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Workshop 1


Quang-Tuan Dang

Nicolina Istrati

Simon Jubert

Achim Napame

Annamaria Ortu

Carlo Scarpa

Practical informations


Institut Montpelliérain Alexander Grothendieck
Place Eugène Bataillon
Université de Montpellier
Place Eugène Bataillon
34095 Montpellier Cedex 5


2-3 June 2022


Local organizer:

Thibaut Delcroix

Scientific committee:

Thibaut Delcroix
Eleonora Di Nezza
Eveline Legendre
Carl Tipler
Tat Dat Tô


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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 \(L\).


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.


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.


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.


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.


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.


Name Institution
Hugues Auvray Université Paris Saclay
Quang-Tuan Dang Université Paul Sabatier
Thibaut Delcroix Université de Montpellier
Eleonora Di Nezza Ecole Polytechnique
Emmanuel Gnandi Université Paul Sabatier
Nicolina Istrati Philipps-Universität Marburg
Simon Jubert Université Paul Sabatier
Eveline Legendre Université Paul Sabatier
Achim Napame Université de Bretagne Occidentale
Tran Trung Nghiem Université de Montpellier
Annamaria Ortu SISSA
Carlo Scarpa SISSA
Carl Tipler Université de Bretagne Occidentale
Tat Dat Tô Sorbonne Université