## Seminario di Geometria

### Informazioni

Il seminario si svolge generalmente il venerdì dalle 13:00 alle 14:00 nell'Aula A del plesso di Matematica (come arrivare). I seminari sono trasmessi in diretta su un apposito canale MS Teams dell'Università di Parma, permettendo la partecipazione da remoto.

### Organizzatori

## Prossimi Seminari

## Seminari Passati

### Joshua Windare (Università di Parma)

### Semistable and Polystable associated with Action of Real Reductive Group

We present a systematic treatment of a Hilbert criterion for stability theory for an action of a real reductive group G on a real submanifold X of a Kahler manifold Z. More precisely, we suppose the action of a compact connected Lie group U extends holomorphically to an action of the complexified group U^\C and that the U-action on Z is Hamiltonian. There is a so called gradient map for G-action on X. We will introduce a large class of actions of G on X (which includes all Hamiltonian actions of U^\C on compact Kahler manifolds) following A. Teleman. And characterize semistability and polystability of a point by a numerical criteria using a G-equivariant function associated with a gradient map, called maximal weight function for such action.

# 8 Aprile 2022

16:00 - Aula A

### Alice Garbagnati (Università di Milano Statale)

### Manifolds with trivial canonical bundle

The manifolds with trivial canonical bundle form a special set of manifolds: they have some peculiar properties, which are not properties of the manifolds which have a non trivial canonical bundle (for example the manifolds with trivial canonical bundle have no a "natural" projective model which encodes their main properties). Viceversa, the manifolds with trivial canonical bundle share some properties (they often have an infinite automorphism group; there is a good construction of their moduli spaces, the Hodge structure of their middle cohomology is very interesting). There are essentially three kinds of manifolds with trivial canonical bundle: the tori, the Hyperkhaeler manifolds, the Calabi--Yau manifolds. The aim of this talk is to describe properties of these manifolds, to relate the different kinds of manifolds with trivial canonical bundle and to construct explicitly examples of manifolds with trivial canonical bundle obtained by considering the quotient of product of manifolds of lower dimension by finite automorphisms. In order to do this one has to discuss some birational invariants of these manifolds and problems related with the desingularization of quotient singularities.

# 11 Marzo 2022

16:00 - Aula A

### Andrea Cattaneo (Università di Parma)

### Positive and negative results for the explicit computation of Kodaira dimension for parallelizable manifolds I, II

# 18 e 25 Febbraio 2022

13:00 - Aula A

### Anna Miriam Benini (Università di Parma)

### Bifurcations in families of meromorphic maps

Bifurcations arise when there is a drastic change in the solutions of some equation depending on a parameter, as the parameter varies. In this talk we study bifurcations in holomorphic families of meromorphic maps with finitely many singular values. The equation(s) that we will study are the equations defining periodic points of period n. Such equations are crucial in complex dynamics because the Julia set (the set on which the dynamics is chaotic) is the closure of repelling periodic points. The celebrated results by Mane-Sad-Sullivan for families of rational maps (and independently by Lyubich, and by Levin for polynomials) show that in a set of parameters where no bifurcations of periodic points occur, the Julia set stays almost the same and so does the dynamics; precisely speaking, all maps are topologically conjugate in such set. Moreover, they establish a precise correlation between bifurcations of periodic points and a change of behaviour in the orbits of singular values. The key new feature that appears for families of meromorphic maps is that periodic points can disappear at infinity at specific parameters, creating a new type of bifurcations. Our work connects this new type of bifurcations with change of behaviour in singular orbits, to establish Mane-Sad-Sullivan's Theorem for meromorphic maps. This is joint work with Matthieu Astorg and Nùria Fagella.

# 28 Gennaio 2022

13:00 - Aula A

### Stefano Marini (Università di Parma)

### CR-relative Khäler manifolds

I will show that two Kähler manifolds which do not share a Kähler submanifold, do not share either a Levi degenerate CR-submanifold (in sense of A.Bejancu-B.Y.Chen) with constant dimension Levi kernel. In particular, they do not share a CR-product. Further, we will see that a Levi degenerate CR-submanifold of the complex euclidian space cannot be isometrically immersed into a flag manifold. This is a joint work with M. Zedda.

# 21 Gennaio 2022

13:00 - Aula A

### Stefano Marini (Università di Parma)

### Finitely Levi non degenerate homogeneous CR-manifolds

A CR-manifold M is a differentiable manifold together with a complex subbundle of the complexification of its tangent bundle, which is formally integrable and has zero intersection with its conjugate bundle. A fundamental invariant of a CR-manifold M is its vector-valuated Levi form. A Levi non degenerate CR-manifold of order k≥1 has a non degenerate Levi form in a higher order sense. For a (locally) homogeneous manifold Levi non degeneracy of order k can be described in terms of its CR-algebra, i.e., a pair of Lie algebras encoding the structure of (locally) homogeneous CR-manifolds. I will introduce these topics presenting some recent results.

# 10 Dicembre 2021

13:00 - Aula A

### Josias Reppekus (Bergische Universität Wuppertal)

### Non-trivial limit sets of Fatou components

A central object of interest when describing the dynamics of a holomorphic self-map F of a complex manifold is the Fatou set of points with stable dynamical behaviour and its connected components, called Fatou components. A Fatou component is invariant, if F maps it inside itself. In one variable, an invariant Fatou component either admits a conjugation of F to a rotation or all its orbits accumulate at a single fixed point of F. In other words, the limit set has either full dimension 1 or trivial dimension 0. One of the many new phenomena in dimension 2 is that the orbits of an invariant Fatou component may accumulate on a limit set of intermediate dimension 1. In this talk, I will present classification results on Fatou components with limit sets of dimension 1 and construct examples not arising as straightforward products of one-dimensional components in the categories of endomorphisms of projective space and automorphisms of \mathbb{C}^2 .

# 16 Dicembre 2021

14:00 - Aula TBD

### Francesco Pediconi (Università di Firenze)

### Locally homogeneous spaces II-III

In this second talk, we survey the theory of locally homogeneous almost-Hermitian spaces. In particular, we will show how to adapt the approach exploited in the previous talk to the almost-Hermitian setting. Time permitting, we will provide explicit formulas for the curvature of all the Gauduchon connections and some explicit examples. This is a joint work with D. Angella.

# 19-26 Novembre 2021

13:00 - Online

### Francesco Pediconi (Università di Firenze)

### Locally homogeneous spaces I

In this talk, we review some basic facts about Riemannian locally homogeneous spaces. In particular, we will compare their metric and algebraic properties. We will also study some different topologies on the space of all the Riemannian locally homogeneous spaces.