Hermitian complex spaces are a large class of singular spaces that include for instance projective varieties endowed with the metric induced by the Fubini-Study metric. Many of the problems raised by Cheeger, Goresky and MacPherson in the case of complex projective varieties admit a natural extension also in this setting. The aim of this talk is to report about some recent results concerning the Hodge-Kodaira Laplacian acting on the canonical bundle of a compact Hermitian complex space. More precisely let $(X,h)$ be a compact and irreducible Hermitian complex space of complex dimension $m$. Consider the Dolbeault operator $\bar{\partial}_{m,0}$ : $L^2 \Omega^{m,0}(reg(X),h) \to L^2\Omega^{m,1}(reg(X),h)$ with domain $\Omega{_c^{m,0}}(reg(X))$ and let $\bar{\mathfrak{d}}_{m,0} : L^2 \Omega^{m,0}(reg(X),h)\to L^2\Omega^{m,1}(reg(X),h)$ be any of its closed extension. Now consider the associated Hodge-Kodaira Laplacian $\bar{\mathfrak{d}^*} \circ\bar{\mathfrak{d}}_{m,0}$ : $L^2 \Omega^{m,0}(reg(X),h)\to L^2\Omega^{m,0}(reg(X),h)$. We will show that the latter operator is discrete and we will provide an estimate for the growth of its eigenvalues. Finally we will prove some discreteness results for the Hodge-Dolbeault operator in the setting of both isolated singularities and complex projective surfaces (without assumptions on the singularities in the latter case).[-]

In this talk I will discuss the existence of complete extremal metrics on the complement of simple normal crossings divisors in compact Kähler manifolds, and stability of pairs, in the toric case. Using constructions of Legendre and Apostolov-Calderbank-Gauduchon, we completely characterize when this holds for Hirzebruch surfaces. In particular, our results show that relative stability of a pair and the existence of extremal Poincaré type/cusp metrics do not coincide. However, stability is equivalent to the existence of a complete extremal metric on the complement of the divisor in our examples. It is the Poincaré type condition on the asymptotics of the extremal metric that fails in general.
This is joint work with Vestislav Apostolov and Hugues Auvray.[-]

A famous conjecture of Kobayashi from the 1970s asserts that a generic algebraic hypersurface of sufficiently large degree $d\geq d_n$ in the complex projective space of dimension $n+1$ is hyperbolic. Yum-Tong Siu introduced several fundamental ideas that led recently to a proof of the conjecture. In 2016, Damian Brotbek gave a new geometric argument based on the use of Wronskian operators and on an analysis of the geometry of Semple jet bundles. Shortly afterwards, Ya Deng obtained effective degree bounds by means of a refined technique. Our goal here will be to explain a drastically simpler proof that yields an improved (though still non optimal) degree bound, e.g. $d_n=[(en)^{2n+2}/5]$. We will also present a more general approach that could possibly lead to optimal bounds.[-]

We study the relationship between metric and algebraic structures on the section ring of a projective manifold and an ample line bundle over it. More precisely, we prove that once the kernel is factored out, the multiplication operator of the section ring becomes an approximate isometry (up to normalization) with respect to the $L^{2}$-norm. We then show that, in fact, those algebraic properties characterise $L^{2}$-norms and describe some applications of this classification. The semiclassical version of Ohsawa-Takegoshi theorem lies at the heart of our approach.[-]