### Complex manifolds, dynamics and birational geometry:

talks and abstracts

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####
Frédéric Campana (Nancy)

The fibres of the Albanese map of projective
`special' manifolds are `special'

This is a joint work with Benoit Claudon.
We prove the statement of the title using
a particular case of the Orbifold C(n,m)-conjecture
established by Birkar-Chen.
####
Benoît Claudon (Nancy)

Kähler and projective groups in the linear case

To determine whether or not the classes of K\"ahler groups
and of the projective ones coincide is a completely open question. In
the linear case the Hodge theory provides us with sufficiently many
powerful tool to handle this problem. In particular, it is possible to
show that linear K\"ahler groups are virtually projective (joint work
with F. Campana and P. Eyssidieux).
####
Sergey Galkin (HSE)

Acyclic line bundles on fake projective planes

On a projective plane there is a unique cubic root of a canonical bundle,
and it is acyclic.
On fake projective planes a cubic root of canonical bundle exists and
unique if there is no 3-torsion, and usually exists but not unique
otherwise.
In 1305.4549 we conjectured that on a fake projective plane a cubic root of
a canonical bundle is acyclic, if it exists.
It would suffice to prove the vanishing of global sections of a tensor
square of this line bundle, but it turned out to be very hard to prove.
I will tell about nine cases proved so far by five different methods,
including my recent work with Ilya Karzhemanov and Evgeny Shinder,
where we exploit the fact that a line bundle is _non_-linearisable to prove
that it is acyclic.
####
Ljudmila Kamenova (Stony Brook)

On Matsushita's conjecture

Matsushita conjectured that the rank of a Lagrangian fibration
on a hyperkahler manifold is either zero or maximal. This talk is about
a joint work with Misha Verbitsky proving Matsushita's conjecture. I will
also mention Claire Voisin's approach towards proving a birational version
of Matsushita's conjecture.
#### Bruno Klingler (Jussieu)

The hyperbolic Ax-Lindemann-Weierstrass conjecture

The hyperbolic Ax-Lindemann-Weierstrass conjecture is a functional
algebraic independence statement for the uniformizing map of an arithmetic
variety. In this talk I will describe the conjecture, its role and its
proof (joint work with E.Ullmo and A. Yafaev).
####
Adrian Langer (Warsaw)

Boundedness for representations of the fundamental group

I will report on a joint work with Hélène Esnault on boundedness
of algebraic flat connections on complex manifolds. I will also
try to report on what is known about related moduli problems
and the Riemann-Hilbert correspondence.
####
Vladimir Lazić (Bonn)

A note on the abundance conjecture

The abundance conjecture and the existence of good models are
the main remaining conjectures in the Minimal Model Program. I will
present a recent progress on these problems obtained in joint work with
Tobias Dorsch.
####
Eyal Markman
(University of Massachussets)

Integral transforms from a K3 surface to a moduli space of stable sheaves on it

Let S be a K3 surface, v an indivisible Mukai vector,
and M(v) the moduli space of stable sheaves on S with Mukai vector v.
The universal sheaf gives rise to an integral functor F from the derived category of
coherent sheaves on S to that on M(v). We show that the functor F is faithful.
The bounded derived category of M(v) is rather mysterious at the moment.
As a first step, we provide a simple conjectural description of its full subcategory whose of objects are images of objects on S via the functor F.
We verify that description whenever M(v) is the Hilbert scheme of points on S.
This work is joint with Sukhendu Mehrotra.

####
Shin-ichi Matsumura (Kagoshima University)

Injectivity theorems with multiplier ideal sheaves and their applications

In this talk, I give an injectivity theorem with multiplier ideal sheaves
of singular metrics with transcendental singularities.
This result can be seen as a generalization of various injectivity and
vanishing theorems.
The proof is based on a combination of the theory of harmonic integrals and
the L^2-method for the dbar-equation.
To treat transcendental singularities, after regularizing a given singular
metric, we study the asymptotic behavior of the harmonic forms with respect
to a family of the regularized metrics.
Moreover we obtain L^2-estimates of solutions of the dbar-equation by using
the Cech complex.

As applications of this injectivity theorem, I give some extension theorems
for holomorphic sections of pluri-log-canonical bundle from subvarieties
to the ambient space.
Moreover, by combining techniques of the minimal model program, we obtain
some results for semi-ampleness related to the abundance conjecture in
birational geometry.

This talk is based on the preprint in arXiv:1308.2033v2 and a joint work
with Y. Gongyo in
arXiv:1406.6132v1.

####
Dmitri Panov (King's Colledge)

Symplectic twistor spaces with S^1-symmetries

Twistor spaces are usually studied because they
have an integrable complex structure. It turns out, that sometimes
twistor spaces have as well a symplectic structure (not necessarily compatible
with a complex one). Imposing an S^1-symmetry drastically
reduces the number of examples, and it is possible to prove that
in dimension 4 only the twistor spaces of S^4 and CP^2 with
opposite orientation have such a structure.
This work is joint with Joel Fine.
####
Dan Popovici (Toulouse)

Positivity Cones in Bidegree $(n-1, n-1)$.

Let $X$ be a compact complex manifold of dimension $n$. Starting
from the duality between the Bott-Chern cohomology in bidegree $(1, 1)$
and the Aeppli cohomology in bidegree $(n-1, n-1)$, we introduce the
Gauduchon cone of $X$ in the latter bidegree whose closure turns out to be
dual to the pseudo-effective cone thanks to Lamari's positivity criterion
and the smaller sG cone consisting of Aeppli cohomology classes associated
with strongly Gauduchon metrics. We shall explain how the study of these
cones
contributes to the understanding of certain fundamental geometric
properties of compact complex manifolds. In the second part of the talk,
we shall describe a new class of compact complex manifolds (the so-called
sGG manifolds) that we have introduced recently in joint work with L.
Ugarte: they are defined by the equality between the sG cone and the
Gauduchon cone which turns out to be a special case of the
$\partial\bar\partial$ lemma and are characterised in purely numerical
terms by certain Betti, Hodge and Bott-Chern numbers. We shall explain the
role they play in the theory of deformations of complex structures and in
a strategy that we have proposed to prove in the long term the conjectured
deformation closedness of Fujiki's class C manifolds.
####
Yuri Prokhorov (Steklov Institute and HSE)

Finite groups of birational automorphisms of algebraic varieties.

We investigate which algebraic varieties have groups of
birational selfmaps satisfying the Jordan property. The talk is based
on joint work with Constantin Shramov.
####
Erwan Rousseau (Marseille)

Curves in Hilbert modular varieties

We will describe some results on the geometry of entire
curves in Hilbert modular varieties of general type in light of the
Green-Griffiths-Lang conjecture. This is a joint work with F. Touzet.
####
Konstantin Shramov
(Steklov Institute and HSE)

Fano threefolds with many symmetries

Varieties with large groups of symmetries are intereseting from many points
of view.
Apart from beautiful intrinsic geometry, they often interact in an
interesting way
with corresponding groups of birational automorphisms and enjoy various
other
particular properties. I will survey some results related to Fano
threefolds with large finite
groups of automorphisms, including rationality/irrationality results,
conjugacy in Cremona
groups, explicit constructions of varieties with maximal number of isolated
singularities etc.
####
Tuyen Truong
(Syracuse University)

Automorphisms of positive entropy on smooth rational threefolds

In contrast to the case of dimension 2, it is difficult to find
(non-trivial) automorphisms of positive entropy on smooth rational
threefolds. In this talk, I will give some explanations for why this is the
case and describe the currently two known examples. Finally, we will
mention some progress on our ongoing project toward a question by K. Oguiso:
Can the two smooth rational threefolds in the known examples be finite
composition of blowups at points or smooth curves, starting from P^3, or
P^2xP^1, or P^1xP^1xP^1?
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