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Research talks;Algebraic and Complex Geometry

Given an algebraic variety defined by a set of equations, an upper bound for its dimension at one point is given by the dimension of the Zariski tangent space. The infinitesimal deformations of a variety $X$ play a somehow similar role, they yield the Zariski tangent space at the local moduli space, when this exists, hence one gets an efficient way to estimate the dimension of a moduli space.
It may happen that this moduli space consists of a point, or even a reduced point if there are no infinitesimal deformations. In this case one says that $X$ is rigid, respectively inifinitesimally rigid.
A basic example is projective space, which is the only example in dimension 1. In the case of surfaces, infinitesimally rigid surfaces are either the Del Pezzo surfaces of degree $\ge$ 5, or are some minimal surfaces of general type.
As of now, the known surfaces of the second type are all projective classifying spaces (their universal cover is contractible), and have universal cover which is either the ball or the bidisk (these are the noncompact duals of $P^2$ and $P^1 \times P^1$ ), or are the examples of Mostow and Siu, or the Kodaira fibrations of Catanese-Rollenske.
Motivated by recent examples constructed with Dettweiller of interesting VHS over curves, which we shall call BCD surfaces, together with ingrid Bauer, we showed the rigidity of a class of surfaces which includes the Hirzebruch-Kummer coverings of the plane branched over a complete quadrangle.
I shall also explain some results concerning fibred surfaces, e.g. a criterion for being a $K(\pi,1)$-space; I will finish mentioning other examples and several interesting open questions.
Given an algebraic variety defined by a set of equations, an upper bound for its dimension at one point is given by the dimension of the Zariski tangent space. The infinitesimal deformations of a variety $X$ play a somehow similar role, they yield the Zariski tangent space at the local moduli space, when this exists, hence one gets an efficient way to estimate the dimension of a moduli space.
It may happen that this moduli space consists of a ...

14J29 ; 14J80 ; 14P25 ; 32G05

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ISBN 978-3-540-41088-1

Lecture notes in mathematics , 1746

Localisation : Collection 1er étage

surface d'Enriques réelle # surface de Del Pezzo # surface algébrique # involution # réseau entier # polyèdre fondamentale # structure quaterniomique # période # modèle d'Horikawa

14P25 ; 14J28 ; 14J15 ; 57S17 ; 58D27 ; 14J80

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ISBN 978-0-8218-2145-9

Graduate studies in mathematics , 0028

Localisation : Disparu

analyse globale # géométrie spinienne # invariant de Seiberg-Witten # physique mathématique # topologie algébrique sur les variétés # variété de dimension 4 # équation elliptique sur les variétés

14J80 ; 53C55 ; 57R15 ; 57R19 ; 57R57 ; 58D27 ; 58J05 ; 58J52

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- 324 p.
ISBN 978-0-19-856763-9

International series of monographs on physics , 0132

Localisation : Ouvrage RdC (TERN)

supersymétrie # théorie quantique # théorie des champs supersymétriques # topologie des surfaces # brisure de symétrie # dualité # théorie de Seiberg-Witten # supergravité

81-02 ; 81T60 ; 81R40 ; 14J80 ; 57R57 ; 81T30 ; 81T16

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- xxiii; 383 p.
ISBN 978-3-540-93912-2

Lecture notes in mathematics , 1972

Localisation : Collection 1er étage

faisceau cohérent # obstruction # théorie géométrique des invariants # champ de Deligne-Mumford # champ de module # paire de Bradlov # faisceau parabolique

14D20 ; 14J60 ; 14J80 ; 14-02

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