A Hermitian metric on a complex manifold of complex dimension $n$ is called {\em astheno-Kähler} if its fundamental $2$-form $F$ satisfies the condition $\partial \overline \partial F^{n - 2} =0$. If $n =3$, then the metric is {\em strong KT}, i.e. $F$ is $\partial \overline \partial$-closed. By using blow-ups and the twist construction, we construct simply-connected astheno-Kähler manifolds of complex dimension $n > 3$. Moreover, we construct a family of astheno-Kähler (non strong KT) $2$-step nilmanifolds of complex dimension $4$ and we study deformations of strong KT structures on nilmanifolds of complex dimension $3$. Finally, we study the relation between astheno-Kähler condition and (locally) conformally balanced one and we provide examples of locally conformally balanced astheno-Kähler metrics on $\T^2$-bundles over (non-Kähler) homogeneous complex surfaces.
In this note we present, for every $n \geq 4$, a non-Kähler compact complex manifold $X$ of complex dimension $n$ admitting a balanced metric and an astheno-Kähler metric which is in addition $k$-th Gauduchon for any $1\leq k\leq n-1$.
We study the existence of three classes of Hermitian metrics on certain types of compact complex manifolds. More precisely, we consider balanced, 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 examples of astheno-Kähler structures on toric bundles over Kähler manifolds. In particular, we find examples of compact complex non-Kähler manifolds which admit a balanced and an astheno-Kähler metrics, thus answering to a question in [52] (see also [24]). One of these…
We provide families of compact astheno-Kähler nilmanifolds and we study the behaviour of the complex blowup of such manifolds. We prove that the existence of an astheno-Kähler metric satisfying an extra differential condition is not preserved by blowup. We also study the interplay between Strong Kähler with torsion metrics and geometrically Bott-Chern metrics. We show that Fino-Parton-Salamon nilmanifolds are geometrically-Bott-Chern-formal, whereas we obtain negative results on the product of two copies of primary Kodaira surface, Inoue surface of type $\mathcal{S}_M$ and on the product of a …
We prove the existence of a (unique) S^1-invariant Ricci-flat Kaehler metric on a neighbourhood of the zero section in the canonical bundle of a real-analytic Kaehler manifold X, extending the metric on X.
We prove that every Kaehler metric, whose potential is a function of the time-like distance in the flat Kaehler-Lorentz space, is of quasi-constant holomorphic sectional curvatures, satisfying certain conditions. This gives a local classification of the Kaehler manifolds with the above mentioned metrics. New examples of Sasakian space forms are obtained as real hypersurfaces of a Kaehler space form with special invariant distribution. We introduce three types of even dimensional rotational hypersurfaces in flat spaces and endow them with locally conformal Kaehler structures. We prove that thes…
Almost hypercomplex manifolds with Hermitian and Norden metrics and more specially the corresponding quaternionic Kaehler manifolds are considered. Some necessary and sufficient conditions the investigated manifolds be isotropic hyper-Kaehlerian and flat are found. It is proved that the quaternionic Kaehler manifolds with the considered metric structure are Einstein for dimension at least 8. The class of the non-hyper-Kaehler quaternionic Kaehler manifold of the considered type is determined.
We give sufficient conditions for the existence of Kaehler-Einstein and constant scalar curvature Kaehler (cscK) metrics on finite ramified Galois coverings of a cscK manifold in terms of cohomological conditions on the Kaehler classes and the branching divisor. This result generalizes previous work on Kaehler-Einstein metrics by Li-Sun [Comm. Math. Phys. 2014], and extends Chen-Cheng's existence results for cscK metrics in [J. Amer. Math. Soc. 2021].
The purpose of this paper is to establish a partial regularity theory on certain homogeneous complex Monge-Ampere equations. As consequences of this new theory, we prove the uniqueness of extremal Kaehler metrics and give an necessary condition for existence of extremal Kaehler metrics.
Using as an underlying manifold an alpha-Sasakian manifold we introduce warped product Kaehler manifolds. We prove that if the underlying manifold is an alpha-Sasakian space form, then the corresponding Kaehler manifold is of quasi-constant holomorphic sectional curvatures with special distribution. Conversely, we prove that any Kaehler manifold of quasi-constant holomorphic sectional curvatures with special distribution locally has the structure of a warped product Kaehler manifold whose base is an alpha-Sasakian space form. Considering the scalar distribution generated by the scalar curvatur…
We apply the results from the article Cahen, Schwachhöfer: Special symplectic connections, to the case of Bochner-Kaehler metrics. We obtain a (local) classification of these based on the orbit types of the adjoint action in $su(n,1)$. The relation between Sasaki and Bochner-Kaehler metrics in cone and transveral metrics constructions is discussed. The connection of the special symplectic and Weyl connections is outlined. The duality between the Ricci-type and Bochner-Kaehler metrics is shown.
We study G-invariant Kaehler metrics on G^C from the Hamiltonian point of view. As an application we show that there exist (GxG)-invariant Ricci-flat Kaehler metrics on G^C for any compact semisimple Lie group G.
A 4-parametric family of 4-dimensional quasi-Kaehler manifolds with Norden metric is constructed on a Lie group. This family is characterized geometrically. The condition for such a 4-manifold to be isotropic Kaehler is given.
A 4n-parametric family of 4n-dimensional quasi-Kaehler manifolds with Killing Norden metric is constructed on a Lie group. This family is characterized geometrically.
Given any closed Kaehler manifold we define, following an idea by Eugenio Calabi, a Riemannian metric on the space of Kaehler metrics regarded as an infinite dimensional manifold. We prove several geometrical features of the resulting space, some of which we think were already known to Calabi. In particular, the space is a portion of an infinite dimensional sphere and admits explicit unique smooth solutions for the Cauchy and the Dirichlet problems for the geodesic equation.
The basic class of the non-integrable almost complex manifolds with Norden metric is considered. Its curvature properties are studied. The isotropic Kaehler type of investigated manifolds is introduced and characterized geometrically.
We prove a regularity result for Monge-Ampère equations degenerate along smooth divisor on Kaehler manifolds in Donaldson's spaces of $β$-weighted functions. We apply this result to study the curvature of Kaehler metrics with conical singularities along divisors and give a geometric sufficient condition on the divisor for its boundedness.
The mobility of a Kaehler 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 Kaehler metric admits a nontrivial hamiltonian 2-form. After summarizing this relationship, we present necessary conditions for a Kaehler metric to have mobility at least three: its curvature must have nontrivial nullity at every point. Using the local classification of Kaehler metrics with hamiltonian 2-forms, we describe explicitly the Kaehler metrics with mobility at least three and hence show that the nullity condition on the …
In this paper, we shall give some affirmative answer to an extremal Kaehler version of the Yau-Tian-Donaldson Conjecture. For a polarized algebraic manifold $(X,L)$, we choose a maximal algebraic torus $T$ in the group of holomorphic automorphisms of $X$. Then the polarization class $c_1(L)$ will be shown to admit an extremal Kaehler metric if $(X,L)$ is strongly K-stable relative to $T$.
A 3-parametric family of 6-dimensional quasi-Kaehler manifolds with Norden metric is constructed on a Lie group. This family is characterized geometrically. The condition for such a 6-manifold to be isotropic Kaehler is given.
In this paper, authors have established Chen's inequalities for the submanifolds of quaternionic Kaehler manifolds characterized by Ricci quarter-symmetric metric connection. Other than these inequalities, we have also derived generalized normalized Casorati curvature inequalities for the same submanifolds.
In this paper, we investigate the Chern number inequalities on $4$-dimensional Kähler manifolds admitting the deformed Hermitian-Yang-Mills metrics under the assumption $\hatθ\in (π,2π)$.
Two Kaehler metrics on one complex manifold are said to be c-projectively equivalent if their J-planar curves, i.e., curves defined by the property that their acceleration is complex proportional to their velocity, coincide. The degree of mobility of a Kaehler metric is the dimension of the space of metrics that are c-projectively equivalent to it. We give the list of all possible values of the degree of mobility of simply connected 2n-dimensional Riemannian Kaehler manifolds. We also describe all such values under the additional assumption that the metric is Einstein. As an application, we de…
We study Kaehlerian manifolds with Norden metric $g$ and develop the theory of their holomorphic hypersurfaces with constant totally real sectional curvatures. We prove a classification theorem for the holomorphic hypersurfaces of $(\mathbb{R}^{2n+2}, g, J)$ with constant totally real sectional curvatures.
The basic class of the non-integrable almost complex manifolds with a pair of Norden metrics are considered. The interconnections between corresponding quantities at the transformation between the two Levi-Civita connections are given. A 4-parametric family of 4-dimensional quasi-Kaehler manifolds with Norden metric is characterized with respect to the associated Levi-Civita connection.
In the present paper, we prove a stability theorem for the Kaehler Ricci flow near the infimum of the functional E_1 under the assumption that the initial metric has Ricci > -1 and |Riem| bounded. At present stage, our main theorem still need a topological assumption (1.2) which we hope to be removed in subsequent papers. The underlying moral is: if a Kaehler metric is sufficiently closed to a Kaehler Einstein metric, then the Kaehler Ricci flow converges to it. The present work should be viewed as a first step in a more ambitious program of deriving the existence of Kaehler Einstein metrics w…
We deal with compact Kaehler manifolds M which are acted on by a semisimple compact Lie group G of isometries with codimension one regular orbits. We provide an explicit description of the standard blow-ups of such manifolds along complex singular orbits, in case b_1(M) = 0 and the regular orbits are Levi nondegenerate. Up to very few exceptions, all the nonhomogeneous manifolds in this class are shown to admit a G-invariant Kaehler-Einstein metric, giving completely new examples of compact Kaehler-Einstein manifolds.
Any constant-scalar-curvature Kaehler (cscK) metric on a complex surface may be viewed as a solution of the Einstein-Maxwell equations, and this allows one to produce solutions of these equations on any 4-manifold that arises as a compact complex surface with b_1 even. It is shown, however, that not all solutions of the Einstein-Maxwell equations on such manifolds arise in this way; new examples can be constructed by means of conformally Kaehler geometry.
A type of almost contact hypersurfaces with Norden metric of a Kaehler manifold with Norden metric is considered. The curvature tensor and the special sectional curvatures are characterized. The canonical connection on such manifolds is studied and the form of the corresponding Kaehler curvature tensor is obtained. Some curvature properties of the manifolds belonging to the widest integrable main class of the considered type of hypersurfaces are given.
We construct left invariant quaternionic contact (qc) structures on Lie groups with zero and non-zero torsion and with non-vanishing quaternionic contact conformal curvature tensor, thus showing the existence of non-flat quaternionic contact manifolds. We prove that the product of the real line with a seven dimensional manifold, equipped with a certain qc structure, has a quaternionic Kaehler metric as well as a metric with holonomy contained in Spin(7). As a consequence we determine explicit quaternionic Kaehler metrics and Spin(7)-holonomy metrics which seem to be new. Moreover, we give expl…
In this paper, we announce the following results: Let M be a Kaehler-Einstein manifold with positive scalar curvature. If the initial metric has nonnegative bisectional curvature and positive at least at one point, then the Kähler-Ricci flow converges exponentially fast to a Kaehler-Einstein metric with constant bisectional curvature.
We describe a family of locally conformal Kaehler metrics on class 1 Hopf surfaces H containing some recent metrics constructed by P. Gauduchon and L. ornea. We study some canonical foliations associated to these metrics, in particular a 2-dimensional foliation E that is shown to be independent of the metric. We elementary prove that E has compact leaves if and only if H is elliptic. In this case the leaves of E give explicitly the elliptic fibration of H, and the natural orbifold structure on the leaf space is illustrated.
In this paper, we construct a set of new functionals of Ricci curvature on any Kaehler manifolds which are invariant under holomorphic transfermations in Kaehler Einstein manifolds and essentially decreasing under the Kaehler Ricci flow. Moreover, if the initial metric has non-negative bisectional curvature, using Tian's inequality, we can prove that each of the functionals has uniform lower bound along the flow which gives a set of integral estimates on curvature. Using this set of integral estimates, we are able to show the following theorem: Let M be a Kaehler-Einstein surface with positive…
In this paper, we study a coupled system of equations on oriented compact 4-manifolds which we call the Bach-Merkulov equations. These equations can be thought of as the conformally invariant version of the classical Einstein-Maxwell equations in general relativity. Inspired by the work of C. LeBrun on Einstein-Maxwell equations on compact Kaehler surfaces, we give a variational characterization of solutions to Bach-Merkulov equations as critical points of the Weyl functional. We also show that extremal Kaehler metrics are solutions to these equations, although, contrary to the Einstein-Maxwel…
The Kaehler manifolds of quasi-constant holomorphic sectional curvatures are introduced as Kaehler manifolds with complex distribution of codimension two, whose holomorphic sectional curvature only depends on the corresponding point and the geometric angle, associated with the section. A curvature identity characterizing such manifolds is found. The biconformal group of transformations whose elements transform Kaehler metrics into Kaehler ones is introduced and biconformal tensor invariants are obtained. This makes it possible to classify the manifolds under consideration locally. The class of…
In this paper we give sufficient conditions on a compact orbifold with an extremal Kaehler metric to admit a resolution with an extremal Kaehler metric. We also complete the Kaehler constant scalar curvature case.
A torsion-free G_2 structure admitting an infinitesimal isometry is shown to give rise to a 4-manifold equipped with a complex symplectic structure and a 1-parameter family of functions and 2-forms linked by second order equations. Reversing the process in various special cases leads to the construction of explicit metrics with holonomy equal to G_2.
In this paper, improving a preceding work, we obtain asymptotic polybalanced kernels associated to extremal Kaehler metrics on polarized algebraic manifolds. As a corollary, we have a stronger asymptotic relative Chow-polystability for extremal Kaehler polarized algebraic manifolds. Finally, related to the Yau-Tian-Donaldson Conjecture for extremal Kaehler metrics, we shall discuss the difference between strong relative K-stability and relative K-stability.
In this note we give a characterization of Kaehler metrics which are both Calabi extremal and Kaehler-Ricci solitons in terms of complex Hessians and the Riemann curvature tensor. We apply it to prove that, under the assumption of positivity of the holomorphic sectional curvature, these metrics are Einstein.
Page's Einstein metric on CP_2 # (-CP_2) is conformally related to an extremal Kaehler metric. Here we construct a family of conformally Kähler solutions of the Einstein-Maxwell equations that deforms the Page metric, while sweeping out the entire Kaehler cone of CP_2 # (-CP_2).The same method also yields analogous solutions on every Hirzebruch surface. This allows us to display infinitely many geometrically distinct families of solutions of the Einstein-Maxwell equations on the smooth 4-manifolds S^2 x S^2 and CP_2 # (-CP_2).
In this paper we investigate the balanced condition (in the sense of Donaldson) and the existence of an Englis expansion for the LeBrun's metrics on $C^2$. Our first result shows that a LeBrun's metric on $C^2$ is never balanced unless it is the flat metric. The second one shows that an Englis expansion of the Rawnsley's function associated to a LeBrun's metric always exists, while the coefficient $a_3$ of the expansion vanishes if and only if the LeBrun's metric is indeed the flat one.
In this paper we address the problem of studying those complex manifolds $M$ equipped with extremal metrics $g$ induced by finite or infinite dimensional complex space forms. We prove that when $g$ is assumed to be radial and the ambient space is finite dimensional then $(M, g)$ is itself a complex space form. We extend this result to the infinite dimensional setting by imposing the strongest assumption that the metric $g$ has constant scalar curvature and is well-behaved (see Definition 1 in the Introduction). Finally, we analyze the radial Kaehler-Einstein metrics induced by infinite dimensi…
In this paper, we first define the complexification of a real analytic map between real analytic Koszul manifolds and show that the complexified map is the holomorphic extension of the original map. Next we define an anti-Kaehler metric compatible with the adapted complex structure on the complexification of a real analytic pseudo-Riemannian manifold. In particular, for a pseudo-Riemannian homogeneous space, we define another complexification and a (complete) anti-Kaehler metric on the complexification. One of main purposes of this paper is to find the interesting relation between these two co…
In this paper, we analyze the asymptotic behaviour of the Hermitian-Yang-Mills flow over a compact non-Kähler manifold $(X,g)$ with the Hermitian metric $g$ satisfying the Gauduchon and Astheno-Kähler condition.
We construct a compact 6-dimensional solvmanifold endowed with a non-trivial invariant generalized Kähler structure and which does not admit any Kähler metric. This is in contrast with the case of nilmanifolds which cannot admit any invariant generalized Kähler structure unless they are tori.
F. Podestà and A. Spiro introduced a class of $G$-manifolds $M$ with a cohomogeneity one action of a compact semisimple Lie group $G$ which admit an invariant Kaehler structure $(g,J)$ (``standard $G$-manifolds") and studied invariant Kaehler and Kaehler-Einstein metrics on $M$. In the first part of this paper, we gave a combinatoric description of the standard non compact $G$-manifolds as the total space $M_φ$ of the homogeneous vector bundle $M = G\times_H V \to S_0 =G/H$ over a flag manifold $S_0$ and we gave necessary and sufficient conditions for the existence of an invariant Kaehler-Eins…
We propose two conjectures about Ricci-flat metrics: Conjecture 1: A Ricci-flat projectively induced metric is flat. Conjecture 2: A Ricci-flat metric on an $n$-dimensional complex manifold such that the $a_{n+1}$ coefficient of the TYZ expansion vanishes is flat. We verify Conjecture 1 (see Theorem 1.1) under the assumptions that the metric is radial and stable-projectively induced and Conjecture 2 (see Theorem 1.2) for complex surfaces whose metric is either radial or complete and ALE. We end the paper by showing, by means of the Simanca metric, that the assumption of Ricci-flatness in Conje…
If a smooth compact 4-manifold M admits a Kaehler-Einstein metric g of positive scalar curvature, Gursky showed that its conformal class [g] is an absolute minimizer of the Weyl functional among all conformal classes with positive Yamabe constant. Here we prove that, with the same hypotheses, [g] also minimizes of the Weyl functional on a different open set of conformal classes, most of which have negative Yamabe constant. An analogous minimization result is then proved for Einstein metrics g which are Hermitian, but not Kaehler.
Let $(M, g)$ be a Kaehler manifold whose associated Kaehler form $ω$ is integral and let $(L, h)\rightarrow (M, ω)$ be a quantization hermitian line bundle. In this paper we study those Kaehler manifolds $(M, g)$ admitting a finite TYCZ expansion. We show that if the TYCZ expansion is finite then $T_{mg}$ is indeed a polynomial in $m$ of degree $n$, $n=dim M$, and the log-term of the Szegö kernel of the disc bundle $D\subset L^*$ vanishes (where $L^*$ is the dual bundle of $L$). Moreover, we provide a complete classification of the Kaehler manifolds admitting finite TYCZ expansion either when …
We discuss two kinds of almost contact metric structures on a one-parameter family of totally umbilical hyperspheres in the nearly Kaehler unit 6-sphere.
In this note we clarify the structure of the moduli space of constant scalar curvature Kaehler metrics as one approaches the boundary of the Kaehler cone on cscK manifolds blown up at finite set of points, in the spirit of the previous work arXiv:math/0504115. Results about which Kaehler classes can be reached and about the position of the blown up points are given.
Four dimensional simply connected Lie groups admitting a pseudo Kähler metric are determined. The corresponding Lie algebras are modelized and the compatible pairs $(J,ω)$ are parametrized up to complex isomorphism (where $J$ is a complex structure and $ω$ is a symplectic structure). Such structure gives rise to a pseudo Riemannian metric $g$ for which $J$ is parallel. It is proved that most of these complex homogeneous spaces admit a pseudo Kähler Einstein metric. Ricci flat and flat metrics are determined. In particular Ricci flat unimodular Kähler Lie algebras are flat in dimension four. Ot…
Indefinite Kaehler solutions of the Einstein equations are studied, and it is almost completely determined which compact complex surfaces admit such metrics.
