ICMS

Sep 22, 2008 - Sep 26, 2008

ICMS, 14 India Street, Edinburgh

Organisers

Name	Institution
Cheltsov, Ivan	University of Edinburgh
Corti, Alessio	Imperial College London
Reid, Miles	University 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:

Presentation Details
Alexeev, Valery
Weighted hyperplane arrangements
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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 Hide 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, Alexey
Mirror symmetry for toric orbifolds
View Abstract Hide Abstract
TBA
Brown, Gavin
Tom and Jerry and Sarkisov links on Fano 3-folds
View Abstract Hide 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
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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, Frederic
Special orbifolds and classification (subject to change)
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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
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TBA
Coates, Tom
Higher genus Gromov-Witten invariants and crepant resolutions
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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
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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
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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
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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
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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
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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, Dimitry
Motivic structures in non-commutative geometry
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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
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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
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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
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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
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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
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We discuss Q-Fano threefolds of large (Fano) index and degree.
Pukhlikov, Aleksandr
Birationally rigid Fano varieties
View Abstract Hide Abstract
I will discuss the linear and quadratic methods of proving birational (super)rigidity of Fano varieties and fibre spaces.
Tziolas, Nikos
Smoothings of schemes with non-isolated singularities
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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
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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
Alexeev, Valery	University of Georgia
Andreatta, Marco	University of Trento
Birkar, Caucher	University of Cambridge
Bondal, Alexey	University of Aberdeen
Brown, Gavin	University of Kent
Bryan, Jim	University of British Columbia
Campana, Frederic	Institut Élie Cartan Nancy
Cascini, Paolo	Imperial College London
Cheltsov, Ivan	University of Edinburgh
Ciliberto, Circo	University of Rome Tor Vergata
Coates, Tom	Imperial College London / Royal Society
Corti, Alessio	Imperial College London
Craw, Alastair	University of Glasgow
de Fernex, Tommaso	University of Utah
Ein, Lawrence	University of Illinois at Chicago
Galkin, Sergey	Steklov Institute of Mathematics
Gasparim, Elizabeth	University of Edinburgh
Hwang, Jun-Muk	Korea Institute for Advanced Study
Iritani, Hiroshi	Imperial College London
Iskovskikh, Vasily	Steklov Mathematical Insitute
Kaledin, Dimitry	Steklov Mathematical Insitute
Kaloghiros, Anne-Sophie	University of Cambridge
Katzarkov, Ludmil	University of Miami
Kirwan, Frances	University of Oxford
Kovalev, Alexei	University of Cambridge
Lamy, Stéphane	Université Claude Bernard Lyon 1
Lazic, Vladimir	University of Cambridge
Logvinenko, Timothy	University of Liverpool
Lopez, Angelo	University of Roma Tre
Lorenzo, Dibiagio	University of Rome, La Sapienza
Mustata, Mircea	University of Michigan
Orlov, Dmitry	Steklov Mathematical Insitute
Panov, Dimitri	Imperial College London
Park, Jihun	Pohang University of Science and Technology
Prokhorov, Yuri	Moscow State University
Pukhlikov, Aleksandr	University of Liverpool
Reid, Miles	University of Warwick
Ryder, Daniel	University of Bristol
Sankaran, Gregory	University of Bath
Tziolas, Nikos	University of Cyprus
Verbitsky, Misha	Institute of Theoretical and Experimental Physics
Wilson, Pelham	University of Cambridge
Wisniewski, Jaroslaw	University of Warsaw