Generalised pairs are, as the name suggests, generalisations of pairs in algebraic geometry. A pair is roughly an algebraic variety together with a boundary divisor. A generalised pair is roughly a pair together with the choice of a "positive" divisor (i.e. nef divisor) on some birational model of the pair. The theory of generalised pairs is very interesting on its own but it has also been instrumental in many advances in birational geometry in recent years including effectivity of Iitaka fibrations, boundedness of Fano varieties, boundedness of complements, boundedness of log Calabi-Yau varieties, moduli of stable Calabi-Yau and stable minimal models, termination of flips and existence of minimal models, connectedness of non-klt loci, etc. The aim of this workshop is to review the theory and some of its applications.

Time: August 26-27, 2021

Invited Speakers:

Caucher Birkar	Tsinghua University
Stefano Filipazzi	École Polytechnique Fédérale de Lausanne
Christopher Hacon	University of Utah
Jingjun Han	Johns Hopkins University
Kenta Hashizume	University of Tokyo
Zhengyu Hu	Chongqing University of Technology
Vladimir Lazic	Saarland University
Jihao Liu	University of Utah
Joaquin Moraga	Princeton University
Thomas Peternell	University of Bayreuth
V. V. Shokurov	Johns Hopkins University
Roberto Svaldi	École Polytechnique Fédérale de Lausanne

Organizers:

Shing-Tung Yau	Tsinghua University
Caucher Birkar	Tsinghua University
Yifei Chen	AMSS
Baohua Fu	MCM, AMSS

Sponsors:

Yau Mathematical Sciences Center, Tsinghua University

Morningside Center of Sciences, Chinese Academy of Sciences

ZOOM:

Zoom ID: 466 356 2952 Password: mcm1234

Schedule:

Date	Chair	Time	Speaker	Title	Video
8.26	Shing-Tung Yau	8:20-8:30	Shing-Tung Yau	Opening Ceremony	Video
	Baohua Fu	8:30-9:30	Caucher Birkar	An overview of generalised pairs	Video& PPT
		10:00-11:00	Christopher Hacon	On the minimal model program for generalized log pairs	Video& PPT
		11:30-12:30	Jihao Liu	Existence of flips for generalized pairs	Video& PPT

8.26	Caucher Birkar	14:00-15:00	Stefano Filipazzi	On the connectedness principle and dual complexes for generalized pairs	Video& PPT
		15:30-16:30	V. V. Shokurov	$n$-complements	Video
		17:00-18:00	Vladimir Lazic	Weak Zariski decompositions and minimal models	Video

8.27	Yifei Chen	8:30-9:30	Joaquin Moraga	Toroidalization principles for generalized klt singularities	Video& PPT
		10:00-11:00	Zhengyu Hu	An abundance theorem for generalised pairs	Video
		11:30-12:30	Kenta Hashizume	Non-vanishing theorem for generalized log canonical pairs with a polarization	Video& PPT

8.27	Caucher Birkar & Yifei Chen	14:00-15:00	Jingjun Han	Fujita's conjecture for pseudo-effective thresholds and Shokurov's conjecture on iterated accumulation points of pseudo-effective thresholds	Video
		15:30-16:30	Roberto Svaldi	A characterization of toricness	Video& PPT
		17:00-18:00	Thomas Peternell	A generalized non-vanishing and abundance conjecture and nef line bundles on K-trivial varieties	Video

8.24 Chen Jiang

8.24 Jingjun Han

Title & Abstract:

Speaker: Caucher Birkar (Tsinghua University)

Title: An overview of generalised pairs

Abstract: In this talk I will give a general overview of the theory of generalised pairs and its applications.

Speaker: Christopher Hacon (University of Utah)

Title: On the minimal model program for generalized log pairs

Abstracts: In this talk I will discuss recent progress in the minimal model program for log canonical generalized pairs.

Speaker: Jihao Liu (University of Utah)

Title: Existence of flips for generalized pairs

Abstracts: Following Prof. Hacon's talk, I will discuss the existence of flips for log canonical generalized pairs in detail. I will talk about some key ideas and philosophy in the proofs of the existence of flips, cone theorem, and contraction theorems for log canonical generalized pairs. I will also discuss some related questions and potential applications. This is joint work with Christopher D. Hacon.

Speaker: Stefano Filipazzi (École Polytechnique Fédérale de Lausanne)

Title: On the connectedness principle and dual complexes for generalized pairs

Abstracts: Let $(X,B)$ be a pair, and let $f \colon X \rightarrow S$ be a contraction with $-(K_X+B)$ nef over S. A conjecture, known as the Shokurov-Koll\'ar connectedness principle, predicts that $f^{-1}(s)$ intersect $\mathrm{Nklt}(X,B)$ has at most two connected components, where $s$ is an arbitrary point in $S$ and $\mathrm{Nklt}(X,B)$ denotes the non-klt locus of $(X,B)$. The conjecture is known in some cases, namely when $-(K_X+B)$ is big over $S$, and when it is $\mathbb{Q}$-trivial over $S$. In this talk, we discuss a proof of the full conjecture and extend it to the case of generalized pairs. Then we apply it to the study of the dual complex of generalized log Calabi--Yau pairs. This is joint work with Roberto Svaldi.

Speaker: V.V. Shokurov (Johns Hopkins University)
Title: $n$-complements
Abstracts: I will recall main results about $n$-complements and
will discuss the role of generalized pairs of Birkar-Zhang
vs Alexeev pairs.

Speaker: Vladimir Lazic (Saarland University)

Title: Weak Zariski decompositions and minimal models

Abstracts: I will present recent results which show that, modulo reasonable assumptions in lower dimensions, the existence of a weak Zariski decomposition of a Q-factorial log canonical generalised pair is equivalent to the existence of a minimal model of the generalised pair. This leads to new unconditional results on the existence of minimal models and Mori fibre spaces of generalised pairs in dimensions at most 5. I will argue that even if one is only interested in the birational geometry of varieties, one cannot avoid the use of generalised pairs. This is joint work with Nikolaos Tsakanikas.

Speaker: Joaquin Moraga (Princeton University)

Title: Toroidalization principles for generalized klt singularities

Abstract: In this talk, I will discuss some recent progress on toroidalization principles for generalized klt singularities. These toroidalizations allow us to prove theorems about the topology of klt singularities and about their minimal log discrepancies. If time permits, I will also explain the relationship between these toroidalization principles and the termination of flips.

Speaker: Zhengyu Hu (Chongqing University of Technology)

Title: An abundance theorem for generalised pairs

Abstract: In this talk I will discuss the finiteness of B-representations for generalised pairs with "general" data. As an application, I will discuss an abundance theorem for generalised dlt pairs, under an extra technical assumption. I will also discuss related problems regarding the abundance theorem.

Speaker: Kenta Hashizume (University of Tokyo)

Title: Non-vanishing theorem for generalized log canonical pairs with a polarization

Abstracts: In this talk, I will deal with generalized pairs with a polarization. I will explain that the non-vanishing theorem holds for generalized pairs with a polarization under assumptions on the nef part and the log canonical part of the generalized pairs. I will also discuss some related topics for generalized pairs with a polarization.

Speaker: Jinjun Han (Johns Hopkins University)

Title: Fujita's conjecture for pseudo-effective thresholds and Shokurov's conjecture on iterated accumulation points of pseudo-effective thresholds

Abstracts: Fujita's conjecture for pseudo-effective thresholds predicts that the set of pseudo-effective thresholds is an ACC set. It is an analogy to ACC for log canonical thresholds. Shokurov's conjecture on iterated accumulation points of pseudo-effective thresholds can be viewed as an analogy to the accumulation points theorem of log canonical thresholds. I will report some progresses towards these two conjectures by using tools from generalized pairs which are developed by Birkar-Zhang. This is based on joint work with Zhan Li.

Speaker: Roberto Svaldi (École Polytechnique Fédérale de Lausanne)

Title: A characterization of toricness

Abstracts: For a log canonical pair (X, D), with -(K_X+D) nef, Shokurov conjectured that a certain numerical quantity, called the complexity, measures how far the pair is from being a toric pair. Shokurov's conjecture actually anticipates a similar behavior in the relative setting, too. In this talk, I will explain how a solution to the above conjecture has emerged in the last few years and how it is related to recent developments in birational geometry. This talk is features joint works with Brown, McKernan, Zong and with Moraga.

Speaker: Thomas Peternell (University of Bayreuth)

Title: A generalized non-vanishing and abundance conjecture and nef line bundles on K-trivial varieties

Abstracts: I will report on joint work, partially in progress, with V. Lazic and K.Oguiso/V.Lazic concerning a nonvanishing/abundance type conjecture which involves a nef line bundle. Special emphasis will be laid on varieties whose canonical bundle is numerically trivial.