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Compact lcK manifolds with parallel vector fields

Andrei Moroianu (2015)

Complex Manifolds

We show that for n > 2 a compact locally conformally Kähler manifold (M2n , g, J) carrying a nontrivial parallel vector field is either Vaisman, or globally conformally Kähler, determined in an explicit way by a compact Kähler manifold of dimension 2n − 2 and a real function.

Compact quotients of large domains in complex projective space

Finnur Lárusson (1998)

Annales de l'institut Fourier

We study compact complex manifolds covered by a domain in n -dimensional projective space whose complement E is non-empty with ( 2 n - 2 ) -dimensional Hausdorff measure zero. Such manifolds only exist for n 3 . They do not belong to the class 𝒞 , so they are neither Kähler nor Moishezon, their Kodaira dimension is - , 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...

Complex structures on product of circle bundles over complex manifolds

Parameswaran Sankaran, Ajay Singh Thakur (2013)

Annales de l’institut Fourier

Let L ¯ i X i be a holomorphic line bundle over a compact complex manifold for i = 1 , 2 . Let S i denote the associated principal circle-bundle with respect to some hermitian inner product on L ¯ i . We construct complex structures on S = S 1 × S 2 which we refer to as scalar, diagonal, and linear types. While scalar type structures always exist, the more general diagonal but non-scalar type structures are constructed assuming that L ¯ i are equivariant ( * ) n i -bundles satisfying some additional conditions. The linear type complex structures...

Non-Kähler compact complex manifolds associated to number fields

Karl Oeljeklaus, Matei Toma (2005)

Annales de l’institut Fourier

For algebraic number fields K with s > 0 real and 2 t > 0 complex embeddings and “admissible” subgroups U of the multiplicative group of integer units of K we construct and investigate certain ( s + t ) -dimensional compact complex manifolds X ( K , U ) . We show among other things that such manifolds are non-Kähler but admit locally conformally Kähler metrics when t = 1 . In particular we disprove a conjecture of I. Vaisman.

On holomorphic maps into compact non-Kähler manifolds

Masahide Kato, Noboru Okada (2004)

Annales de l’institut Fourier

We study the extension problem of holomorphic maps σ : H 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 Ω σ of extension for σ over Δ is contained in a subdomain of Δ . For such manifolds, we define, in this paper, an invariant Hex n ( X ) using the Hausdorff dimensions of the singular sets of σ ’s and study its properties to deduce informations on the complex structure of X .

The Fujiki class and positive degree maps

Gautam Bharali, Indranil Biswas, Mahan Mj (2015)

Complex Manifolds

We show that a map between complex-analytic manifolds, at least one ofwhich is in the Fujiki class, is a biholomorphism under a natural condition on the second cohomologies. We use this to establish that, with mild restrictions, a certain relation of “domination” introduced by Gromov is in fact a partial order.

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