Serre problem and Inoue-Hirzebruch surfaces

Zaffran, Dan

doi:10.1007/pl00004440

Cited by 14 publications

(13 citation statements)

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“…But a complete answer should also tell when two domains in the list are actually isomorphic. For n = 2, it actually follows from the above and from the literature about Inoue-Hirzebruch surfaces that considering "even Dloussky matrices" (defined in [20]) already yields all examples of D's not in S.…”

Section: Geometric Description Of Domains Not In Smentioning

confidence: 86%

“…Examples of domains not in S are: 1) Any D such that log D is an octant delimited by the X i 's. This example is the three-dimensional generalization of [1] and [20]. 2) Another basic example is given as follows: pick a point p = x 1 X 1 +x 2 X 2 + x 3 X 3 with x 1 x 2 x 3 = 0.…”

Section: Geometric Description Of Domains Not In Smentioning

confidence: 99%

“…The only (easy) extra work is to check that log D contains an R-orbit of the form {A t x y : t ∈ R} for some x y ∈ log D (see Section 3 for more details). The proof from [20] can be adapted as well. We conclude that D ∈ S.…”

Section: Proof Of Theorem 1 Formentioning

confidence: 99%

“…Thus D CL ∈ S. Their D CL has the following properties (see Figure 1): D CL has the Reinhardt symmetry, i.e., it is invariant by the map (z 1 , z 2 ) → (α 1 z 1 , α 2 z 2 ) for any complex numbers α 1 and α 2 of modulus 1; D CL has an automorphism g of the form g(z 1 , z 2 ) = (z The second named author generalized their construction and gave a better understanding of those bundles. In [20], a key point is the existence of a g-equivariant open dense embedding D CL → D CL , where D CL \ D CL is an infinite chain of rational curves. Roughly speaking, the non-Steinness of E CL is "explained" by what happens near that infinite chain.…”

Section: Introduction and Notationsmentioning

confidence: 99%

“…It reveals that in dimension two, Coeuré-Loeb's examples D CL (for all A ∈ SL 2 (Z)) are essentially the only Reinhardt bounded domains not in S: all other examples are provided by g-invariant subdomains of some D CL . Moreover, it is easily checked that the interior of the closure in D CL of such a subdomain still contains the infinite chain of curves, so from the point of view of [20], it is natural that those domains do not belong to S. Indeed, proofs in [1] and [20] apply almost verbatim to show that they do not belong to S. We shall see that both methods and results become more complicated in dimension three.…”

Section: Introduction and Notationsmentioning

confidence: 99%

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Steinness of bundles with fiber a Reinhardt bounded domain

Oeljeklaus¹,

Zaffran

2006

Bul. Soc. Math. France

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Abstract.-Let E denote a holomorphic bundle with fiber D and with basis B. Both D and B are assumed to be Stein. For D a Reinhardt bounded domain of dimension d = 2 or 3, we give a necessary and sufficient condition on D for the existence of a non-Stein such E (Theorem 1); for d = 2, we give necessary and sufficient criteria for E to be Stein (Theorem 2). For D a Reinhardt bounded domain of any dimension not intersecting any coordinate hyperplane, we give a sufficient criterion for E to be Stein (Theorem 3).

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Section: Geometric Description Of Domains Not In Smentioning

confidence: 86%

Section: Geometric Description Of Domains Not In Smentioning

confidence: 99%

Section: Proof Of Theorem 1 Formentioning

confidence: 99%

Section: Introduction and Notationsmentioning

confidence: 99%

Section: Introduction and Notationsmentioning

confidence: 99%

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Steinness of bundles with fiber a Reinhardt bounded domain

Oeljeklaus¹,

Zaffran

2006

Bul. Soc. Math. France

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Flexibility Properties of Complex Manifolds and Holomorphic Maps

Forstnerič

2017

Stein Manifolds and Holomorphic Mappings

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Holomorphic functions on bundles over annuli

Zaffran

2008

Math. Ann.

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ABSTRACT. We consider a family˘Em(D, M )¯of holomorphic bundles constructed as follows: to any given M ∈ GLn(Z), we associate a "multiplicative automorphism" ϕ of (C * ) n . Now let D ⊆ (C * ) n be a ϕ-invariant Stein Reinhardt domain. Then Em(D, M ) is defined as the flat bundle over the annulus of modulus m > 0, with fiber D, and monodromy ϕ.We show that the function theory on Em(D, M ) depends nontrivially on the parameters m, M and D. Our main result is thatwhere ρ(M ) denotes the max of the spectral radii of M and M −1 .As corollaries, we: -obtain a classification result for Reinhardt domains in all dimensions; -establish a similarity between two known counterexamples to a question of J.-P. Serre; -suggest a potential reformulation of a disproved conjecture of Siu Y.-T.Let D be a Stein manifold. We say that D belongs to S when: for any Stein manifold B and any locally trivial bundle E → B, the manifold E is also Stein. A famous question of Serre can be formulated as: "Are all manifolds in S?" Skoda answered it in the negative, by proving that C 2 ∈ S (cf. Here we study holomorphic functions on a family of bundles {E m (D, M )} over annuli, depending on a non necessarily bounded Reinhardt domain Date: September 24, 2017. zaffran@fudan.edu.cn Fudan University, Shanghai. Academia Sinica, Taipei.

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Serre problem and Inoue-Hirzebruch surfaces

Cited by 14 publications

References 13 publications

Steinness of bundles with fiber a Reinhardt bounded domain

Steinness of bundles with fiber a Reinhardt bounded domain

Flexibility Properties of Complex Manifolds and Holomorphic Maps

Holomorphic functions on bundles over annuli

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