# Program

Talks will be held in PK-5115 (just besides the 5th floor elevator), and coffee breaks in PK-5675.

(*) indicates that the talk will be given virtually.

Time | Monday | Tuesday | Wednesday | Thursday | Friday |
---|---|---|---|---|---|

8:30-9:30 | Carlo Gasbarri* | Gianluca Pacienza | Lie Fu* | Lie Fu* | Julie Wang* |

9:30-10:00 | Break | Break | Break | Break | Break |

10:00-11:30 | Gianluca Pacienza | Carlo Gasbarri* | Matt Satriano* | Matt Satriano* | Matt Satriano* |

11:30-14:00 | Lunch | Lunch | Lunch | Lunch | Lunch |

14:00-15:30 | Nathan Grieve | Nathan Grieve | Gregoire Menet | Gregoire Menet | Poster sessions and informal discussions |

15:30-16:00 | Break | Break | Break | Break | Break |

16:00-17:30 | Problem/tutorial sessions and informal discussions, or invited speaker | Invited speaker: Amos Turchet | Invited speaker: Laure Flapan | Invited speaker: Ljudmila Kamenova* (16:00-17:00) | Problem/tutorial sessions and informal discussions |

# Talks

**Laure Flapan**

**Kodaira dimension of some moduli spaces of hyperkähler manifolds**

We study the geometry of some moduli spaces of polarized hyperkähler manifolds. We use techniques of Gritsenko-Hulek-Sankaran involving the Borcherds modular form to determine a bound on the degree of the polarization after which these moduli spaces are always of general type. This is joint work with I. Barros, P. Beri, and E. Brakkee.

**Lie Fu**

**Orbifold products and hyper-Kähler resolution conjectures**

About 20 years ago, Chen and Ruan developed an orbifold Gromov-Witten theory and invented the so-called orbifold cohomology theory. Their construction received adaptations, refinements and reinterpretations in the algebraic setting thanks to the work of Fantachi-Göttsche, Jarvis-Kaufmann-Kimura, and Abramovich-Graber-Vistoli, etc. In my lectures, I will explain the construction of the orbifold product, formulate Ruan's hyper-Kähler resolution conjecture in the original cohomological settting, as well as in the motivic and K-theoretic settings, and then provide some evidences and applications. The talk is partially based on series of joint work with Manh Toan Nguyen, Zhiyu Tian and Charles Vial.

**Carlo Gasbarri**

**An informal (and hopefully friendly) introduction to stacks and orbifolds**

In this talk we will introduce the main geometrical ideas underlying the concept of stacks and orbifolds. Instead of focusing on the rigorous approach to them, we will try to describe DM stacks as natural geometrical objects not so different from varieties and schemes. We will try to explain why we need to introduce stacks, the functor of points, the basic ideas of descent theory and we will describe explicitly the construction of quotient and root stacks.

We will try to describe motivating examples, in particular the case of stacky curves (with, if there is time, some classical arithmetic applications due to Darmon and Granville).

**Nathan Grieve**

**Two introductory lectures about sites, stacks and orbifolds**

Lecture 1: I will recall various scheme and sheaf theoretic prerequisites which are needed in order to define and develop the theory of Grothendieck topologies, sites and Algebraic Spaces. I will also give context, motivation and examples for the general theory.

Lecture 2: I will give a reasonably self contained treatment of the most basic results from the theory of Deligne-Mumford stacks and the existence problem for coarse moduli spaces. I also intend to discuss in some detail the case of (global) quotient stacks, including a treatment of their Picard groups and category of quasi-coherent sheaves, the root stack construction and the construction of toric stacks via the theory of stacky fans. Finally, I intend to outline the main points in the construction of the Moduli stack of genus g >= 2 curves.

**Ljudmila Kamenova**

**Non-hyperbolicity of primitive symplectic varieties and Kobayashi's conjectures**

The Kobayashi pseudometric $d_M$ on a complex manifold \(M\) is the maximal pseudometric such that any holomorphic map from the Poincare disk to \(M\) is distance-decreasing. Kobayashi conjectured that this pseudometric vanishes on Calabi-Yau manifolds, and in particular, Calabi-Yau manifolds have "entire curves". Using ergodicity of complex structures, together with S. Lu and M. Verbitsky we proved this conjecture for all K3 surfaces and for many classes of hyperkähler manifolds. Here we give a generalization of these results to primitive symplectic varieties. This is a work in progress with Christian Lehn.

**Gregoire Menet**

**On compact hyperkähler orbifolds**

In order to classify the complex varieties, the hyperkähler manifolds have appeared as important objects because of their role of elementary bricks in the Bogomolov decomposition theorem. More recently this theory has been generalized to the singular setting. In this context, the hyperkähler orbifolds can be seen as one of the simplest generalization of the hyperkähler manifolds. In this lecture, we will see that many results from the smooth case have been generalized in the orbifold setting. Moreover, we will explain how to construct examples of hyperkähler orbifolds.

**Gianluca Pacienza**

**An invitation to compact hyperkähler manifolds**

In these lectures I will provide a gentle introduction to the theory of compact hyperkähler (HK) manifolds, with a view towards the advanced lectures that will be given by Lie Fu and Grégoire Menet. We will start from their place in the classification theory and provide several motivating examples. We will then insist on the (local and global) period map which makes the beauty and the richness of their theory. In passing we will of course survey the Beauville-Bogomolov-Fujiki quadratic form, the deformation theory of HK manifolds and the twistor families. If time permits we will touch upon some more geometric aspects of their theory such as the automorphisms of HK manifolds, their fibrations and some special subvarieties (eg rational curves and the loci they cover).

**Matt Satriano**

**Outline for lectures:**

Lecture 1: We will define root stacks, as introduced by Cadman as well as Abramovich, Graber and Vistoli. We will discuss the Chevalley-Shepahrd-Todd Theorem which characterizes when a quotient by a finite group is smooth. Following Vistoli's construction, we will then use this theorem to construct canonical stacks of finite quotient singularities.

Lecture 2: Making use of our constructions from the previous lecture, we will prove the "Bottom-up Theorem" which is a structure result for smooth Deligne-Mumford stacks. We will give several examples illustrating the key points of the theorem and, time permitting, discuss an application to non-commutative geometry.

Lecture 3: We discuss other "bottom-up" phenomena which extend beyond the Deligne-Mumford case such as the class of fantastacks. We will discuss connections with Grothendieck-Riemann-Roch as well as a recent resolution of singularities algorithm introduced by Abramovich and Quek.

**Amos Turchet**

**Non-special varieties and hyperbolicity**

We show how to construct several examples of non-special varieties where no étale cover admits a dominant map into a variety of general type (i.e. that are weakly-special). We show that these varieties possess Hyperbolicity properties that give evidence towards Campana's conjecture in the function field and in the analytic setting. The key input is a truncated version of the Ru-Vojta theorem. This is based on a series of joint works with Erwan Rosseau and Julie Wang.

**Julie Wang**

**Some cases of Campana’s orbifold conjecture**

We study some cases of Campana's orbifold conjecture for \(\mathbb P^2\) and finite ramified covers of \(\mathbb P^2\) with three components admitting sufficiently large multiplicities. We also prove a truncated second main theorem of level one for analytic maps into \(\mathbb P^2\) intersecting the coordinate lines in sufficiently high multiplicities.

This is a joint work with Ji Guo.