Analytic methods in algebraic geometry

The $\partial\bar{\partial}$ lemma plays an important role in complex geometry. In the first part of this talk, I will explain a twisted log version of the classical $\partial\bar{\partial}$ lemma. In the second part, I will discuss some applications in complex geometry. This talk is based on some joint works with Mihai Păun and on a recent work in progress with Ya Deng, C. Hacon and Mihai Păun.

In 1995, Kollár conjectured the positivity of Euler characteristic $\chi(K_X)$ for complex projective varieties $X$ with "big" fundamental groups. I will explain how non-abelian Hodge theory and $L^2$-methods can be combined to prove Kollár’s conjecture when $\pi_1(X)$ is linear. If time permits, I will also briefly discuss our recent work on a conjecture of De Oliveira–Katzarkov–Ramachandran concerning deformation openness of such varieties, as well as the Chern–Hopf–Thurston conjecture regarding the sign of $\chi(X)$ for varieties $X$ with "large" fundamental groups. This talk is based on a series of joint works with Chikako Mese and Botong Wang.

The Mabuchi functional M was introduced by Mabuchi in the 80's in relation to the existence of canonical metrics on a compact Kähler manifold. The critical points of M are indeed constant scalar curvature Kähler (cscK) metrics. Recently, Chen and Cheng proved that the existence of a (smooth) cscK metric is equivalent to the properness of such functional. In order to look for singular metrics, it is then natural to study the properties of the Mabuchi functional in singular settings. In this talk we prove that this functional is (almost) convex in the very general "big case". This is a joint work with S. Trapani and A. Trusiani

I will report on work in progress in collaboration with Tristan Collins. We study various geometric properties of log Calabi-Yau manifolds, i.e. pairs $(X,D)$ where $X$ is a projective manifold, $D$ is a smooth (or SNC) reduced divisor such that $K_X+D$ is trivial. I'll explain some vanishing theorems, stability properties for the logarithmic tangent bundle and will highlight the stark differences between the smooth and SNC cases. The methods rely ultimately on the existence of complete Ricci flat Kähler metrics on the complement of $D$.

The MMP for compact Kähler manifolds, initiated by works of Campana, Peternell and myself more than ten years ago, has made a lot of progress recently due to work of Das, Fujino, Hacon, Ou and Paun. In this talk I will survey some of these results and speak about potential applications to geometric problems for example for manifolds with nef anticanonical class.

We say that the formal principle with convergence holds for a compact complex submanifold A in a complex manifold X, if a formal isomorphism between the formal neighborhood of A in X and that of another embedding of A in any complex manifold always converges, namely, the formal isomorphism can be extended to a biholomorphic map between suitable Euclidean neighborhoods. Hirschowitz and Commichau- Grauert showed that the formal principle with convergence holds if the normal bundle of the submanifold is sufficiently positive. We discuss the problem when the normal bundle is only weakly positive, but the submanifold satisfies certain geometric conditions. Our main interest is when the submanifold is a general minimal rational curve in a uniruled projective manifold, such as a general line on a rational homogeneous space or a projective hypersurface of low degree. This is a joint work with Jaehyun Hong.

Given a compact Kähler variety X, proving that X is algebraic is essentially equivalent to producing sufficiently many algebraic subvarieties in it. We will discuss how the presence of positive cohomology classes enables this, with a particular focus on threefolds.

A central problem in geometry is to understand the fibrations naturally associated with complex varieties and to decompose these varieties into their fundamental building blocks. I will begin by reviewing the structure theorems developed in the 1990s and their subsequent extensions for compact Kähler manifolds with "non-negative curvature" in various senses, including non-negative holomorphic (bi)sectional curvature and nef (or pseudo-effective) tangent bundles. Roughly speaking, these results decompose such manifolds into Ricci-positive and Ricci-flat blocks. I will then present a result showing that any compact Kähler manifold with nef anticanonical bundle admits a locally trivial fibration whose fiber is rationally connected and whose base is Calabi-Yau, extending the projective case established by Cao and Horing. A key ingredient is a flatness criterion for pseudo-effective sheaves with vanishing first Chern class. This is joint work with Juanyong Wang, Xiaojun Wu, and Qimin Zhang.

We study harmonic maps into Euclidean buildings, allowing targets that are not necessarily locally finite. Our main technical result establishes that such maps have singular sets of Hausdorff codimension at least two, thereby extending the regularity theory of Gromov and Schoen to this broader setting. As an application, we obtain a superrigidity theorem for algebraic groups over fields with non-Archimedean valuations. This generalizes the rank-one p-adic superrigidity results of Gromov and Schoen and places the Bader–Furman extension of Margulis’ higher-rank superrigidity theorem in a geometric framework. In addition, we prove an existence theorem for pluriharmonic maps from Kähler manifolds to Euclidean buildings.

Kähler-Ricci shrinkers are differential geometric objects arising from the study of singularities of Ricci flows on Kähler manifolds. We will explain a foundational result connecting Kähler-Ricci shrinkers and Fano fibrations in algebraic geometry. Various conjectures and open questions will be discussed. Based on joint work with Junsheng Zhang.

I will talk about the singularities of the canonical $L^2$-metrics of direct image sheaves of adjoint type. Let $f : X \to Y$ be a "fibration", and let $(L, h)$ be a semi-positive line bundle on $X$. We can define a canonical $L^2$-metric $g$ on $f_*(K_{X/Y} \otimes L)$ by fiberwise integration. This Hermitian metric $g$ may have singularities when $f$ is not smooth. We show that $g$ is a good metric in the sense of Mumford, assuming that $Y$ is one-dimensional and $f$ is semi-stable.