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Georges Dloussky

Publications and source records attributed to Georges Dloussky.

14 recordsLinked to original sources
arXiv2024

On Classification of compact complex surfaces of class VII

Let $S$ be a minimal compact complex surface with Betti numbers $b_1(S)=1$ and $b_2(S)\ge 1$ i.e. a compact surface in class VII$_0^+$. We show that if there exists a twisted logarithmic 1-form $\tau\in H^0(S,\Omega^1(\log D)\otimes \mathcal L_\lambda)$, where $D$ is a non zero divisor and $\mathcal L\in H^1(S,\mathbb C^\star)$, then $S$ is a Kato surface. It is known that $\lambda$ is in fact real and we show that $\lambda\ge 1$ and unique if $S$ is not a Inoue-Hirzebruch surface. Moreover $\lambda=1$ if and only if $S$ is a Enoki surface. When $\lambda>1$ these conditions are equivalent to the existence of a negative PSH function $\hat \tau$ on the cyclic covering $p:\hat S\to S$ of $S$ which is PH outside $\hat D:=p^{-1}(D)$ with automorphy constant being the same automorphy constant $\lambda$ for a suitable automorphism of $\hat S$. With previous results obtained with V.Apostolov it suggests a strategy to prove the GSS conjecture.

math.CV↗
arXiv2022

Twisted differentials and Lee classes of locally conformally symplectic complex surfaces

We study the set of deRham classes of Lee $1$-forms of the locally conformally symplectic (LCS) structures taming the complex structure of a compact complex surface in the Kodaira class VII, and show that the existence of non-trivial upper/lower bounds with respect to the degree function correspond respectively to the existence of certain negative/non-negative PSH functions on the universal cover. We use this to prove that the set of Lee deRham classes of taming LCS is connected, as well as to obtain an explicit negative upper bound for this set on the hyperbolic Kato surfaces. This leads to a complete description of the sets of Lee classes on the known examples of class VII complex surfaces, and to a new obstruction to the existence of bi-hermitian structures on the hyperbolic Kato surfaces of the intermediate type. Our results also reveal a link between bounds of the set of Lee classes and non-trivial logarithmic holomorphic $1$-forms with values in a flat holomorphic line bundle.

math.DG↗
arXiv2020

Non Kählerian surfaces with a cycle of rational curves

Let $S$ be a compact complex surface in class VII$_0^+$ containing a cycle of rational curves $C=\sum D_j$. Let $D=C+A$ be the maximal connected divisor containing $C$. If there is another connected component of curves $C'$ then $C'$ is a cycle of rational curves, $A=0$ and $S$ is a Inoue-Hirzebruch surface. If there is only one connected component $D$ then each connected component $A_i$ of $A$ is a chain of rational curves which intersects a curve $C_j$ of the cycle and for each curve $C_j$ of the cycle there at most one chain which meets $C_j$. In other words, we do not prove the existence of curves other those of the cycle $C$, but if some other curves exist the maximal divisor looks like the maximal divisor of a Kato surface with perhaps missing curves. The proof of this topological result is an application of Donaldson theorem on trivialization of the intersection form and of deformation theory. We apply this result to show that a twisted logarithmic $1$-form has a trivial vanishing divisor.

math.AG↗
arXiv2020

Smooth deformations of singular contractions of class VII surfaces

We consider normal compact surfaces $Y$ obtained from a minimal class VII surface $X$ by contraction of a cycle $C$ of $r$ rational curves with $C^2<0$. Our main result states that, if the obtained cusp is smoothable, then $Y$ is globally smoothable. The proof is based on a vanishing theorem for $H^2(Θ_Y)$. If $r<b_2(X)$ any smooth small deformation of $Y$ is rational, and if $r=b_2(X)$ (i.e. when $X$ is a half-Inoue surface) any smooth small deformation of $Y$ is an Enriques surface. The condition "the cusp is smoothable" in our main theorem can be checked in terms of the intersection numbers of the cycle, using the Looijenga conjecture (which has recently become a theorem). Therefore this is a "decidable" condition. We prove that this condition is always satisfied if $r<b_2(X)\leq 11$. Therefore the singular surface $Y$ obtained by contracting a cycle $C$ of $r$ rational curves in a minimal class VII surface $X$ with $r<b_2(X)\leq 11$ is always smoothable by rational surfaces. The statement holds even for unknown class VII surfaces.

math.CV↗
arXiv2016

On the Lee classes of locally conformally symplectic complex surfaces

We prove that the deRham cohomology classes of Lee forms of locally conformally symplectic structures taming the complex structure of a compact complex surface $S$ with first Betti number equal to $1$ is either a non-empty open subset of $H^1_{dR}(S, \mathbb R)$, or a single point. In the latter case, we show that $S$ must be biholomorphic to a blow-up of an Inoue-Bombieri surface. Similarly, the deRham cohomology classes of Lee forms of locally conformally Kähler structures of a compact complex surface $S$ with first Betti number equal to $1$ is either a non-empty open subset of $H^1_{dR}(S, \mathbb R)$, a single point or the empty set. We give a characterization of Enoki surfaces in terms of the existence of a special foliation, and obtain a vanishing result for the Lichnerowicz-Novikov cohomology groups on the class ${\rm VII}$ compact complex surfaces with infinite cyclic fundamental group.

math.DG↗
arXiv2016

Special birational structures on non-Kähler complex surfaces

We investigate the following conjecture: all compact non-Kähler complex surfaces admit birational structures. After Inoue-Kobayashi-Ochiai, the remaining cases to study are essentially surfaces in class VII_0^+. In case of Kato surfaces with a cycle and one branch of rational curves we show that they have a special birational structure given by new normal forms of contracting germs in Cremona group Bir(P^2(C)). In particular all surfaces S with GSS and 0<b_2(S)<4 admit a birational structure. From the existence of a special birational structure we deduce meromorphic mappings from the universal cover of S to the projective plane which blow down an infinite number of rational curves.

math.CV↗
arXiv2015

On Bott-Chern cohomology of compact complex surfaces

We study Bott-Chern cohomology on compact complex non-Kähler surfaces. In particular, we compute such a cohomology for compact complex surfaces in class $\text{VII}$ and for compact complex surfaces diffeomorphic to solvmanifolds.

math.DG↗
arXiv2013

From non-Kählerian surfaces to Cremona group of P^2(C)

For any minimal compact complex surface S with n=b_2(S)>0 containing global spherical shells (GSS) we study the effectiveness of the 2n parameters given by the n blown up points. There exists a family of surfaces with GSS which contains as fibers S, some Inoue-Hirzebruch surface and non minimal surfaces, such that blown up points are generically effective parameters. These families are versal outside a non empty hypersurface T. We deduce that, for any configuration of rational curves, there is a non empty open set in the Oeljeklaus-Toma moduli space such that the corresponding surfaces are defined by a contracting germ in Cremona group, in particular admit a birational structure.

math.CV↗
arXiv2011

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 {\it explosion of the area} phenomenon: the area of a curve $C_b\subset X_b$ in a fixed 2-homology class can diverge as $b\to b_0$. 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 $z\to 0$. 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 a minimal surface with global spherical shell, it is given by an infinite series of {\it 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 {\it 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.

math.CV↗
arXiv2008

Quadratic forms and singularities of genus one or two

We study singularities obtained by the contraction of the maximal divisor in compact (non kaehlerian) surfaces which contain global spherical shells. These singularities are of genus 1 or 2, may be Q-Gorenstein, numerically Gorenstein or Gorenstein. A family of polynomials depending on the configuration of the curves computes the discriminant of the quadratic forms of these singularities. We introduce a multiplicative branch topological invariant which determines the twisting of a non-vanishing holomorphic 1-form on the complement of the singular point.

math.CV↗
arXiv2007

Bihermitian metrics on Hopf surfaces

Inspired by a construction due to Hitchin, we produce strongly bihermitian metrics on certain Hopf complex surfaces, which integrate the locally conformally Kaehler metrics found by Gauduchon and Ornea. We also show that the Inoue complex surfaces with zero second Betti number do not admit bihermitian metrics. This completes the classification of the compact complex surfaces admitting strongly bihermitian metrics.

math.DG↗
arXiv2002

Class VII surfaces with $b_2$ curves

We give an affirmative answer to a conjecture of Ma. Kato, namely that every compact complex surface $S$ in Kodaira's class $VII_0$ with $b_2(S) > 0$ and $b_2(S)$ rational curves, admits a global spherical shell.

math.CV↗