure.

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42

Zedda, Michela. "Strongly not relatives Kähler manifolds." Complex Manifolds 4, no. 1 (February 23, 2017): 1–6. http://dx.doi.org/10.1515/coma-2017-0001.

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Abstract In this paper we study Kähler manifolds that are strongly not relative to any projective Kähler manifold, i.e. those Kähler manifolds that do not share a Kähler submanifold with any projective Kähler manifold even when their metric is rescaled by the multiplication by a positive constant. We prove two results which highlight some relations between this property and the existence of a full Kähler immersion into the infinite dimensional complex projective space. As application we get that the 1-parameter families of Bergman-Hartogs and Fock-Bargmann-Hartogs domains are strongly not relative to projective Kähler manifolds.

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Carmona Jiménez, José Luis, and Marco Castrillón López. "Reduction of Homogeneous Pseudo-Kähler Structures by One-Dimensional Fibers." Axioms 9, no. 3 (August 1, 2020): 94. http://dx.doi.org/10.3390/axioms9030094.

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We study the reduction procedure applied to pseudo-Kähler manifolds by a one dimensional Lie group acting by isometries and preserving the complex tensor. We endow the quotient manifold with an almost contact metric structure. We use this fact to connect pseudo-Kähler homogeneous structures with almost contact metric homogeneous structures. This relation will have consequences in the class of the almost contact manifold. Indeed, if we choose a pseudo-Kähler homogeneous structure of linear type, then the reduced, almost contact homogeneous structure is of linear type and the reduced manifold is of type C5⊕C6⊕C12 of Chinea-González classification.

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Li, Yan, Gang Tian, and Xiaohua Zhu. "Singular Kähler-Einstein metrics on $ \mathbb Q $-Fano compactifications of Lie groups." Mathematics in Engineering 5, no. 2 (2022): 1–43. http://dx.doi.org/10.3934/mine.2023028.

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<abstract><p>In this paper, we prove an existence result for Kähler-Einstein metrics on $ \mathbb Q $-Fano compactifications of Lie groups by the variational method, provided their moment polytopes satisfy a <italic>fine</italic> condition. As an application, we prove that there is no $ \mathbb Q $-Fano $ {\rm SO}_4(\mathbb C) $-compactification which admits a Kähler-Einstein metric with the same volume as that of a smooth K-unstable Fano $ {\rm SO}_4(\mathbb C) $-compactification.</p></abstract>

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45

CHEN, BO-YONG. "THE BERGMAN METRIC ON TEICHMÜLLER SPACE." International Journal of Mathematics 15, no. 10 (December 2004): 1085–91. http://dx.doi.org/10.1142/s0129167x04002697.

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46

JOHANSEN, A. "TWISTING OF N=1 SUSY GAUGE THEORIES AND HETEROTIC TOPOLOGICAL THEORIES." International Journal of Modern Physics A 10, no. 30 (December 10, 1995): 4325–57. http://dx.doi.org/10.1142/s0217751x9500200x.

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It is shown that D=4N=1 SUSY Yang-Mills theory with an appropriate supermultiplet of matter can be twisted on a compact Kähler manifold. The conditions for cancellation of anomalies of BRST charge are found. The twisted theory has an appropriate BRST charge. We find a nontrivial set of physical operators defined as classes of the cohomology of this BRST operator. We prove that the physical correlators are independent of the external Kähler metric up to a power of a ratio of two Ray-Singer torsions for the Dolbeault cohomology complex on a Kähler manifold. The correlators of local physical operators turn out to be independent of antiholomorphic coordinates defined with a complex structure on the Kähler manifold. However, a dependence of the correlators on holomorphic coordinates can still remain. For a hyper-Kähler metric the physical correlators turn out to be independent of all coordinates of insertions of local physical operators.

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47

Cheltsov, Ivan, and Jesus Martinez-Garcia. "Stable Polarized del Pezzo Surfaces." International Mathematics Research Notices 2020, no. 18 (July 26, 2018): 5477–505. http://dx.doi.org/10.1093/imrn/rny182.

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Abstract We give a simple sufficient condition for $K$-stability of polarized del Pezzo surfaces and for the existence of a constant scalar curvature Kähler metric in the Kähler class corresponding to the polarization.

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Tønnesen-Friedman, Christina W. "Kähler Yamabe minimizers on minimal ruled surfaces." MATHEMATICA SCANDINAVICA 90, no. 2 (June 1, 2002): 180. http://dx.doi.org/10.7146/math.scand.a-14369.

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It is shown that if a minimal ruled surface $\mathrm{P}(E) \rightarrow \Sigma$ admits a Kähler Yamabe minimizer, then this metric is generalized Kähler-Einstein and the holomorphic vector bundle $E$ is quasi-stable.

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Santoro, Bianca. "Existence of complete Kähler Ricci-flat metrics on crepant resolutions." Communications in Contemporary Mathematics 16, no. 03 (May 26, 2014): 1450003. http://dx.doi.org/10.1142/s0219199714500035.

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In this note, we obtain existence results for complete Ricci-flat Kähler metrics on crepant resolutions of singularities of Calabi–Yau varieties. Furthermore, for certain asymptotically flat Calabi–Yau varieties, we show that the Ricci-flat metric on the resolved manifold has the same asymptotic behavior as the initial variety.

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50

YAN, RONGMU. "COMPLEX BERWALD MANIFOLDS WITH VANISHING HOLOMORPHIC SECTIONAL CURVATURE." Glasgow Mathematical Journal 50, no. 2 (May 2008): 203–8. http://dx.doi.org/10.1017/s001708950800414x.

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AbstractIn this paper, we prove that a strongly convex and Kähler-Finsler metric is a complex Berwald metric with zero holomorphic sectional curvature if and only if it is a complex locally Minkowski metric.

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