Non-Kähler manifolds and GIT-quotients

Abstract

Bosio generalized the construction by López de Medrano-VerjovskyMeersseman (LVM) of a family of non-algebraic compact complex manifolds of any dimension. We describe how to construct the generalized family from certain Geometric Invariant Theory (GIT) quotients. We show that Bosio's generalization parallels exactly the extension from Mumford's GIT to the more general GIT developed by Białynicki-Birula andŚwiȩcicka. This point of view yields new results on the geometry of LVM and Bosio's manifolds.López de Medrano and Verjovsky discovered in 1997 a way to construct many compact complex manifolds (cf. [14]). They start with a C-action on CP n induced by a diagonal linear vector field (satisfying certain properties), and find an invariant open dense subset U ⊂ CP n where the action is free, proper and cocompact, so the quotient N = U/C is a compact complex manifold. Their construction was extended to C m -actions by Meersseman in [15], yielding a vast family of non-Käh-ler compact manifolds, called LVM-manifolds. These manifolds lend themselves very well to various computations, and a thorough study of their properties is conducted in [15]. Furthermore, they are (deformations of) a very natural generalization of Calabi-Eckmann manifolds. Finally, the topology of LVM-manifolds can be extraordinarily complicated: we refer to [5] for the most recent results about a study started off in [20] and [14].

Group actions, non-Kähler complex manifolds and SKT structures

“…Any LVMB manifold satisfying condition (K) (cf. [16]), and all the examples in Section 3 are of the form X/L as above. Further, in all these examples the corresponding group N acts with at most finite stabilizers so that the quotient X/N has the structure of a complex analytic orbifold.…”

“…By the description in [16], an LVMB manifold satisfying a certain rationality condition (K) admits a Seifert principal fibration over an orbifold toric variety. If the base of the fibration is a nonsingular toric variety then the corresponding LVMB manifolds may be recovered as a special case of our construction.…”

“…It is possible to look at all LVMB manifolds that satisfy condition (K) as part of a more general phenomenon. It is shown in [16] that any such LVMB manifold can be identified to an orbit space X/L, where X is a complex manifold that admits a proper holomorphic action of (C * ) 2m with finite stabilizers, and L is a torsion-free closed co-compact subgroup of (C * ) 2m . In Section 8, we consider the more general situation of any proper holomorphic action of a complex linear algebraic group G on a complex manifold X.…”

Complex geometry of moment-angle manifolds

“…One way to define a transverse Kähler form on Z K is to use a modification of an argument of Loeb and Nicolau [30]; it works only for normal fans: The condition that Σ is a normal fan in Proposition 4.4 is important. Indeed, it was shown in [14] that general LVMB-manifolds do not admit transverse Kähler forms. A similar argument proves that there are no transverse Kähler forms on the quotients U (K)/C for general complete simplicial fans.…”

Intersection of Quadrics in ℂ n $$\mathbb {C}^n$$ , Moment-Angle Manifolds, Complex Manifolds and Convex Polytopes

“…In fact V is the image under the canonical projection C n Ñ P n´1 C of the set of orbits in C n that do not accumulate to the origin (a sort of Kempf-Ness condition). In a very pretty paper [19] Stéphanie Cupit-Foutou and Dan Zaffran describe how to construct the generalized family of LVMB manifolds from certain Geometric Invariant Theory (GIT) quotients.…”