What complex analytic spaces can be obtained as the universal covering of a complex algebraic variety? Motivated by this question, Shafarevich asked whether the universal covering of any smooth projective variety X is necessarily holomorphically convex. In other words, is there a proper holomorphic map from the universal covering of X to a Stein analytic space? Although still open, Shafarevich's question has received partial positive answers, for example when the fundamental group of X admits a faithful complex linear representation (Eyssidieux-Kaztarkov-Pantev-Ramachandran). In my talk, I will discuss the generalisation of Shafarevich's question to non-compact algebraic varieties. This is joint work with Ben Bakker and Jacob Tsimerman.

A complex projective manifold X is `weakly special' if none of its finite etale covers maps surjectively onto a variety of general type. X is `potentially dense' if (defined over a number field k, and if) its set of k'-rational points is Zariski dense for some larger number field k'. Lang's conjecture and Chevalley-Weil theorem imply that X is weakly special if potentially dense. The converse `weak specialness conjecture' was formulated in 2000. The orbifold version of Mordell conjecture claims that orbifold curves of general type over a number field always have a finite set of (suitably defined) rational points. It is open, implied by abc. Using a threefold constructed by G. Lafon in 2007, we observe that the `weak specialness conjecture' conflicts with orbifold Mordell, and thus with abc. The analog statement for entire curves conflicts unconditionnally with Nevanlinna second Main Theorem. The issue lies in the difference between `specialness' and `weak specialness'.

In this talk, I will explain a very general ddbar lemma, together with a decomposition theorem for currents with values in a (singular) Hermitian line bundle. As applications, we will explain a Kähler version on an injectivity theorem due to O. Fujino in the projective case, as well as some application for the invariance of plurigenera. It is based on joint works with Mihai Păun.

The K3 period domain contains a 57-dimensional family of quadratically embedded smooth rational curves. Among those of these curves that are defined over the real numbers, the generic curve is obtained as the image of the twistor family built from a Kähler-Einstein metric on the K3-manifold. I will discuss how to use the rich complex geometry of the parameter space of such curves (the "cycle space") to establish a very satisfactory moduli theory for marked families of K3 surfaces lying over curves inside the period domain. This in particular encompasses an unobstructedness result and a Local Torelli Theorem for such families. I will explain that the moduli space of marked families is Hausdorff and furthermore admits a quotient by the natural "change-of-marking" group action; this will yield a coarse moduli space. Finally, I will point towards open questions and directions for further research, e.g. related to Huybrechts' work on "brilliant families" of K3 surfaces or to automorphic cohomology classes on the K3 period domain.

I will talk about recent joint work with Mihai Paun where we prove that given a stable, reflexive $\mathbb Q$-sheaf $F$ on a compact Kähler threefold with log terminal singularities, the orbifold Chern classes of $F$ satisfy the Bogomolov-Gieseker inequality.

Degenerations of projective spaces are a classical subject of complex algebraic geometry: if the central fibre is smooth, it is isomorphic to the projective space by a well-known result of Siu. Similar results hold if we assume that the hyperplane class extends as an ample Cartier divisor to the central fibre. In this talk I will discuss what happens if we assume that the central fibre is a Fano variety with klt singularities. We will see that there are many possibilities and their geometry depends on the stability of the tangent sheaf. This is joint work with Thomas Peternell.

A number of moduli problem are, via Hodge theory, closely related to ball quotients. In this situation there is often a choice of possible compactifications such as the GIT compactification´and its Kirwan blow-up or the Baily-Borel compactification and the toroidal compactificatikon. The relationship between these compactifications is subtle and often geometrically interesting. In this talk I will discuss several cases, including cubic surfaces and threefolds and Deligne-Mostow varieties. This discussion links several areas such as birational geometry, moduli spaces of pointed curves, modular forms and derived geometry. This talk is based on joint work with S. Casalaina-Martin, S. Grushevsky, S. Kondo, R. Laza and Y. Maeda.

Almost twenty years ago, Campana introduced C-pairs to complex geometry. Interpolating between compact and non-compact geometry, C-pairs capture the notion of "fractional positivity" in the "fractional logarithmic tangent bundle". Today, they are an indispensible tool in the study of hyperbolicity, higher-dimensional birational geometry and several branches of arithmetic geometry. This talk reports on joint work with Erwan Rousseau. We clarify the notion of a "morphism of C-pairs", define (and prove the existence of) a "C-Albanese variety", and discuss the beginnings of a Nevanlinna theory for "orbifold entire curves".

In this talk I will first review KSB/A stability, especially their local version and then discuss new results, joint with János Kollár, showing that it is enough to check these conditions, including flatness, up to codimension 2. This implies that we have a very good understanding of this stability condition in general, because local KSB-stability is trivial at codimension 1 points, and quite well understood at codimension 2 points, since we have a complete classification of 2-dimensional slc singularities.

The Abundance conjecture predicts that on a minimal projective klt pair $(X,D)$, the adjoint divisor $K_X+D$ is semiample. When the Euler-Poincaré characteristic of $X$ does not vanish, I will present a necessary and sufficient condition for the conjecture to hold in terms of the asymptotic behaviour of multiplier ideals of currents with minimal singularities of small twists of $K_X+D$. I will also give strong evidence that an important class of currents with minimal singularities - supercanonical currents - is central to the completion of the proof of the Abundance conjecture for minimal klt pairs with non-vanishing Euler-Poincaré characteristic.

In a seminal work Eugenio Calabi introduced in 1953 the diastasis and proved powerful extension theorems for germs of holomorphic isometries from Kähler manifolds into complex space forms such as the projective space equipped with the Fubini-Study metric, which led to extension theorems in the interior for germs of holomorphic isometries (understood to be up to scaling constants ) between bounded domains $(U,ds_U^2)$ equipped with the Bergman metric. Among bounded domains there are the bounded symmetric domains $\Omega \Subset \mathbb C^n$ in their Harish-Chandra realizations, which have semi-algebraic boundaries. In 2012 we proved an algebraicity theorem for germs of holomorphic isometries $f: (\Omega, \lambda ds^2_\Omega;x_0) \longrightarrow (\Omega',ds^2_{\Omega'};x_0')$, $\lambda > 0$. In this lecture I will explain, using the rescaling method for (complex-analytic families of) algebraic subvarieties of $\Omega$: (1) how the study of holomorphic isometries led to the Ax-Lindemann-Weierstrass Theorem for quotients of $\,\mathbb B^n$ by arbitrary lattices, (2) how the study of the asymptotic behavior of holomorphic isometries of the Poincar\'e disk into $\Omega$ led to a uniformization theorem for projective varieties covered by algebraic subvarieties $Z \subset \Omega$ , and (3) how the latter serves as a starting point for a proof of the Ax-Schanuel theorem for quotients of $\Omega$ by arbitrary cocompact lattices.

I would like to explain some roles of Calabi-Yau threefolds with $c_2$-contraction, among abelian or elliptically fibered Calabi-Yau threefold with an automorphism of positive entropy, as well as a more arithmetic problem on Kawaguchi-Silverman Conjecture for automorphisms of Calabi-Yau threefolds with $c_2$-contraction (for a better understanding of a result due to Lesieutre and Satriano), especially, non-trivially and non-vacuously hold cases with descriptions of their full automorphism groups. My study of fibered Calabi-Yau threefolds was strongly motivated by an impressive work of Professor Thomas Peternell in the 90s.

I will describe recent progress in understanding the filtered de Rham (or Du Bois) complex of a complex algebraic variety, both in terms of general properties, and as a tool for the definition and study of refined classes of singularities. I will also explain how one can use these developments to deduce basic results on the topology of singular varieties.

A conjecture by Campana and Peternell says that if a positive multiple of $K_X$ is linearly equivalent to an effective divisor D plus a pseudo-effective divisor, then the Kodaira dimension of X should be at least as big as the Iitaka dimension of D. This is a very useful generalization of the non-vanishing conjecture (which is the case D = 0). I will explain why the Campana-Peternell conjecture is (almost) equivalent to the non-vanishing conjecture, using recent work on singular metrics on pluri-adjoint bundles.

There are two parts in the Fujita conjecture. One is the freeness part which since 1995 has been proved with various weaker bounds. The other is the very ampleness part, which remains open even with weaker bounds. In this talk we will present a proof of the very ampleness part of the Fujita conjecture with a weaker bound by the method of multiplier ideal sheaves of higher order and other new techniques.

Thanks to Faltings, Arakelov and Parshin’s solution to Mordell’s conjecture we know that smooth complex projective curves of genus at least equal to 2 have finite number of rational points. A key input in the proof of this fundamental result is the boundedness of families of smooth projective curves of a fixed genus (greater than 1) over a fixed base scheme. The latter was generalized by the combined spectacular results of Kovács-Lieblich and Bedulev-Viehweg to higher dimensional analogues of such curves; the so-called canonically polarized projective manifolds. In this talk I will discuss our recent extension of this boundedness result to the case of families of varieties with semiample canonical bundle (for example Calabi-Yaus). This is based on joint work with Kenneth Ascher.

Borel and Haefliger asked in 1961 whether any cycle on a smooth projective complex variety is smoothable, that is, cohomologous to a cycle which is a combination of smooth subvarieties. There are many negative results on this subject, when the codimension is not greater than the dimension,and I will survey them quickly. The same question with rational coefficients is still open in this case. Finally, I will sketch the proof of the smoothability of cycles when the dimension is smaller than the codimension, which I obtained with J. Kollár.