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Papers by Georges Dloussky

Research paper thumbnail of Quadratic forms and singularities of genus one or two

Annales de la faculté des sciences de Toulouse Mathématiques, 2011

Research paper thumbnail of Colmatage de surfaces holomorphes et classification des surfaces compactes

Annales de l’institut Fourier, 1993

Research paper thumbnail of Analycite separee et prolongements analytiques

Lecture Notes in Mathematics, 2000

Research paper thumbnail of Complex surfaces with Betti numbers 𝑏₁=1,𝑏₂&𝑔𝑡;0 and finite quotients

Contemporary Mathematics, 2001

Research paper thumbnail of Special birational structures on non-Kählerian complex surfaces

Journal de Mathématiques Pures et Appliquées, 2016

Research paper thumbnail of Prolongements d'applications analytiques

Lecture Notes in Mathematics, 2000

Sans rsum

Research paper thumbnail of Structure des surfaces de Kato

Research paper thumbnail of 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... more 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>0 and a flat line bundle F such that -mK\otimes F has nontrivial sections, then S contains a Global Spherical Shell. We apply this last result to complete classification of bihermitian surfaces.

Research paper thumbnail of From non-Kählerian surfaces to Cremona group of P2(C)

Research paper thumbnail of Une construction elementaire des surfaces d'Inoue-Hirzebruch

Mathematische Annalen, 1988

Research paper thumbnail of Sur la classification des germes d'applications holomorphes contractantes

Mathematische Annalen, 1988

Research paper thumbnail of Analyticit� s�par�e et prolongement analytique

Mathematische Annalen, 1990

Research paper thumbnail of On surfaces of class VII with numerically anticanonical divisor

American Journal of Mathematics, 2006

We consider minimal compact complex surfaces S with Betti numbers b 1 = 1 and n = b 2 > 0. A theo... more 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.

Research paper thumbnail of Infinite bubbling in non-Kählerian geometry

Mathematische Annalen, 2012

In a holomorphic family (X b ) b∈B of non-Kählerian compact manifolds, the holomorphic curves rep... more In a holomorphic family (X b ) b∈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 ⊂ 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∈D\{0} , so one obtains non-proper families of exceptional divisors E z ⊂ 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 E z of E z in the universal cover X z does converge to an effective divisor E 0 in X 0 , but this limit divisor is not compact. We prove that this limit divisor is always bounded towards the pseudo-convex end of 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.

Research paper thumbnail of Infinite bubbling in non-K

In a holomorphic family (Xb)binB(X_b)_{b\in B}(Xb)binB of non-K\"ahlerian compact manifolds, the holomorphic cu... more In a holomorphic family (Xb)binB(X_b)_{b\in B}(Xb)binB of non-K\"ahlerian 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-K\"ahler geometry is the {\it explosion of the area} phenomenon: the area of a curve CbsubsetXbC_b\subset X_bCbsubsetXb in a fixed 2-homology class can diverge as btob0b\to b_0btob0. This phenomenon occurs frequently in the deformation theory of class VII surfaces. For instance it is well known that any minimal GSS surface X0X_0X0 is a degeneration of a 1-parameter family of simply blown up primary Hopf surfaces (Xz)zinDsetminus0(X_z)_{z\in D\setminus\{0\}}(Xz)zinDsetminus0, so one obtains non-proper families of exceptional divisors EzsubsetXzE_z\subset X_zEzsubsetXz whose area diverge as zto0z\to 0zto0. 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 widetildeEz\widetilde E_zwidetildeEz of EzE_zEz in the universal cover widetildeXz\widetilde X_zwidetildeXz does converge to an effective divisor widetildeE_0\widetilde E_0widetildeE0 in widetildeX0\widetilde X_0widetildeX0, but this limit divisor is not compact. We prove that this limit divisor is always bounded towards the pseudo-convex end of widetildeX0\widetilde X_0widetildeX0 and that, when X0X_0X_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.

Research paper thumbnail of Surfaces of class VII 0 and Hénon automorphisms

Comptes Rendus De L Academie Des Sciences Series I Mathematics, 1999

Research paper thumbnail of Classification of singular germs of mappings and deformations of compact surfaces of class VII 0

Annales Polonici Mathematici, 1998

Research paper thumbnail of Locally Conformally Symplectic Structures on Compact Non-Kähler Complex Surfaces

International Mathematics Research Notices, 2015

ABSTRACT We prove that every compact complex surface with odd first Betti number admits a locally... more ABSTRACT We prove that every compact complex surface with odd first Betti number admits a locally conformally symplectic 222-form which tames the underlying almost complex structure.

Research paper thumbnail of On separately algebraic sets

Research paper thumbnail of From locally conformally Kähler to bi-Hermitian structures on non-Kähler complex surfaces

Mathematical Research Letters, 2015

We prove that locally conformally Kähler metrics on certain compact complex surfaces with odd fir... more We prove that locally conformally Kähler metrics on certain compact complex surfaces with odd first Betti number can be deformed to new examples of bi-Hermitian metrics. 1 2 V. APOSTOLOV, M. BAILEY, AND G. DLOUSSKY (ii) Everywhere on M , J + = J − (resp. J + = −J − ), but for at least one x ∈ M , J + (x) = −J − (x) (resp. J + (x) = J − (x), though-by replacing J − with −J − if necessary-we can assume without loss of generality that in this class J + and J − never agree but J + and −J − sometimes do); (iii) There are points on M where J + = J − and also points where J + = −J − . Recall [35, 36, 12, 29] that on a compact complex surface S = (M, J) a Kähler metric exists if and only if the first Betti number is even. Similarly, by [4, Cor. 1 and Prop. 4], a bi-Hermitian conformal structure (c, J + , J − ) corresponds to a generalized Kähler structure for some g ∈ c if and only if b 1 (M ) is even. Furthermore, in this case the flat holomorphic line bundle L mentioned above is trivial ([3, Lemma 4]) and the bi-Hermitian structures are either of type (i) or (ii) ([3, Prop. 4]). The first case corresponds to Kähler surfaces with trivial canonical bundle (see [3]), i.e. tori and K3 surfaces. The classification in the second case follows by [3, 6] and a recent result in [19]: S must be then a Kähler surface of negative Kodaira dimension whose anticanonical bundle K * S has a non-trivial section and any Kähler metric on S = (M, J + ) can be deformed to a non-trivial bi-Hermitian structure (c, J + , J − ) of the class (ii).

Research paper thumbnail of Quadratic forms and singularities of genus one or two

Annales de la faculté des sciences de Toulouse Mathématiques, 2011

Research paper thumbnail of Colmatage de surfaces holomorphes et classification des surfaces compactes

Annales de l’institut Fourier, 1993

Research paper thumbnail of Analycite separee et prolongements analytiques

Lecture Notes in Mathematics, 2000

Research paper thumbnail of Complex surfaces with Betti numbers 𝑏₁=1,𝑏₂&𝑔𝑡;0 and finite quotients

Contemporary Mathematics, 2001

Research paper thumbnail of Special birational structures on non-Kählerian complex surfaces

Journal de Mathématiques Pures et Appliquées, 2016

Research paper thumbnail of Prolongements d'applications analytiques

Lecture Notes in Mathematics, 2000

Sans rsum

Research paper thumbnail of Structure des surfaces de Kato

Research paper thumbnail of 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... more 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>0 and a flat line bundle F such that -mK\otimes F has nontrivial sections, then S contains a Global Spherical Shell. We apply this last result to complete classification of bihermitian surfaces.

Research paper thumbnail of From non-Kählerian surfaces to Cremona group of P2(C)

Research paper thumbnail of Une construction elementaire des surfaces d'Inoue-Hirzebruch

Mathematische Annalen, 1988

Research paper thumbnail of Sur la classification des germes d'applications holomorphes contractantes

Mathematische Annalen, 1988

Research paper thumbnail of Analyticit� s�par�e et prolongement analytique

Mathematische Annalen, 1990

Research paper thumbnail of On surfaces of class VII with numerically anticanonical divisor

American Journal of Mathematics, 2006

We consider minimal compact complex surfaces S with Betti numbers b 1 = 1 and n = b 2 > 0. A theo... more 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.

Research paper thumbnail of Infinite bubbling in non-Kählerian geometry

Mathematische Annalen, 2012

In a holomorphic family (X b ) b∈B of non-Kählerian compact manifolds, the holomorphic curves rep... more In a holomorphic family (X b ) b∈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 ⊂ 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∈D\{0} , so one obtains non-proper families of exceptional divisors E z ⊂ 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 E z of E z in the universal cover X z does converge to an effective divisor E 0 in X 0 , but this limit divisor is not compact. We prove that this limit divisor is always bounded towards the pseudo-convex end of 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.

Research paper thumbnail of Infinite bubbling in non-K

In a holomorphic family (Xb)binB(X_b)_{b\in B}(Xb)binB of non-K\"ahlerian compact manifolds, the holomorphic cu... more In a holomorphic family (Xb)binB(X_b)_{b\in B}(Xb)binB of non-K\"ahlerian 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-K\"ahler geometry is the {\it explosion of the area} phenomenon: the area of a curve CbsubsetXbC_b\subset X_bCbsubsetXb in a fixed 2-homology class can diverge as btob0b\to b_0btob0. This phenomenon occurs frequently in the deformation theory of class VII surfaces. For instance it is well known that any minimal GSS surface X0X_0X0 is a degeneration of a 1-parameter family of simply blown up primary Hopf surfaces (Xz)zinDsetminus0(X_z)_{z\in D\setminus\{0\}}(Xz)zinDsetminus0, so one obtains non-proper families of exceptional divisors EzsubsetXzE_z\subset X_zEzsubsetXz whose area diverge as zto0z\to 0zto0. 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 widetildeEz\widetilde E_zwidetildeEz of EzE_zEz in the universal cover widetildeXz\widetilde X_zwidetildeXz does converge to an effective divisor widetildeE_0\widetilde E_0widetildeE0 in widetildeX0\widetilde X_0widetildeX0, but this limit divisor is not compact. We prove that this limit divisor is always bounded towards the pseudo-convex end of widetildeX0\widetilde X_0widetildeX0 and that, when X0X_0X_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.

Research paper thumbnail of Surfaces of class VII 0 and Hénon automorphisms

Comptes Rendus De L Academie Des Sciences Series I Mathematics, 1999

Research paper thumbnail of Classification of singular germs of mappings and deformations of compact surfaces of class VII 0

Annales Polonici Mathematici, 1998

Research paper thumbnail of Locally Conformally Symplectic Structures on Compact Non-Kähler Complex Surfaces

International Mathematics Research Notices, 2015

ABSTRACT We prove that every compact complex surface with odd first Betti number admits a locally... more ABSTRACT We prove that every compact complex surface with odd first Betti number admits a locally conformally symplectic 222-form which tames the underlying almost complex structure.

Research paper thumbnail of On separately algebraic sets

Research paper thumbnail of From locally conformally Kähler to bi-Hermitian structures on non-Kähler complex surfaces

Mathematical Research Letters, 2015

We prove that locally conformally Kähler metrics on certain compact complex surfaces with odd fir... more We prove that locally conformally Kähler metrics on certain compact complex surfaces with odd first Betti number can be deformed to new examples of bi-Hermitian metrics. 1 2 V. APOSTOLOV, M. BAILEY, AND G. DLOUSSKY (ii) Everywhere on M , J + = J − (resp. J + = −J − ), but for at least one x ∈ M , J + (x) = −J − (x) (resp. J + (x) = J − (x), though-by replacing J − with −J − if necessary-we can assume without loss of generality that in this class J + and J − never agree but J + and −J − sometimes do); (iii) There are points on M where J + = J − and also points where J + = −J − . Recall [35, 36, 12, 29] that on a compact complex surface S = (M, J) a Kähler metric exists if and only if the first Betti number is even. Similarly, by [4, Cor. 1 and Prop. 4], a bi-Hermitian conformal structure (c, J + , J − ) corresponds to a generalized Kähler structure for some g ∈ c if and only if b 1 (M ) is even. Furthermore, in this case the flat holomorphic line bundle L mentioned above is trivial ([3, Lemma 4]) and the bi-Hermitian structures are either of type (i) or (ii) ([3, Prop. 4]). The first case corresponds to Kähler surfaces with trivial canonical bundle (see [3]), i.e. tori and K3 surfaces. The classification in the second case follows by [3, 6] and a recent result in [19]: S must be then a Kähler surface of negative Kodaira dimension whose anticanonical bundle K * S has a non-trivial section and any Kähler metric on S = (M, J + ) can be deformed to a non-trivial bi-Hermitian structure (c, J + , J − ) of the class (ii).