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We introduce a family of spaces, parametrized by positive real numbers, that includes all of the Hirzebruch surfaces. Each space is viewed from two distinct perspectives. First, as a leaf space of a compact, complex, foliated manifold, following [BZ1]. Second, as a symplectic cut of the manifold $\mathbb{C}\times S^2$ in a possibly nonrational direction, following [BP2]., Comment: 15 pages, 10 figures, final version, to appear in Boll. Unione Mat. Ital

We present a summary of some results from our article (Battaglia and Zaffran, Int. Math. Res. Not. IMRN 2015 no. 22 (2015), 11785–11815) and other recent results on the so-called LVMB manifolds. We emphasize some features by taking a different point of view. We present a simple variant of the Delzant construction, in which the group that is used to perform the symplectic reduction can be chosen of arbitrarily high dimension, and is always connected.

We consider a family E_m(D,M) of holomorphic bundles constructed as follows: to any given M in GL_n(Z), we associate a "multiplicative automorphism" f of (C*)^n. Now let D be a f-invariant Stein Reinhardt domain in (C*)^n. Then E_m(D,M) is defined as the flat bundle over the annulus of modulus m>0, with fiber D, and monodromy f. We show that the function theory on E_m(D,M) depends nontrivially on the parameters m, M and D. Our main result is that E_m(D,M) is Stein if and only if m log(r(M))

Bosio generalized the construction by Lopez de Medrano-Verjovsky- Meersseman (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 Bialynicki-Birula and Świȩcicka. This point of view yields new results on the geometry of LVM and Bosio’s manifolds.

On montre que parmi les surfaces compactes complexes de classe VII 0 avec b 2 > 0, les surfaces d'Inoue-Hirzebruch sont caracterisees par le fait qu'elles possedent deux champs de vecteurs tordus. Ce resultat est un pas vers la comprehension des feuilletages sur les surfaces VII 0 .