### Complex manifolds, dynamics and birational geometry: talks and abstracts

#### November 10-14, 2014, Laboratory of Algebraic Geometry, Moscow

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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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