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.

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-Lindemann-Weierstrass theorem 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.

We survey several rationality notions and present a result on retract irrational hypersurfaces which improves earlier results of the speaker and Moe. Joint work with Jan Lange.

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.