Hopf Surfaces Research Articles

Pseudoconvex non-Stein domains in primary Hopf surfaces

We describe pseudoconvex non-Stein domains in primary Hopf surfaces using techniques developed by Hirschowitz.

Localization on Hopf surfaces

We discuss localization of the path integral for supersymmetric gauge theories with an R-symmetry on Hermitian four-manifolds. After presenting the localization locus equations for the general case, we focus on backgrounds with S^1 x S^3 topology, admitting two supercharges of opposite R-charge. These are Hopf surfaces, with two complex structure moduli p,q. We compute the localized partition function on such Hopf surfaces, allowing for a very large class of Hermitian metrics, and prove that this is proportional to the supersymmetric index with fugacities p,q. Using zeta function regularisation, we determine the exact proportionality factor, finding that it depends only on p,q, and on the anomaly coefficients a, c of the field theory. This may be interpreted as a supersymmetric Casimir energy, and provides the leading order contribution to the partition function in a large N expansion.

Classification of real analytic Levi flat hypersurfaces of 1-concave type in Hopf surfaces

A compact Levi flat hypersurface in a complex manifold is said to be of q-concave type if it admits a neighborhood system consisting of q-concave manifolds in the sense of Andreotti and Grauert. The real analytic Levi flat hypersurfaces of 1-concave type in Hopf surfaces are classified.

The Chern–Ricci flow on complex surfaces

AbstractThe Chern–Ricci flow is an evolution equation of Hermitian metrics by their Chern–Ricci form, first introduced by Gill. Building on our previous work, we investigate this flow on complex surfaces. We establish new estimates in the case of finite time non-collapsing, analogous to some known results for the Kähler–Ricci flow. This provides evidence that the Chern–Ricci flow carries out blow-downs of exceptional curves on non-minimal surfaces. We also describe explicit solutions to the Chern–Ricci flow for various non-Kähler surfaces. On Hopf surfaces and Inoue surfaces these solutions, appropriately normalized, collapse to a circle in the sense of Gromov–Hausdorff. For non-Kähler properly elliptic surfaces, our explicit solutions collapse to a Riemann surface. Finally, we define a Mabuchi energy functional for complex surfaces with vanishing first Bott–Chern class and show that it decreases along the Chern–Ricci flow.

On compact complex surfaces of Kähler rank one

The Kähler rank of compact complex surfaces was introduced by Harvey and Lawson in their 1983 paper on Kähler manifolds as a measure of “kählerianity”. Here we give a partial classification of compact complex surfaces of Kähler rank 1. These are either elliptic surfaces, or Hopf surfaces, or they admit a holomorphic foliation of a very special type. As a consequence we give an affirmative answer to the question raised by Harvey and Lawson whether the Kähler rank is a bimeromorphic invariant.

Vector bundles on blown-up Hopf surfaces

We show that certain moduli spaces of vector bundles over blown-up primary Hopf surfaces admit no compact components. These are the moduli spaces used by Andrei Teleman in his work on the classification of class VII surfaces.

Infinite bubbling in non-Kählerian geometry

In a holomorphic family $${(X_b)_{b\in B}}$$ of non-Kählerian compact manifolds, the holomorphic curves representing a fixed 2-homology class do not form a proper family in general. The deep source of this fundamental difficulty in non-Kähler geometry is the explosion of the area phenomenon: the area of a curve $${C_b\subset X_b}$$ in a fixed 2-homology class can diverge as b → b 0. This phenomenon occurs frequently in the deformation theory of class VII surfaces. For instance it is well known that any minimal GSS surface X 0 is a degeneration of a 1-parameter family of simply blown up primary Hopf surfaces $${(X_z)_{z\in D{\setminus}\{0\}}}$$ , so one obtains non-proper families of exceptional divisors $${E_z\subset X_z}$$ whose area diverge as z → 0. Our main goal is to study in detail this non-properness phenomenon in the case of class VII surfaces. We will prove that, under certain technical assumptions, a lift $${\widetilde E_z}$$ of E z in the universal cover $${\widetilde X_z}$$ does converge to an effective divisor $${\widetilde E_0}$$ in $${\widetilde X_0}$$ , but this limit divisor is not compact. We prove that this limit divisor is always bounded towards the pseudo-convex end of $${\widetilde X_0}$$ and that, when X 0 is a minimal surface with global spherical shell, it is given by an infinite series of compact rational curves, whose coefficients can be computed explicitly. This phenomenon—degeneration of a family of compact curves to an infinite union of compact curves—should be called infinite bubbling. We believe that such a decomposition result holds for any family of class VII surfaces whose generic fiber is a blown up primary Hopf surface. This statement would have important consequences for the classification of class VII surfaces.

Complement to explicit description of Hopf surfaces and their automorphism groups

In the previous paper we determined Aut(X) of each Hopf surface X W G with W C 2 (0, 0) so that its holomorphic automorphism group is given by Aut(X) Aut(X) G. We calculate the group of connected components 0(Aut(X)) by reviewing the classification. A Hopf surface X is a compact complex surface whose universal covering space is W C 2 (0, 0). So, X W G by denoting its covering transformation group by G. A Hopf surface is called primary if its fundamental group G is isomorphic to the group Z of integers, and secondary otherwise. In Theorem 1 of [3] we determined Aut(X) of each secondary Hopf surface X W G so that its holomorphic automorphism group is given by Aut(X) Aut(X) G, where Aut(X) is the normalizer of G in the holomorphic automorphism group of C 2 fixing (0, 0). Moreover, in the following cases (2) and (3) it co incides with the normalizer of G in GL(2, C). We calculate 0(Aut(X)) including primary Hopf surfaces as a continuation by correcting some parts of [3]. Before stating Theorem 2 we review the classification of Hopf surfaces. Except the special case (0) that G is not given by any subgroup of GL(2, C), we may assume that G GL(2, C) and G is an extension of H g G det g 1 by Z. Let a be

Instantons and curves on class VII surfaces

We develop a general strategy, based on gauge theoretical methods, to prove existence of curves on class VII surfaces. We prove that, for b 2 = 2, every minimal class VII surface has a cycle of rational curves hence, by a result of Nakamura, is a global deformation of a one parameter family of blown up primary Hopf surfaces. The case b 2 = 1 was solved in a previous article. The fundamental object intervening in our strategy is the moduli space M pst (0, K) of polystable bundles & with c 2 (ℰ) = 0, det(&) = k. For large b 2 the geometry of this moduli space becomes very complicated. The case b 2 = 2 treated here in detail requires new ideas and difficult techniques of both complex geometric and gauge theoretical nature. We explain the substantial obstacles which must be overcome in order to extend our methods to the case b 2 > 3.

A Parabolic Flow of Pluriclosed Metrics

We define a parabolic flow of pluriclosed metrics. This flow is of the same family introduced by the authors in [15]. We study the relationship of the existence of the flow and associated static metrics to topological information on the underlying complex manifold. Solutions to the static equation are automatically Hermitian-symplectic, a condition we define herein. These static metrics are classified on K3 surfaces, complex toroidal surfaces, nonminimal Hopf surfaces, surfaces of general type, and class VII+ surfaces. To finish, we discuss how the flow may potentially be used to study the topology of class VII+ surfaces.

Locally homogeneous structures on Hopf surfaces

We study holomorphic locally homogeneous geometric structures modelled on line bundles over the projective line. We classify these structures on primary Hopf surfaces. We write out the developing map and holonomy morphism of each of these structures explicitly on each primary Hopf surface.

On surfaces of class VII + 0 with numerically anticanonical divisor

We consider minimal compact complex surfaces S with Betti numbers b 1 = 1 and n = b 2 > 0. A theorem of Donaldson gives n exceptional line bundles. We prove that if in a deformation, these line bundles have sections, S is a degeneration of blown-up Hopf surfaces. Besides, if there exists an integer m ≥ 1 and a flat line bundle F such that H 0 ( S,-mK ⊗ F ) ≠ 0, then S contains a Global Spherical Shell. We apply this last result to complete classification of bihermitian surfaces.

A note on logarithmic transformations on the Hopf surface

In this note we study logarithmic transformations in the sense of differential topology on two fibers of the Hopf surface. It is known that such transformations are susceptible to yield exotic smooth structures on 4-manifolds. We will show here that this is not the case for the Hopf surface, all integer homology Hopf surfaces we obtain are diffeomorphic to the standard Hopf surface. To cite this article: R. Zentner, C. R. Acad. Sci. Paris, Ser. I 342 (2006).

Free circle actions with contractible orbits on symplectic manifolds

We prove that closed symplectic four-manifolds do not admit any smooth free circle actions with contractible orbits, without assuming that the actions preserve the symplectic forms. In higher dimensions such actions by symplectomorphisms do exist, and we give explicit examples based on a construction of Fernandez, Gray and Morgan. We also prove that the only compact complex surfaces admitting free circle actions with contractible orbits are primary Hopf surfaces.

DEFORMATIONS OF KODAIRA MANIFOLDS

Using compact simple Lie groups and Heisenberg groups, we combine and generalize the constructions of complex structures on Kodaira surfaces and Hopf surfaces. We identify locally complete parameter spaces of deformations of these spaces and analyze the deformation of Kodaira manifolds in details.

Hopf surfaces: locally conformal Kähler metrics and foliations

In this paper we describe a family of locally conformal Kähler metrics on class 1 Hopf surfaces H α,β containing some recent metrics constructed in [GO98]. We study some canonical foliations associated to these metrics, in particular a 2-dimensional foliation ℰα,β that is shown to be independent of the metric. We prove with elementary tools that ℰα,β has compact leaves if and only if α m =β n for some integers m and n, namely in the elliptic case. In this case we prove that the leaves of ℰα,β explicitly give the elliptic fibration of H α,β, and we describe the natural orbifold structure on the leaf space.

Periodic points of holomorphic maps via Lefschetz numbers

In this paper we study the set of periods of holomorphic maps on compact manifolds, using the periodic Lefschetz numbers introduced by Dold and Llibre, which can be computed from the homology class of the map. We show that these numbers contain information about the existence of periodic points of a given period; and, if we assume the map to be transversal, then they give us the exact number of such periodic orbits. We apply this result to the complex projective space of dimension n n and to some special type of Hopf surfaces, partially characterizing their set of periods. In the first case we also show that any holomorphic map of C P ( n ) {\mathbb CP}(n) of degree greater than one has infinitely many distinct periodic orbits, hence generalizing a theorem of Fornaess and Sibony. We then characterize the set of periods of a holomorphic map on the Riemann sphere, hence giving an alternative proof of Baker’s theorem.

Explicit description of Hopf surfaces and their automorphism groups

Explicit description of Hopf surfaces and their automorphism groups

On the metric structure of non-Kähler complex surfaces

We give a characterization of a locally conformally Kahler (l.c.K.) metric with parallel Lee form on a compact complex surface. Using the Kodaira classification of surfaces, we classify the compact complex surfaces admitting such structures. This gives a classification of Sasakian structures on compact three-manifolds. A weak version of the above mentioned characterization leads to an explicit construction of l.c.K. metrics on all Hopf surfaces. We characterize the locally homogeneous l.c.K. metrics on geometric complex surfaces, and we prove that some Inoue surfaces do not admit any l.c.K. metric.

Algebraic reduction of twistor spaces of Hopf surfaces

Let Z be a compact connected complex manifold of dimension three. Let a(Z) be the algebraic dimension of Z, i.e., the transcendence degree of the meromorphic function field of Z. We have 0 Y onto a projective nonsingular curve Y, which induces by the pull-back an isomorphism of the meromorphic function fields. We call such a map / a meromorphic algebraic reduction of Z. Some restrictions on the possible structure of general fibers of / are known as in Ueno [10, 12.4, 12.5]. But the question as to which surfaces actually appear in such an algebraic reduction seems still open except in the Kahler case. (See [2] for the general results in the Kahler case.) Now twistor spaces of compact anti-self-dual surfaces provide an interesting family of non-Kahler compact complex threefolds whose algebraic dimensions can take any values from zero to three. So it should be quite natural to consider the above question for these twistor spaces. The purpose of this note is to answer the question in one of the simplest cases of the twistor spaces of Hopf surfaces, which, we hope, should serve as useful examples in understanding the situation in the general case. We show for instance that in case a(Z) = 1 either 1) a general fiber is a Hopf surface or 2) the normalization of a general fiber, which is in general non-normal, is a nonsingular ruled surface of genus one. Note that Gauduchon [4] has already determined the algebraic dimensions of the twistor spaces of Hopf surfaces (cf. Proposition 1.1 below), generalizing the previous result of Pontecorvo [8], who has shown that the twistor spaces of special Hopf surfaces of the form S(r, r) (cf. below) are of algebraic dimension two and has determined the structure of their algebraic reductions. Thanks are due to the referee for his careful reading of this note.