Higher Dimensional Algebraic Geometry: Classification, Minimal Models, Fano Varieties and Stacks

Home > What's on > Workshops > Higher Dimensional Algebraic Geometry: Classification, Minimal Models, Fano Varieties and Stacks

Higher Dimensional Algebraic Geometry: Classification, Minimal Models, Fano Varieties and Stacks

 22 - 26 Sep 2008
 ICMS, 14 India Street, Edinburgh

Organiser

Name
Institution
Cheltsov, IvanUniversity of Edinburgh
Corti, AlessioImperial College London
Reid, MilesUniversity of Warwick

Short Report

The two main objectives of this workshop were to put into focus rapid recent developments in the higher dimensional minimal model program (HD MMP), and to consolidate advances in the geometry of Fano and Calabi-Yau 3-folds that have arisen over the last decade. In addition, the meeting helped plan how to exploit new results, methods and problems arising from these developments.

The last few years have seen dramatic advances in HD MMP, with the proof of the existence of minimal models under appropriate conditions, and the prospect within a few years of having a complete generalisation of the minimal model program and the classification of varieties in all dimensions, comparable to the known foundational framework of the theory or surfaces and 3-folds. At the same time, other topics in algebraic geometry are enjoying prominent advances; these include birational geometry (especially the geometry of Fano varieties), the geometry of moduli spaces, resolution of singularities, orbifold cohomology, etc.

These topics feature in several international programs and conferences, including the 2007-08 Warwick EPSRC algebraic geometry symposium and the MSRI half-year program on Algebraic geometry Jan-May 2009. Some of the problem areas we covered are fast moving, but we also considered a number of more established questions.

Link to full report

Arrangements

Participation
Participation is by invitation only. The workshop will begin with registration at on Monday 22 September and finish on Friday 26 September 2008.

UK Visas
If you are travelling from overseas you may require an entry visa. A European visa does not guarantee entry to the UK. Please use this link to the UK Visas site to find out if you need a visa and if so how to apply for one. If you do require a visa, ICMS can provide a signed invitation letter.

Venue
The workshop will take place at the head-quarters of ICMS, 14 India Street, Edinburgh. This house is the birthplace of James Clerk Maxwell and is situated in the historic New Town of Edinburgh, near the city centre.

The ICMS travel pages contain advice on how to travel to Edinburgh. For local information the finding ICMS page shows the location of ICMS and contains useful maps of the city centre.

The seminar room at ICMS has whiteboards, 2 overhead projectors, a data projector and laptop.

Wireless access is available throughout the ICMS building. There are also five public PCs which may be used at any time for internet access and to check email.

Accommodation
ICMS will arrange single en-suite rooms in local guest houses for those who require it. Accommodation is typically about 15 to 30 minutes walk from ICMS. Participants are also free to make their own arrangements and may claim back the cost, with receipts, up to a maximum of £45.00 per night bed and breakfast. A list of Edinburgh accommodation of various sorts and prices is available here . Sections 1-3 are particularly relevant.

Meals and Refreshments
Buffet lunches will be provided on Monday 22 September and Friday 26 September only. For the remainder of the days, participants are free to go out for lunch and explore the many cafes, restaurants, sandwich shops and bars in the surrounding area. On arrival we will provide you with a ‘welcome’ pack which will contain information about eating places nearby.

Morning and afternoon refreshments will be provided throughout the workshop.

There will be an informal wine reception after the close of lectures on Monday 22 September.

On Tuesday 23 September there will be an informal dinner at a local restaurant.

The workshop dinner will take place on the evening of Thursday 25 September.

Registration
Registration will take place on Monday 22 September from 9.00 to 10.00.

Financial Arrangements
Unless otherwise specified in your invitation letter, the workshop grant will cover the cost of your bed and breakfast accommodation, tea/coffee throughout the workshop, lunch on Monday and Friday, the wine reception, the informal supper on Tuesday and the Workshop Dinner on Thursday evening.

If we have agreed to pay some of your travel costs, you will be informed in the 'final information' email. Reimbursement will take place after the workshop. At Registration you will be given an expenses claim form and this should be submitted to ICMS, with receipts. Please note that we cannot reimburse any item without a receipt. It would be helpful if you could bring your bank details to the workshop. In addition to the bank account number, participants from the USA and Canada will require their bank’s routing number, those from the UK will be asked for the bank sort code, and those from Europe and the rest of the world should supply their IBAN and SWIFT/BIC code.

Under the terms of our EPSRC funding we are required to charge a 30.00 GBP registration fee to cover costs not admissible under the grant. The fee will be payable on arrival at the workshop payment may be by cash, sterling cheque or credit/debit card. If you anticipate any difficulty covering the fee, please let me know.

Programme

PROVISIONAL (subject to change)

Monday 22 September

09.00 - 10.00

Registration

10.00 - 11.00

Jim Bryan (University of British Columbia)
The crepant resolution conjecture for Donaldson-Thomas theory and the orbifold topological vertex

11.00 - 11.30

Tea/Coffee

11.30 - 12.30

Frances Kirwan (University of Oxford)
Quotients by non-reductive group actions

12.30 - 14.00

Buffet lunch

14.00 - 15.00

Aleksandr Pukhlikov (University of Liverpool)
Birationally rigid Fano varieties

15.00 - 15.30

Tea/Coffee

15.30 - 16.30

Caucher Birkar (University of Cambridge)
Boundedness vs thresholds and flips

16.30 - 17.30

Nikos Tziolas (University of Cyprus)
Smoothings of schemes with non-isolated singularities

17.30 - 18.30

Wine reception at ICMS, 14 India Street

 

Tuesday 23 September

09.30 - 10.30

Hiroshi Iritani (Imperial College London)
Fourier-Mukai transformations and toric quantum cohomology
(Coates, Corti, Iritani and Tseng I)

10.30 - 11.00

Tea/Coffee

11.00 - 12.00

Tom Coates (Imperial College London / Royal Society)
Higher genus Gromov-Witten invariants and crepant resolutions
(Coates, Corti, Iritani and Tseng II)

12.00 - 13.00

Misha Verbitsky (Institute of Theoretical and Experimental Physics)
Nef classes on hyperkaehler manifolds

13.00 - 14.30

Lunch

14.30 - 15.30

Lawrence Ein (University of Illinois at Chicago)
Regularities of projective varieties

15.30 - 16.00

Tea/Coffee

16.00 - 17.00

Paolo Cascini (Imperial College London)
On the minimal model program

19.00 -

Informal evening meal at Nargile Turkish Restaurant, 73 Hanover Street

 

Wednesday 24 September

09.30 - 10.30

Valery Alexeev (University of Georgia)
Weighted hyperplane arrangements

10.30 - 11.00

Tea/Coffee

11.00 - 12.00

Gavin Brown (University of Kent)
Tom and Jerry and Sarkisov links on Fano 3-folds

12.00 - 13.00

Alexey Bondal (Aberdeen University)
Mirror symmetry for toric orbifolds

13.00 -

Free afternoon

 

Thursday 25 September

09.30 - 10.30

Vasily Iskovskikh (Steklov Mathematical Insitute)
On elements of prime order in the Cremona group of the plane over a perfect field

10.30 - 11.00

Tea/Coffee

11.00 - 12.00

Frederic Campana (Institut Élie Cartan Nancy)
Special orbifolds and classification

12.00 - 13.00

Dmitry Orlov (Steklov Mathematical Insitute)
Generators and dimensions of triangulated categories

13.00 - 14.30

Lunch

14.30 - 15.30

Jun-Muk Hwang (Korea Institute for Advanced Study)
Base manifolds for fibrations of projective irreducible symplectic manifolds

15.30 - 16.00

Tea/Coffee

16.00 - 17.00

Yuri Prokhorov (Moscow State University)
On Q-Fano threefolds

19.00 -

Workshop dinner at First Coast Restaurant, 99-101 Dalry Road, Edinburgh

 

Friday 26 September

09.30 - 10.30

Alessio Corti (Imperial College London)
Towards finite generation without the MMP

10.30 - 11.00

Tea/Coffee

11.00 - 12.00

Ludmil Katzarkov (University of Miami)
Generalized HMS and rationality questions

12.00 - 13.00

Dimitry Kaledin (Steklov Mathematical Insitute)
Motivic structures in non-commutative geometry

13.00 - 14.30

Buffet lunch

14.30 - 15.30

Mircea Mustata (University of Michigan)
An inductive approach to singularities of pairs

15.30 - 16.00

Tea/Coffee

16.00 - 17.00

TBA



Presentations

Alexeev, Valery
Weighted hyperplane arrangements
View Abstract
We give a common generalization of (1) Hassett's weighted stable curves, and (2) Hacking-Keel-Tevelev's stable hyperplane arrangements.
Birkar, Caucher
Boundedness vs thresholds and flips
View Abstract
I will describe a joint work in progress with Shokurov concerning several 3 or 4 dimensional conjectures on boundedness of Fano's, ACC, termination, and related problems.
Bondal, Alexei
Mirror symmetry for toric orbifolds
View Abstract
TBA
Brown, Gavin
Tom and Jerry and Sarkisov links on Fano 3-folds
View Abstract
I explain how to use Kustin-Miller unprojection (the original special case of Type I unprojection of Papadakis-Reid) to construct Fano 3-folds embedded anticanonically in codimension 4 in weighted projective space. For each numerical family, there are at least two distinct ways of setting up the unprojection data: Tom and Jerry. The construction is birational, and it provides the first step of a Sarkisov link. The conclusion of this link distinguishes the two cases.
Bryan, Jim
The crepant resolution conjecture for Donaldson-Thomas theory and the orbifold topological vertex
View Abstract
We formulate a conjectural relationship between the Donaldson-Thomas invariants of a three dimensional Calabi-Yau orbifold and the Donaldson-Thomas invariants of its canonical crepant resolution. We develop an orbifold version of the topological vertex formalism and use it to prove our crepant resolution conjecture in the case of toric orbifolds with transverse A_n singularities. Connections with the Gromov-Witten crepant resolution conjecture and implications for the orbifold GW/DT correspondence will be discussed, time permitting.
Campana, Frédéric
Special orbifolds and classification (subject to change)
View Abstract
Because of multiple fibres, birational classification of projective manifolds needs to consider the larger category of geometric orbifolds. The objects are the same as the pairs of the log-minimal model program. We define however morphisms only for smooth orbifolds. Special orbifolds are those not mapping rationally onto any positive-dimensional orbifold of general type. These are orbifold combination of those with vanishing Kodaira dimension and those which are rationally connected (in a weak sense). By one fibration, any orbifold decomposes canonically into its antithetical parts: special (the orbifold fibres) and general type (the orbifold base). This decomposition conjecturally splits the orbifold at the arithmetic and hyperbolic levels as well.
Cascini, Paolo
On the minimal model program
View Abstract
TBA
Coates, Tom
Higher genus Gromov-Witten invariants and crepant resolutions
View Abstract
Let X be a Gorenstein orbifold and Y a crepant resolution of X. Suppose that the quantum cohomology algebra of Y is semisimple. We describe joint work with Iritani which shows that in this situation the genus-zero crepant resolution conjecture implies a higher-genus version of the crepant resolution conjecture. We expect that the higher-genus version in fact holds without the semisimplicity hypothesis.
Corti, Alessio
Towards finite generation without the MMP
View Abstract
I will introduce work of Vlad Lazic on an approach to prove the finite generation of the canonical ring by induction on the dimension without usinig minimal model theory.
Ein, Lawrence
Regularities of projective varieties
View Abstract
Let X be a closed subscheme of P^N. The regularity of X measures the complexity in computing the syzygies of X. By a well known result of Castelnuovo and Mumford, the regularity of X is controlled by vanishing of higher cohomologies. Using Nadel's vanishing theorem and inversion of adjunction, we (joint work with Tommaso deFernex) show that we can a sharp upper bound for the regularities of X, if X has log canonical singularities.
Hwang, Jun-Muk
Base manifolds for fibrations of projective irreducible symplectic manifolds
View Abstract
Given a projective irreducible symplectic manifold M of dimension 2n, a projective manifold X and a surjective holomorphic map f:M to X with connected fibres of positive dimension, we prove that X is biholomorphic to the projective space of dimension n. The proof is obtained by exploiting two geometric structures at general points of X: the affine structure arising from the action variables of the Lagrangian fibration f and the structure defined by the variety of minimal rational tangents on the Fano manifold X.
Iritani, Hiroshi
Fourier-Mukai transformations and toric quantum cohomology
View Abstract
Based on the K-theory integral structure in quantum cohomology, I study the functoriality of toric quantum cohomology under birational morphisms. Using this, I will see how the Fourier-Mukai transformations are related to Yongbin Ruan's crepant resolution/flop conjecture.
Iskovskikh, Vasily
On elements of prime order in the Cremona group of the plane over a perfect field
View Abstract
We show that the Cremona group of the plane Cr(2,k) over a perfect field k contains an element of order l > 5 not equal to the characteristic if and only if there exists a 2-dimensional algebraic torus T over k such that T(k) contains an element of order l. If the characteristic is equal to zero and k does not contain a primitive l-th root of unity, we show that there are no elements of order l > 7 in Cr(2,k) and all elements of order 7 are conjugate.
Kaledin, Dmitry
Motivic structures in non-commutative geometry
View Abstract
Non-commutative, or "categorical" geometry asks which invariants of an algebraic variety X only depend on the category of coherent sheaves on X, and can be generalized to other abelian or A_infty categories. It has been known from early 80s that this is largely the case for the de Rham cohomology: it can computed as the "cyclic homology" of the category of sheaves. However, de Rham cohomology carries important additional structures -- a Hodge structure in characteristic 0, and a "filtered Dieudonne module" structure in characteristic p or over Z_p. I am going to report on a recent discovery: at least the latter can be defined on cyclic homology in complete generality. One corollary is a proof of a non-commutative Hodge-to-de Rham degeneration theorem by the method of Deligne-Illusie.
Katzarkov, Ludmil
Generalized HMS and rationality questions
View Abstract
In this talk we will discuss foundations of generalized homological mirror symmetry and its relation to birational geometry. Many examples will be worked out.
Kirwan, Frances
Quotients by non-reductive group actions
View Abstract
Mumford's geometric invariant theory (GIT) provides a method for constructing (projective completions of) quotient varieties for linear actions of reductive groups on projective varieties. This talk will discuss possible ways to extend some of the ideas of GIT to non-reductive group actions.
Mustata, Mircea
An inductive approach to singularities of pairs
View Abstract
I will explain how the description of singularities of pairs via spaces of arcs allows an approach to singularities of pairs by induction on dimension.
Orlov, Dmitry
Generators and dimensions of triangulated categories
View Abstract
We show that the dimension of the bounded derived category of coherent sheaves on a smooth quasi-projective curve is equal to one.
Prokhorov, Yuri
On Q-Fano threefolds
View Abstract
We discuss Q-Fano threefolds of large (Fano) index and degree.
Pukhlikov, Aleksandr
Birationally rigid Fano varieties
View Abstract
I will discuss the linear and quadratic methods of proving birational (super)rigidity of Fano varieties and fibre spaces.
Tziolas, Nikolaos (Nikos)
Smoothings of schemes with non-isolated singularities
View Abstract
I will describe the deformation and Q-Gorenstein deformation theory of a scheme X with positive dimensional singular locus and obtain criteria for the existence of smoothings and Q-Gorenstein smoothings of X.
Verbitsky, Misha
Nef classes on hyperkaehler manifolds
View Abstract
The SYZ conjecture claims that any non-ample nef bundle on a holomorphic symplectic manifold is semiample, and corresponds to a Lagrangian fibration. When a nef bundle L admits a metric with semipositive curvature, cohomology of L are in correspondence with holomorphic L-valued forms on M. Using stability and divisorial Zariski decomposition due to S. Boucksom, we show that some power of L is effective. This method can be modified using multiplier ideals and Nadel vanishing to show that any nef bundle on a holomorphic symplectic manifold has a tensor power which is effective.

Participants

Name
Institution
Valery, AlexeevUniversity of Georgia
Marco, AndreattaUniversity of Trento
Caucher, BirkarUniversity of Cambridge
Alexei, BondalSteklov Mathematical Institute/Higher School of Economics
Gavin, BrownUniversity of Kent
Jim, BryanUniversity of British Columbia
Frédéric, CampanaUniversité de Lorraine
Paolo, CasciniImperial College London
Ivan, CheltsovUniversity of Edinburgh
Circo, CilibertoUniversity of Rome Tor Vergata
Tom, CoatesImperial College London
Alessio, CortiImperial College London
Alastair, CrawUniversity of Bath
Tommaso, de FernexUniversity of Utah
Lawrence, EinUniversity of Illinois at Chicago
Sergey, GalkinSteklov Institute of Mathematics
Elizabeth, GasparimUniversity of Edinburgh
Jun-Muk, HwangKorea Institute for Advanced Study
Hiroshi, IritaniImperial College London
Vasily, IskovskikhSteklov Mathematical Insitute
Dmitry, KaledinIndependent University of Moscow
Anne-Sophie, KaloghirosBrunel University London
Ludmil, KatzarkovUniversity of Miami
Frances, KirwanUniversity of Oxford
Alexei, KovalevUniversity of Cambridge
Stéphane, LamyUniversity of Warwick
Vladimir, LazićUniversität Bonn
Timothy, LogvinenkoUniversity of Liverpool
Angelo, LopezUniversity of Roma Tre
Dibiagio, LorenzoUniversity of Rome, La Sapienza
Mircea, MustataUniversity of Michigan
Dmitry, OrlovSteklov Mathematical Insitute
Dmitri, PanovKing's College London
Jihun, ParkPohang University of Science and Technology
Yuri, ProkhorovMoscow State University
Aleksandr, PukhlikovUniversity of Liverpool
Miles, ReidUniversity of Warwick
Daniel, RyderUniversity of Bristol
Gregory, SankaranUniversity of Bath
Nikolaos (Nikos), TziolasUniversity of Cyprus
Misha, VerbitskyInstitute of Theoretical and Experimental Physics
Pelham, WilsonUniversity of Cambridge
Jaroslaw, WisniewskiUniversity of Warsaw