Volume 9, Issue 1

We give a complete classification of left invariant para-Kähler structures on four-dimensional simply connected Lie groups up to an automorphism. As an application we discuss some curvatures properties of the canonical connection associated to these structures as flat, Ricci flat and existence of Ricci solitons.

Let ( M , ∇, 〈 , 〉 ) be a manifold endowed with a flat torsionless connection r and a Riemannian metric 〈 , 〉 and ( T k M ) k ≥1 the sequence of tangent bundles given by T k M = T ( T k −1 M ) and T 1 M = TM . We show that, for any k ≥ 1, T k M carries a Hermitian structure ( J k , g k ) 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 T k M such that ( J k , g k , ∇ 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 , . . ., x n ), 〈 ∂xi,∂xj 〉=∂2φ∂xi∂xj \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 ( T k M , J k , g k ) 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, ( T k M , J k , g k ) 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.

In this continuation of [4], we investigate the deformations of holomorphic Cartan geometries where the underlying complex manifold is allowed to move. The space of infinitesimal deformations of a flat holomorphic Cartan geometry is computed. We show that the natural forgetful map, from the infinitesimal deformations of a flat holomorphic Cartan geometry to the infinitesimal deformations of the underlying flat principal bundle on the topological manifold, is an isomorphism.

We characterize the k -th Gauduchon condition and by applying its characterization, we reprove that a compact k -th Gauduchon, semi-Kähler manifold becomes quasi-Kähler, which tells us that in particular, a compact almost pluriclosed, semi-Kähler manifold is quasi-Kähler.

Using a quasi-linear version of Hodge theory, holomorphic vector bundles in a neighbourhood of a given polystable bundle on a compact Kähler manifold are shown to be (poly)stable if and only if their corresponding classes are (poly)stable in the sense of geometric invariant theory with respect to the linear action of the automorphism group of the bundle on its space of in˝nitesimal deformations.

This paper is an introduction to cosymplectic topology. Through it, we study the structures of the group of cosymplectic diffeomorphisms and the group of almost cosymplectic diffeomorphisms of a cosymplectic manifold ( M , ω , η ) : ( i )− we define and present the features of the space of almost cosymplectic vector fields (resp. cosymplectic vector fields); ( ii )− we prove by a direct method that the identity component in the group of all cosymplectic diffeomorphisms is C 0 −closed in the group Diff ∞ ( M ) (a rigidity result), while in the almost cosymplectic case, we prove that the Reeb vector field determines the almost cosymplectic nature of the C 0 −limit ϕ of a sequence of almost cosymplectic diffeomorphisms (a rigidity result). A sufficient condition based on Reeb’s vector field which guarantees that ϕ is a cosymplectic diffeomorphism is given (a ˛exibility condition), the cosymplectic analogues of the usual symplectic capacity-inequality theorem are derived and the cosymplectic analogue of a result that was proved by Hofer-Zehnder follows.

We give an exposition of Sullivan’s theorem on realizing rational homotopy types by closed smooth manifolds, including a discussion of the necessary rational homotopy and surgery theory, adapted to the realization problem for almost complex manifolds: namely, we give a characterization of the possible simply connected rational homotopy types, along with a choice of rational Chern classes and fundamental class, realized by simply connected closed almost complex manifolds in real dimensions six and greater. As a consequence, beyond demonstrating that rational homotopy types of closed almost complex manifolds are plenty, we observe that the realizability of a simply connected rational homotopy type by a simply connected closed almost complex manifold depends only on its cohomology ring. We conclude with some computations and examples.

This paper shows that given 0 < p < 3 and a complex Borel measure µ on the unit disk 𝔻 the inhomogeneous Cauchy-Riemann ̄∂-equation ∂z¯u(z)=dμ(z)(2πi)-1dz¯∧dz {\partial _{\bar z}}u\left( z \right) = {{d\mu \left( z \right)} \over {{{\left( {2\pi i} \right)}^{ - 1}}d\bar z \wedge dz}} − a complex Gauss curvature of the weighted disk (𝔻, µ ) ᗄ z ∈ 𝔻, has a distributional solution (initially defined on ̄𝔻 = 𝔻 ∪ 𝕋) u ∈ ℒ 2, p (𝕋) (formed of: (i) Morrey’s space M 2,0< p <1 (𝕋); (ii) John-Nirenberg’s space BMO (𝕋) = 𝒧 2,1 (𝕋); (iii) Hölder-Lipschitz’s space C C0<p-12<1 {C^{0 < {{p - 1} \over 2} < 1}} (𝕋)), if and only if 𝔻¯∋z↦∫𝔻(1-zw¯)-1dμ¯(w) \mathbb{D} z \mapsto \int\limits_\mathbb{D} {{{\left( {1 - z\bar w} \right)}^{ - 1}}d\bar \mu } \left( w \right) belongs to the analytic Campanato space ϱ𝒜 p (𝔻), thereby not only extending Carleson’s corona & Wolff’s ideal theorems to the algebra M ϱ𝒜 p (𝔻) of all analytic pointwise multiplications of ϱ𝒜 p (𝔻), but quadratically generalizing Brownawell’s result on Hilbert’s Nullstellensatz for the analytic polynomial class 𝒫(ℂ).

We describe examples showing the sharpness of Fujita’s conjecture on adjoint bundles also in the general type case, and use these examples to formulate related bold conjectures on pluricanonical maps of varieties of general type.

The purpose in this paper is to determine a locally conformal Kähler solvmanifold such that the nilradical of the solvable Lie group is constructed by a Heisenberg Lie group.

In this work we show that the Weil-Petersson volume (which coincides with the CM degree) in the case of weighted points in the projective line is continuous when approaching the Calabi-Yau geometry from the Fano geometry. More specifically, the CM volume computed via localization converges to the geometric volume, computed by McMullen with different techniques, when the sum of the weights approaches the Calabi-Yau geometry.

We investigate the Fu-Yau equation on compact almost astheno-Kähler manifolds and show an a priori C 0 -estiamte for a smooth solution of the equation.

We compute almost-complex invariants h∂¯p,o h_{\bar \partial }^{p,o} , hDolp,o h_{Dol}^{p,o} and almost-Hermitian invariants hδ¯p,o h_{\bar \delta }^{p,o} on families of almost-Kähler and almost-Hermitian 6-dimensional solvmanifolds. Finally, as a consequence of almost-Kähler identities we provide an obstruction to the existence of a compatible symplectic structure on a given compact almost-complex manifold. Notice that, when ( X , J , g , ω ) is a compact almost Hermitian manifold of real dimension greater than four, not much is known concerning the numbers h∂¯p,q h_{\bar \partial }^{p,q} .

Hyperkähler quotients by non-free actions are typically singular, but are nevertheless partitioned into smooth hyperkähler manifolds. We show that these partitions are topological stratifications, in a strong sense. We also endow the quotients with global Poisson structures which recover the hyperkähler structures on the strata. Finally, we give a local model which shows that these quotients are locally isomorphic to linear complex-symplectic reductions in the GIT sense. These results can be thought of as the hyperkähler analogues of Sjamaar–Lerman’s theorems for singular symplectic reduction. They are based on a local normal form for the underlying complex-Hamiltonian manifold, which may be of independent interest.

We study symmetric minimal surfaces in the three-dimensional Heisenberg group Nil 3 using the generalized Weierstrass type representation, the so-called loop group method. In particular, we will present a general scheme for how to construct minimal surfaces in Nil 3 with non-trivial geometry. Special emphasis will be put on equivariant minimal surfaces. Moreover, we will classify equivariant minimal surfaces given by one-parameter subgroups of the isometry group Iso ◦ (Nil 3 ) of Nil 3 .

We propose a new method to construct degenerate 3-( α , δ )-Sasakian manifolds as fiber products of Boothby-Wang bundles over hyperkähler manifolds. Subsequently, we study homogeneous degenerate 3-( α , δ )-Sasakian manifolds and prove that no non-trivial compact examples exist aswell as that there is exactly one family of nilpotent Lie groups with this geometry, the quaternionic Heisenberg groups.

We investigate the pseudo-hyperkähler geometry of higher degree rational curves in the twistor space of a hyperkähler 4-manifold.

Recently, Dinew and Popovici introduced and studied an energy functional F acting on the metrics in the Aeppli cohomology class of a Hermitian-symplectic metric and showed that in dimension 3 its critical points (if any) are Kähler. In this article we further investigate the critical points of this functional in higher dimensions and under holomorphic deformations. We first prove that being a critical point for F is a closed property under holomorphic deformations. We then show that the existence of a Kähler metric ω in the Aeppli cohomology class is an open property under holomorphic deformations. Furthermore, we consider the case when the (2, 0)-torsion form ρ ω 2, 0 of ω is ∂ -exact and prove that this property is closed under holomorphic deformations. Finally, we give an explicit formula for the differential of F when the (2, 0)-torsion form ρ ω 2, 0 is ∂ -exact.