Complex geometry of moment-angle manifolds

We describe the basic cohomology ring of the canonical holomorphic foliation on a moment-angle manifold, LVMB-manifold, or any complex manifold with a maximal holomorphic torus action. Namely, we show that the basic cohomology has a description similar to the cohomology ring of a complete simplicial toric variety due to Danilov and Jurkiewicz. This settles a question of Battaglia and Zaffran, who previously computed the basic Betti numbers for the canonical holomorphic foliation in the case of a shellable fan. Our proof uses an Eilenberg–Moore spectral sequence argument; the key ingredient is the formality of the Cartan model for the torus action on a moment-angle manifold. We develop the concept of transverse equivalence as an important tool for studying smooth and holomorphic foliated manifolds. For an arbitrary complex manifold with a maximal torus action, we show that it is transverse equivalent to a moment-angle manifold and therefore has the same basic cohomology.

In this paper, we introduce the notion of maximal actions of compact tori on smooth manifolds and study compact connected complex manifolds equipped with maximal actions of compact tori. We give a complete classification of such manifolds, in terms of combinatorial objects, which are triples ( Δ , 𝔥 , G ) {(\Delta,\mathfrak{h},G)} of nonsingular complete fan Δ in 𝔤 {\mathfrak{g}} , complex vector subspace 𝔥 {\mathfrak{h}} of 𝔤 ℂ {\mathfrak{g}^{\mathbb{C}}} and compact torus G satisfying certain conditions. We also give an equivalence of categories with suitable definitions of morphisms in these families, like toric geometry. We obtain several results as applications of our equivalence of categories; complex structures on moment-angle manifolds, classification of holomorphic nondegenerate ℂ n {\mathbb{C}^{n}} -actions on compact connected complex manifolds of complex dimension n, and construction of concrete examples of non-Kähler manifolds.

We show that the moment-angle manifolds corresponding to complete simplicial fans admit non-Kaehler complex-analytic structures. This generalises the known construction of complex-analytic structures on polytopal moment-angle manifolds, coming from identifying them as LVM-manifolds. We proceed by describing Dolbeault cohomology and some Hodge numbers of moment-angle manifolds by applying the Borel spectral sequence to holomorphic principal bundles over toric varieties.

We study the geometry of compact complex manifolds M equipped with a maximal action of a torus T = (S 1) k . We present two equivalent constructions that allow one to build any such manifold on the basis of special combinatorial data given by a simplicial fan Σ and a complex subgroup H ⊂ T ℂ = (ℂ*) k . On every manifold M we define a canonical holomorphic foliation F and, under additional restrictions on the combinatorial data, construct a transverse Kähler form ω F . As an application of these constructions, we extend some results on the geometry of moment-angle manifolds to the case of manifolds M.

For compact complex manifolds with vanishing first Chern class that are compact torus principal bundles over Kähler manifolds, we prove that all holomorphic geometric structures on them, of affine type, are locally homogeneous. For a compact simply connected complex manifold in Fujiki class [Formula: see text], whose dimension is strictly larger than the algebraic dimension, we prove that it does not admit any holomorphic rigid geometric structure, and also it does not admit any holomorphic Cartan geometry of algebraic type. We prove that compact complex simply connected manifolds in Fujiki class [Formula: see text] and with vanishing first Chern class do not admit any holomorphic Cartan geometry of algebraic type.

It is well known that the Laplace operator plays an important role in the theory of harmonic integrals and the Bochner technique both in Riemannian and Kahler manifolds. In recent years, under the initiation of S.S. Chern, the global differential geometry of real and complex Finsler manifolds has gained a great development ([1], [2], [3], [4]). A lot of results about the Laplacian and its applications have been obtained in a real Finsler manifold ([5], [6]). But up to now there are no results for the Laplacian and its applications in a complex Finsler manifold. The key point in the theory of the Bochner technique and harmonic integrals is to define a suitable Laplace operator. In the case of Finsler manifolds the difficulty is that the Finsler metric depends on the fibre coordinates. Using the idea that the Laplacian on Euclidean space or on a Riemannian manifold measures the average value of a function around a point, P. Centore ([7]) generalizes the Laplacian on a Riemannian manifold to a real Finsler manifold. Considering a complex manifold as a real manifold, there is a one-to-one correspondence between the real coordinates and the complex coordinates ([8]). In this paper, we use the mean value idea to define the Laplacian on a strongly Kahler-Finsler manifold, first for functions and then for forms, and we derive some remarkable properties for the Laplacian for functions and extend the Laplacian to arbitrary forms. Indeed our Laplacian on strongly Kahler-Finsler manifolds generalizes the Kahlerian Laplacian. And it is worth to remark that using the osculating Kahler metric — which we obtain in the following to define the pointwise and global inner product when we define the Hodge-Laplace operator of (p, q)-forms — is more natural than using the fundamental tensor of the Finsler metric and can avoid many complicated calculations.

These are notes for the CIME school on Complex non-Kahler geometry from July 9th to July 13th of 2018 in Cetraro, Italy. It is an overview of different properties of a class of non-Kahler compact complex manifolds called LVMB manifolds, obtained as the Hausdorff space of leaves of systems of commuting complex linear equations in an open set in complex projective space \({\mathbb P}_{\mathbb {C}}^{n-1}\).

We study the topology of Hamiltonian-minimal Lagrangian submanifolds N in C^m constructed from intersections of real quadrics in a work of the first author. This construction is linked via an embedding criterion to the well-known Delzant construction of Hamiltonian toric manifolds. We establish the following topological properties of N: every N embeds as a submanifold in the corresponding moment-angle manifold Z, and every N is the total space of two different fibrations, one over the torus T^{m-n} with fibre a real moment-angle manifold R, and another over a quotient of R by a finite group with fibre a torus. These properties are used to produce new examples of Hamiltonian-minimal Lagrangian submanifolds with quite complicated topology.

We study the geometry of submanifolds of complex Hopf manifolds endowed with the (locally conformal Kaehler) Boothby metric. 1. Generalized Hopf manifolds and the Boothby metric. Let fleC, 0 l , generated by z^az, z(=W. Then Ga acts freely and properly discontinuously on W, see [28], vol. II, p. 137, so that the quotient space Hna=W/Ga becomes in a natural way a complex w-dimensional manifold. This is the well known complex Hopf manifold. In their attempt to construct complex structures on products SxL, where S is the unit circle and L an odd dimensional homotopy sphere, E. Brieskorn & A. Van de Ven, [3], have generalized Hopf manifolds a follows. Let n > l and (fe0, ••• , bn)^Z , bj^l, 0£j£n. Let (z0, ••• , zn) be the natural complex coordinates on C n + 1 . Define X(b) = X2n(b0, ••• , fc»)cC n+1 by the equation: Then X(b) ia an aίϊine algebraic variety with one singular point at the origin of C if bj^2, / = 0 , - n (and without singularities if b3— for at least one /). Next B{b)—X(b)— {0} is a complex n-dimensional manifold, referred hereafter as the Brieskorn manifold determined by the integers b0, ••• , bn. See [2]. There is a natural holomorphic action of C on B(b) given by: t(Zo, —, ^ n ) = ( ^ O e X p ( y ^ ) , ••• , Zn ΘXp (—-y^ ) ) (1) where f e C , wa=— log \a \— iΦa, Φα=arctan(/m(α)//?β(fl)), —π/2 D x 5»(« defined by f(a, jc)=(α, ί/αx), for any a e D 1 , xe=B(b), where £/αe=GL(n+l, C) is the matrix: Note that / is an automorphism of DxBφ). The action of GL(n+l, C) on C n + 1 induces an action of Z « { / 7 m e Z } on DxBφ). Let: be the quotient space. We establish the following: THEOREM 1. X is a complex n-dimensional manifold. Moreover, if n—2, then there exists a surjective holomorphic map π: X-^D which makes X into a complex analytic family of compact complex surfaces, for any a^D there is a diffeomorphism between π~\a) and Hl(b). We recall that a triple (X, π, M) is a complex analytic family of compact complex manifolds if X, M are complex manifolds and π: X-*M is a proper holomorphic map which is of maximal rank at all points of X. Then each fibre π~\a G G M , is a compact complex manifold. Note that the action (1) of Z on Bφ) generalizes slightly the one in [3], p. 390. There B(l, ••• , 1)/Z is diffeomorphic to W/G1/e. The proof of Theorem 1 is organized in several steps, as follows. STEP 1. Z acts freely on DxB(b). Let (a, x) be a fixed point of / m , m e Z . Thus Ufx = x, and consequently: for O^tj ^n. Since at least one z3 is non-zero, it follows that m=0. STEP 2. {//meZ} is a properly discontinuous group of analytic transformations of DxBψ). Let KdD, LcB(b) be compact subsets. It is enough to show that the set of all m e Z with the property:

We prove that any pluriharmonic map from a compact complex manifold with positive first Chern class (defined outside a certain singularity set of codimension at least two) into a complex Grassmann manifold of rank two is explicitly constructed from a rational map into a complex projective space. Under some restrictions on dimension and rank of the domain manifold and the target manifold, respectively, we also prove that similar results hold for other complex Grassmann manifolds as targets. Introduction. Let φ: M -> N be a smooth map from a complex manifold into a Riemannian manifold. Then, φ is said to be pluriharmonic if the (0, l)-exterior covariant derivative Ddφ of the (1, O)-differential dφ of φ vanishes identically. Let V be the pull-back connection on the pull-back bundle φ~^TN. We have (0.1) (Pdφ){X, Y) = V$dφ(Y)-dφ(dχY), X, where Γ M 1 0 is the holomorphic tangent bundle of M. If φ~TN has the Koszul-Malgrange holomorphic structure, that is, the (0, l)-part of V coincides with the δ-operator, we may say that φ is pluriharmonic if and only if φ sends any holomorphic section of TM to a holomorphic section of φ~TN. It is easily seen that if φ is holomorphic and TV is a Kahler manifold then φ~TN' has the Koszul-Malgrange holomorphic structure, hence any holomorphic map is pluriharmonic. Note that an anti-holomorphic map is also pluriharmonic if N is a Kahler manifold. Conversely, the existence of the Koszul-Malgrange holomorphic structure OIK/) TN is ensured if φ is pluriharmonic and Nhas nonnegative or nonpositive curvature operator. In this case, if N is a Kahler manifold, then φ~TN' has the Koszul-Malgrange holomorphic structure (cf. [O-U2]). From the point of view of Riemannian geometry, the most interesting property of pluriharmonic maps is that it Partially supported by the Grants-in-Aid for Scientific Research, The Ministry of Education, Science and Culture, Japan. 1991 Mathematics Subject Classification. Primary 58E20; Secondary 53C42.

The object of study in the present dissertation are some topics in differential geometry of smooth manifolds with additional tensor structures and metrics of Norden type. There are considered four cases depending on the dimension of the manifold: 2n, 2n + 1, 4n and 4n + 3. The studied tensor structures, which are counterparts in the different related dimensions, are the almost complex/contact/hypercomplex structure and the almost contact 3-structure. The considered metric on the 2n-dimensional case is the Norden metric, and the metrics in the other three cases are generated by it. The purpose of the dissertation is to carry out the following: 1. Further investigations of almost complex manifolds with Norden metric including studying of natural connections with conditions for their torsion and invariant tensors under the twin interchange of Norden metrics. 2. Further investigations of almost contact manifolds with B-metric including studying of natural connections with conditions for their torsion and associated Schouten-van Kampen connections as well as a classification of affine connections. 3. Introducing and studying of Sasaki-like almost contact complex Riemannian manifolds. 4. Further investigations of almost hypercomplex manifolds with Hermitian-Norden metrics including studying of integrable structures of the considered type on 4-dimensional Lie algebra and tangent bundles with the complete lift of the base metric; introducing of associated Nijenhuis tensors in relation with natural connections having totally skew-symmetric torsion as well as quaternionic Kahler manifolds with Hermitian-Norden metrics. 5. Introducing and studying of manifolds with almost contact 3-structures and metrics of Hermitian-Norden type and, in particular, associated Nijenhuis tensors and their relationship with natural connections having totally skew-symmetric torsion.

Chapter 6 Complex differential geometry

A complex manifold X is called homogeneous if there exists a connected complex or real Lie group G acting transitively on X as a group of biholomorphic transformations. The goal is a general classification of homogeneous complex manifolds. Since the class of homogeneous complex manifolds is much too big for any serious attempt of complete classification, it is necessary to impose further conditions. For example E. Cartan classified in [Ca] symmetric homogeneous domains in ℂn. Here we will require that X is of small dimension. For dim ℂ(X) = 1 the classification follows from the uniformization Theorem. In 1962 J. Tits classified the compact homogeneous complex manifolds in dimension two and three [Ti1]. In 1979 J. Snow classified all homogeneous manifolds X = G/H with dim ℂ(X) ≤ 3, G being a solvable complex Lie group and H discrete [SJ1]. The classification of all complex-homogeneous (i.e. G is a complex Lie group) twodimensional manifolds was completed in 1981 by A. Huckleberry and E. Livorni [HL]. Next, in 1984 K. Oeljeklaus and W. Richthofer classified all those homogeneous two-dimensional complex manifolds X = G/H where G is only a real Lie group [OR]. The classification of three-dimensional complex-homogeneous manifolds was completed in 1985 [W1]. Finally in 1987 the general classification of the three-dimensional homogeneous complex manifolds was given by our Dissertation [W2]. The purpose of this note is to describe these manifolds and briefly outline the methods involved in the classification.

A complex manifold X is called homogeneous if there exists a connected complex or real Lie group G acting transitively on X as a group of biholomorphic transformations. The goal is a general classification of homogeneous complex manifolds. Since the class of homogeneous complex manifolds is much too big for any serious attempt of complete classification, it is necessary to impose further conditions. For example É. Cartan classified in [Ca] symmetric homogeneous domains in ℂn. Here we will require that X is of small dimension. For dim ℂ(X) = 1 the classification follows from the uniformization Theorem. In 1962 J. Tits classified the compact homogeneous complex manifolds in dimension two and three [Ti1]. In 1979 J. Snow classified all homogeneous manifolds X = G/H with dim ℂ(X) ≤ 3, G being a solvable complex Lie group and H discrete [SJ1]. The classification of all complex-homogeneous (i.e. G is a complex Lie group) twodimensional manifolds was completed in 1981 by A. Huckleberry and E. Livorni [HL]. Next, in 1984 K. Oeljeklaus and W. Richthofer classified all those homogeneous two-dimensional complex manifolds X = G/H where G is only a real Lie group [OR]. The classification of three-dimensional complex-homogeneous manifolds was completed in 1985 [W1]. Finally in 1987 the general classification of the three-dimensional homogeneous complex manifolds was given by our Dissertation [W2]. The purpose of this note is to describe these manifolds and briefly outline the methods involved in the classification.

Toric varieties form an important class of examples in algebraic geometry, as they admit a complete description in terms of combinatorial data, so-called lattice fans. In Section 2.1, we briefly recall this description and also some of the basic facts in toric geometry. Then we present Cox's construction of the characteristic space of a toric variety in terms of a defining fan and discuss the basic geometry around this. Section 2.2 is pure combinatorics. We introduce the notion of a “bunch of cones” and show that, in an appropriate setting, this is the Gale dual version of a fan. Under this duality, the normal fans of polytopes correspond to bunches of cones arising canonically from the chambers of the so-called Gelfand–Kapranov–Zelevinsky decomposition. In Section 2.3, we discuss the geometric meaning of bunches of cones: they encode the maximal separated good quotients for subgroups of the acting torus on an affine toric variety. In Section 2.4, we specialize these considerations to toric characteristic spaces, that is, to the good quotients arising from Cox's construction. This leads to an alternative combinatorial description of toric varieties in terms of “lattice bunches,” which turns out to be particularly suitable for phenomena around divisors. Toric varieties Toric varieties and fans We introduce toric varieties and their morphisms and recall that this category admits a complete description in terms of lattice fans. Definition 2.1.1.1 A toric variety is an irreducible, normal variety X together with an algebraic torus action T × X → X and a base point x 0 ∈ X such that the orbit map T → X , t → t · x 0 is an open embedding.