Page 105 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 105
ARITHMETIC AND GEOMETRY OF ALGEBRAIC SURFACES (MS-45)
Rigid complex manifolds via product quotients
Christian Gleissner, christian.gleissner@uni-bayreuth.de
University of Bayreuth, Germany
Product quotients give rise to a large class of varieties equipped with a lot of symmetry. They are
particularly useful as a tool to provide examples of complex manifolds with special properties.
In this talk I will explain how to construct in each dimension dimension n (at least three) an
example of a rigid n-manifold of Kodaira dimension one, as a resolution of a singular product
quotient. Their existence was conjectured by I. Bauer and F. Catanese and constructed in a joint
project with I. Bauer.
Subbundles of the Hodge bundle, fibred surfaces and the Coleman-Oort
conjecture
Víctor González Alonso, gonzalez@math.uni-hannover.de
Leibniz Universität Hannover, Germany
The Hodge bundle of a semistable family of complex projective curves has two nested sub-
bundles: the flat unitary subbundle (spanned by flat sections with respect to the Gauss-Manin
connection), and the kernel of the Higgs field. The latter contains the flat subbundle, but it was
not clear how strict the inclusion could be. In this talk I will present some techniques to estimate
the ranks of both subbundles and to construct families where they are arbitrarily different. I will
also discuss some implications of this result for the classification of fibred surfaces, as well as
some connections with the Coleman-Oort conjecture on the (non-)existence of totally geodesic
subvarieties in the Torelli locus of principally polarized abelian varieties. This is joint work
with Sara Torelli.
Infinitesimal Torelli for elliptic surfaces revisited
Remke Kloosterman, klooster@math.unipd.it
University of Padova, Italy
In this talk we consider the infinitesimal Torelli problem for elliptic surfaces without multiple
fibers.
We give a new proof for the case of elliptic surfaces without multiple fibers with Euler
number at least 24 and nonconstant j-invariant and with Euler number at least 72 and constant
j-invariant.
For all of the remaining cases we will indicate whether infinitesimal Torelli holds, does not
hold or our methods are insufficient to decide.
This solve an issue raised by Atsushi Ikeda in a recent paper, in connection with his con-
struction a counterexample to infinitesimal Torelli with pg = q = 1.
103
Rigid complex manifolds via product quotients
Christian Gleissner, christian.gleissner@uni-bayreuth.de
University of Bayreuth, Germany
Product quotients give rise to a large class of varieties equipped with a lot of symmetry. They are
particularly useful as a tool to provide examples of complex manifolds with special properties.
In this talk I will explain how to construct in each dimension dimension n (at least three) an
example of a rigid n-manifold of Kodaira dimension one, as a resolution of a singular product
quotient. Their existence was conjectured by I. Bauer and F. Catanese and constructed in a joint
project with I. Bauer.
Subbundles of the Hodge bundle, fibred surfaces and the Coleman-Oort
conjecture
Víctor González Alonso, gonzalez@math.uni-hannover.de
Leibniz Universität Hannover, Germany
The Hodge bundle of a semistable family of complex projective curves has two nested sub-
bundles: the flat unitary subbundle (spanned by flat sections with respect to the Gauss-Manin
connection), and the kernel of the Higgs field. The latter contains the flat subbundle, but it was
not clear how strict the inclusion could be. In this talk I will present some techniques to estimate
the ranks of both subbundles and to construct families where they are arbitrarily different. I will
also discuss some implications of this result for the classification of fibred surfaces, as well as
some connections with the Coleman-Oort conjecture on the (non-)existence of totally geodesic
subvarieties in the Torelli locus of principally polarized abelian varieties. This is joint work
with Sara Torelli.
Infinitesimal Torelli for elliptic surfaces revisited
Remke Kloosterman, klooster@math.unipd.it
University of Padova, Italy
In this talk we consider the infinitesimal Torelli problem for elliptic surfaces without multiple
fibers.
We give a new proof for the case of elliptic surfaces without multiple fibers with Euler
number at least 24 and nonconstant j-invariant and with Euler number at least 72 and constant
j-invariant.
For all of the remaining cases we will indicate whether infinitesimal Torelli holds, does not
hold or our methods are insufficient to decide.
This solve an issue raised by Atsushi Ikeda in a recent paper, in connection with his con-
struction a counterexample to infinitesimal Torelli with pg = q = 1.
103
