Young researchers in Kähler and Hermitian geometry

June 2

14:00 Stability of equivariant logarithmic tangent sheaves on smooth toric varieties, Achim Napame

Abstract: For an equivariant log pair \((X, D)\) where \(X\) is a smooth toric variety and \(D\) a simple normal crossing divisor, we will study the slope-stability of the logarithmic tangent sheaf \(T_X(- \log D)\). We will give a necessary condition on \(D\) which ensures the existence of polarizations \(L\) on \(X\) such that \(T_X(- \log D)\) is (semi)-stable with respect to \(L\).

ArXiv:2111.15387

15:00 Pluripotential Chern-Ricci flows, Quang-Tuan Dang

Abstract: We develope a parabolic pluripotential theory on compact Hermitian manifolds, providing a parabolic analogue of the fundamental work of Bedford-Taylor. We study weak solutions of degenerate parabolic complex Monge-Ampère equations and apply it to the study of the Chern-Ricci flow on varieties with log terminal singularities.

ArXiv:2201.01150

16:30 Holomorphic submersions and special Kähler metric, Annamaria Ortu

Abstract: Proper holomorphic submersions of Kähler manifolds can be thought of as both a generalisation of holomorphic vector bundles and as a way of studying the behaviour of Kähler manifolds in families. We will consider fibrations whose fibres are K-semistable varieties that admit a degeneration to Kähler manifolds with constant scalar curvature, in a way compatible with the fibration structure. On such fibrations, we will describe a condition, called optimal symplectic connection condition, which gives a canonical choice of a relatively Kähler metric and a generalisation of the Hermite-Einstein condition on vector bundles.

ArXiv:2201.12562

June 3

09:00 The Calabi problem on semisimple principal fibrations, Simon Jubert

Abstract: Semisimple principal fibrations are a certain class of holomorphic fibrations \(Y\) over a product of constant scalar curvature Kähler manifold with fiber a compact Kähler manifold \(X\). One of the main assets of these fibrations is that they come equipped with a connection which allows defining, from any Kähler metrics on \(X\), a Kähler metric on \(Y\), called compatible metric. A Kähler class containing a compatible metric is said to be compatible. In this talk, after giving details of the notions above, I will explain how to translate the Calabi problem on a compatible Kähler class on \(Y\), to a weighted cscK problem (in the sense of Lahdili) on the corresponding fiber \(X\). This is a joint work with V. Apostolov and A. Lahdili.

ArXiv:2104.09709

10:00 Scalar curvature and deformations of complex structures, Carlo Scarpa

Abstract: We consider a system of partial differential equations on a compact Kähler manifold, whose variables are a Kähler form and a first-order deformation of the complex structure. If the deformation vanishes, the system reduces to the constant scalar curvature condition for the Kähler metric. Starting from a moment map interpretation for this system, we will describe a generalization of \(K\)-stability that is conjectured to characterize the existence of solutions.

ArXiv:2202.00429

11:30 On the Bott-Chern Cohomology of Vaisman manifolds, Nicolina Istrati

Abstract: Vaisman manifolds are complex manifolds which can be endowed with a special type of a Hermitian structure, namely a locally conformally Kähler metric with parallel Lee form. The geometry of Vaisman manifolds is closely related to Kählerian geometry, as these manifolds come endowed with a natural transversally Kähler foliation. However, Vaisman manifolds do not satisfy the dd^c-lemma, therefore it is interesting to study their Bott-Chern cohomology, which is then a refined invariant. In this talk, I will explain how one can express this cohomology in terms of the basic cohomology with respect to the foliation, and in particular show that the numerical obstructions to the dd^c-lemma can be arbitrarily high. This is based on joint work with Alexandra Otiman.