### A new example of a compactification of C3.

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We prove that Kummer threefolds $T/G$ with algebraic dimension $0$ have Kodaira dimension 0.

We construct new families of non-Kähler compact complex threefolds belonging to Kato's Class L. The construction uses certain polynomial automorphisms of C3. We also study basic properties of our manifolds.

We study compact complex manifolds covered by a domain in $n$-dimensional projective space whose complement $E$ is non-empty with $(2n-2)$-dimensional Hausdorff measure zero. Such manifolds only exist for $n\ge 3$. They do not belong to the class $\mathcal{C}$, so they are neither Kähler nor Moishezon, their Kodaira dimension is $-\infty $, their fundamental groups are generalized Kleinian groups, and they are rationally chain connected. We also consider the two main classes of known 3-dimensional examples: Blanchard manifolds, for which...

We study the extension problem of holomorphic maps $\sigma :H\to X$ of a Hartogs domain $H$ with values in a complex manifold $X$. For compact Kähler manifolds as well as various non-Kähler manifolds, the maximal domain ${\Omega}_{\sigma}$ of extension for $\sigma $ over $\Delta $ is contained in a subdomain of $\Delta $. For such manifolds, we define, in this paper, an invariant Hex${}_{n}\left(X\right)$ using the Hausdorff dimensions of the singular sets of $\sigma $’s and study its properties to deduce informations on the complex structure of $X$.

Based on the results of the first two parts to this paper, we prove that the canonical bundle of a minimal Kähler threefold (i.e.${K}_{X}$ is nef) is good,i.e.its Kodaira dimension equals the numerical Kodaira dimension, (in particular some multiple of ${K}_{X}$ is generated by global sections); unless $X$ is simple. “Simple“ means that there is no compact subvariety through the very general point of $X$ and $X$ not Kummer. Moreover we show that a compact Kähler threefold with only terminal singularities whose canonical...

We study relatively semi-stable vector bundles and their moduli on non-Kähler principal elliptic bundles over compact complex manifolds of arbitrary dimension. The main technical tools used are the twisted Fourier-Mukai transform and a spectral cover construction. For the important example of such principal bundles, the numerical invariants of a 3-dimensional non-Kähler elliptic principal bundle over a primary Kodaira surface are computed.