Bogomolov and Mumford proved that every complex projective K3 surface contains a rational curve. Since then, a lot of progress has been made by Bogomolov, Chen, Hassett, Li, Liedtke, Tschinkel and others, towards the stronger statement that any such surface in fact contains infinitely many rational curves. In this talk I will present joint work with Xi Chen and Christian Liedtke completing the remaining cases of this conjecture, reproving some of the main previously known cases more conceptually and extending the result to arbitrary genus in a suitable sense.[-]

Almost one decade ago, Poonen constructed the first examples of algebraic varieties over global fields for which Skorobogatov's etale Brauer-Manin obstruction does not explain the failure of the Hasse principle. By now, several constructions are known, but they all share common geometric features such as large fundamental groups.
This talk discusses a construction of simply connected fourfolds over global fields of positive characteristic for which the Brauer-Manin machinery fails. Contrary to earlier work in this direction, our construction does not rely on major conjectures. Instead, we establish a new diophantine result of independent interest: a Mordell-type theorem for Campana's "geometric orbifolds" over function fields of positive characteristic. Along the way, we also construct the first example of simply connected surface of general type over a global field with a non-empty, but non-Zariski dense set of rational points.
Joint work with Pereira and Smeets.[-]

We study locally the representation varieties of fundamental groups of smooth complex algebraic varieties. These are schemes whose complex points parametrize such representations into linear algebraic groups. At a given representation, the structure of the formal local ring to the representation variety tells about the obstructions to deform formally this representation, which is ultimately related to topological obstructions to the possible fundamental groups of complex algebraic varieties. This was first described by Goldman and Millson in the case of compact Kähler manifold, using formal deformation theory and differential graded Lie algebras. We review this using methods of Hodge theory and of derived deformation theory and we are able to describe locally the representation variety for non-compact smooth varieties and representations underlying a variation of Hodge structure.[-]

The geometric P=W conjecture is a conjectural description of the asymptotic behavior of a celebrated correspondence in non-abelian Hodge theory. In particular, it is expected that the dual boundary complex of the compactification of character varieties is a sphere. In a joint work with Enrica Mazzon and Matthew Stevenson, we manage to compute the first non-trivial examples of dual complexes in the compact case. This requires to develop a new theory of essential skeletons over a trivially-valued field. As a byproduct, inspired by these constructions, we show that certain character varieties appear in degenerations of compact hyper-Kähler manifolds. In this talk we will explain how these new non-archimedean techniques can shed new light into classical algebraic geometry problems.[-]

In diophantine geometry, the Batyrev-Manin-Peyre conjecture originally concerns rational points on Fano varieties. It describes the asymptotic behaviour of the number of rational points of bounded height, when the bound becomes arbitrary large.
A geometric analogue of this conjecture deals with the asymptotic behaviour of the moduli space of rational curves on a complex Fano variety, when the 'degree' of the curves 'goes to infinity'. Various examples support the claim that, after renormalisation in a relevant ring of motivic integration, the class of this moduli space may converge to a constant which has an interpretation as a motivic Euler product.
In this talk, we will state this motivic version of the Batyrev-Manin-Peyre conjecture and give some examples for which it is known to hold : projective space, more generally toric varieties, and equivariant compactifications of vector spaces. In a second part we will introduce the notion of equidistribution of curves and show that it opens a path to new types of results.[-]

Positivity of the tangent bundle and its top exterior power (namely, the anticanonical bundle) is the subject of extensive literature, and several open problems. Notably, Campana and Peternell predict that if $-K_{X}$ is strictly nef, then $X$ is a Fano variety: this conjecture is proven up to dimension 3 by the work of Maeda and Serrano. In this talk, we investigate positivity of the intermediate exterior powers of the tangent bundle. We prove that if $X$ is a smooth projective n-fold and the third, fourth or (n-1)-th exterior power of $T_{X}$ is strictly nef, then $X$ is a Fano variety. Moreover, we classify smooth projective varieties of Picard number at least two with third or fourth exterior power of $T_{X}$ strictly nef. This work is actually slightly more general, as it boils down to classifying rationally connected varieties such that the degree of the anticanonical bundle on rational curves is quite large.[-]

Reid's recipe is an equivalent of the McKay correspondence in dimension three. It marks interior line segments and lattice points in the fan of the G-Hilbert scheme (a specific crepant resolution of $\mathbb{C}^{3} / G$ for $G \subset S L(3, \mathbb{C})$ ) with characters of irreducible representations of $G$. Our goal is to generalise this by marking the toric fan of a crepant resolution of any affine Gorenstein singularity, in a way that is compatible with both the G-Hilbert case and its categorical counterpart known as Derived Reid's Recipe. To achieve this, we foray into the combinatorial land of quiver moduli spaces and dimer models. This is joint work with Alastair Craw and Jesus Tapia Amador.[-]

The classical Hodge conjecture for smooth, projective varieties has been open for over 70 years, although it has been proven for some specific varieties. In this talk I will discuss a cohomological version of the Hodge conjecture for singular varieties. I will give a sufficient condition in terms of Mumford-Tate groups for a variety to satisfy the singular Hodge conjecture. If time allows I will give explicit examples of such varieties. This is joint work with Ananyo Dan.[-]

An lc-trivial fibration $f : (X, B) \to Y$ is a fibration such that the log-canonical divisor of the pair $(X, B)$ is trivial along the fibres of $f$. As in the case of the canonical bundle formula for elliptic fibrations, the log-canonical divisor can be written as the sum of the pullback of three divisors: the canonical divisor of $Y$; a divisor, called discriminant, which contains informations on the singular fibres; a divisor, called moduli part, that contains informations on the variation in moduli of the fibres. The moduli part is conjectured to be semiample. Ambro proved the conjecture when the base $Y$ is a curve. In this talk we will explain how to prove that the restriction of the moduli part to a hypersurface is semiample assuming the conjecture in lower dimension. This is a joint work with Vladimir Lazić.[-]