Orbifolds and Birational Geometry
Lead Research Organisation:
University of Warwick
Department Name: Mathematics
Abstract
Our main object of study are orbifolds - roughly speaking, spaces which are locally the quotient M/G of a complex manifold M by the action of a finite group G.Subgroups of linear groups play a prominent role in geometry and algebra at all levels: finite subgroups of rotations of 3-space appear as the symmetry groups of regular solids, that is, the regular cylinders and the famous Platonic solids. The finite subgroups of SL(2,\C) are classified as an ADE scheme, with cyclic groups, binary groups and groups obtained as spinor double covers of the Platonic groups. The quotients of complex 2-space \C^2 by these groups are a famous family of surface singularities studied by Felix Klein around 1870, and by Coxeter and Du Val in the 1930s; each of these singularities has a resolution by a surface containing a bunch of algebraic curves (spheres) with configuration graph the same ADE diagram. As a notable example, the binary icosahedral group gives the singular hypersurface (x^2+y^3+z^5 = 0) in complex 3-space \C^3, that appears in many guises in topology and algebra, and whose resolution is the Dynkin diagram E8.John McKay observed in the 1980s that the ADE resolution graph of the Klein singularities is given by the McKay quiver in the representation theory of the abstract group G. This observation was translated into geometry by Gonzales-Sprinberg and Verdier in 1978: an irreducible representation of G gives a tautological vector bundle on the resolved orbifold, and the classes of these bundles base the K theory. Conjectures of Reid from the early 1990s generalized this McKay correspondence to higher dimensions, and the subject has grown from then into a whole field of study, exploring the rich and intricate relations between the equivariant geometry of M (that is, primarily, the representation theory of G) and the geometry of the resolved orbifold. Reid's 1999 Bourbaki seminar includes a colloquial summary of these matters; the correspondence takes place on many levels - cohomology, K-theory, derived categories, motivic integration, as moduli spaces, as stacks, and so on.Orbifold geometry appears in many quite different contexts, including the 3-fold terminal and canonical singularities of minimal model theory, moduli space problems, the geometry of varieties in weighted projective spaces or other toric ambient spaces and the representation theory surrounding the McKay correspondence, as exploited notably by Mark Haiman. More generally, one need not insist that the local cover M is nonsingular, leading for example to the hyperquotient singularities (hypersurface singularity divided by a group action) that play an important role in Mori and Reid's study of 3-fold terminal singularities.
Planned Impact
Based on the existing influence of Reid's work on birational geometry of 3-folds and on the McKay correspondence, we confidently expect that our project will have major impact in each of the following areas: (1) the higher dimensional minimal model program (2) the classification of higher dimensional varieties (3) the biregular treatment in explicit terms of Fano varieties, canonical n-folds and Calabi-Yau varieties (4) the birational geometry of uniruled higher dimensional varieties (5) the resolution of orbifold singularities and the McKay correspondence (6) singularity theory (7) toric geometry (8) derived categories (9) representation theory Mori theory and the existence of relative minimal and canonical models have traditionally provided important insight into the study of singularities of varieties and their deformation theory. The orbifold RR formula, which deals with weighted projective phenomomena is an important component of this theory. Working systematically with orbifolds, both in the stacky and the birational language provides a bridge between equivariant methods and resolution of singularities. The McKay correspondence relates equivariant geometry of orbifolds and K-trivial resolution of singularities; it exists in many different languages, notably the motivic integration form and the derived category form of Bridgeland, King and Reid, and is an essential link between representation theory and geometry.
Organisations
Publications
Anno R
(2012)
On adjunctions for Fourier-Mukai transforms
in Advances in Mathematics
Anno R
(2016)
Orthogonally spherical objects and spherical fibrations
in Advances in Mathematics
Anno R
(2017)
Spherical DG-functors
in The Journal of the European Mathematical Society
Anno Rina
(2010)
Orthogonally spherical objects and spherical fibrations
in arXiv e-prints
Anno Rina
(2013)
Spherical DG-functors
in arXiv e-prints
Buckley A
(2013)
Ice cream and orbifold Riemann-Roch
in Izvestiya: Mathematics
Cautis S
(2017)
Derived Reid's recipe for abelian subgroups of SL 3 (C)
in Journal für die reine und angewandte Mathematik (Crelles Journal)
Cautis S
(2014)
Erratum to A derived approach to geometric McKay correspondence in dimension three (J. reine angew. Math. 636 (2009), 193-236)
in Journal für die reine und angewandte Mathematik (Crelles Journal)
Cautis S
(2012)
Derived Reid's recipe for abelian subgroups of SL3(C)
Davis Sarah Elizabeth
(2012)
Crepant resolutions and A-Hilbert schemes in dimension four
Description | (1) Timothy Logvinenko, with Ali Craw and Sabin Cautis, had successfully extended derived Reid's recipe, the threefold generalisation of the classical McKay correspondence, for finite Abelian groups G in SL(3,C) from the isolated singularties to the fully general case and published the results in the paper "Derived Reid's recipe for Abelian subgroups of SL(3,C)". (2) Reid's EPSRC funded student Sarah Davis (together with Reid and Logvinenko) worked on understanding the geometry of G-Hilb(CC^n) scheme for the groups of 1/n(a,b,1,,1) type. This is the first time that the G-Hilbert scheme was understood for a whole class of groups in the dimension greater than 3. The most striking result gives infinitely many examples of groups for which a crepant resolution exists, and infinitely many examples fo which no such resolution exists. It strongly suggests that the probability of a crepant resolution existing in dimension >= 4 is less that 1/2^k, where k is any measure of the complexity of the group. The results are in progress, with a preliminary draft available "How to calculate A-Hilb C^n for 1/r(a,b,1,,1)". (3) Timothy Logvinenko and Rina Anno laid down foundations for general theory of spherical functors, beginning with basics on Fourier-Mukai adjunctions in the paper "On adjunctions for Fourier-Mukai transforms", then treating the case of categorical fibrations in "Orthogonally spherical objects and spherical fibrations", and finally giving a full treatment of the general case in "Spherical DG-functors". (4) Reid's Korean PhD student JUNG Seung-Jo showe that, similar to G-Hilb, the well-known "economic" resolution of a 3-fold terminal quotient singularity can be obtained as a moduli space of theta-stable G-constellations. This appeared as the paper "Terminal Quotient Singularities in Dimension three via variation of GIT". (5) JUNG also worked on extending the Craw-Ishii result that for G in SL(3,CC) every relative minimal model of a quotient abelian singularity CC^3/G can be realised as a moduli space theta-stable G-constellations. Seung-Jo tried to extend this to those G in GL(3,C) where relative minimal models are smoooth. Partial results appeared in his paper "On the Craw-Ishii conjecture"." (6) Together with a number of PhD and u/g project students, Reid worked on understanding the G-Hilb scheme for the trihedral subgroups of SL(3,C), generalising the G-graphs of Nakamura to "trihedral boats", a new phenomena characteristic to the G-Hilb of trihedral subgroups of SL(3,CC). |
Exploitation Route | Concrete understanding of the geometry of G-Hilb scheme and of the derived Reid's recipe is of huge importance to anyone trying to understand the birational geometry of orbifolds. The papers which take forward ideas studied under this grant include "Geometric Reid's recipe for dimer models" by Raf Bocklandt, Alastair Craw, and Alexander Quintero Velez, "Flops and mutations for crepant resolutions of polyhedral singularities" by Alvaro Nolla de Celis and Yuhi Sekiya, and the thesis-in-progress of EPSRC funded PhD student Sara Muhvic. |
Sectors | Other |
URL | http://www.cantab.net/users/t.logvinenko |
Description | Chancellor's scholarship |
Amount | £80,000 (GBP) |
Organisation | University of Warwick |
Sector | Academic/University |
Country | United Kingdom |
Start | 10/2010 |
End | 09/2013 |
Description | Korean Government scholarship |
Amount | £30,000 (GBP) |
Organisation | Government of South Korea |
Department | Ministry of Education |
Sector | Public |
Country | Korea, Republic of |
Start | 10/2010 |
End | 09/2012 |
Description | Newton International Fellowship |
Amount | £63,000 (GBP) |
Organisation | The Royal Society |
Sector | Charity/Non Profit |
Country | United Kingdom |
Start | 01/2013 |
End | 12/2014 |
Description | Royal Society International Joint Project |
Amount | £12,000 (GBP) |
Organisation | The Royal Society |
Sector | Charity/Non Profit |
Country | United Kingdom |
Start | 07/2011 |
End | 06/2012 |
Description | University of Warwick Institute for Advanced Studies |
Amount | £15,000 (GBP) |
Funding ID | IAVS 1103 |
Organisation | University of Warwick |
Department | Institute of Advanced Study |
Sector | Academic/University |
Country | United Kingdom |
Start | 06/2012 |
End | 06/2012 |
Description | Warwick EPSRC Symposium on Derived Categories and Applications |
Amount | £159,849 (GBP) |
Funding ID | EP/L018314/1 |
Organisation | Engineering and Physical Sciences Research Council (EPSRC) |
Sector | Public |
Country | United Kingdom |
Start | 09/2014 |
End | 08/2015 |
Description | AGTP: School on Algebraic Geometry and Theoretical Physics |
Form Of Engagement Activity | Participation in an activity, workshop or similar |
Part Of Official Scheme? | No |
Geographic Reach | International |
Primary Audience | Professional Practitioners |
Results and Impact | AGTP: School on Algebraic Geometry and Theoretical Physics Mon 9th-Sat 14th July 2012, University of Warwick Partly funded by an interdisciplinary grant of GBP9000 from University of Warwick Institute for Advanced Studies Organised by Timothy Logvinenko and Miles Reid Lecture courses by Paul Aspinwall (Duke), Mark Gross (San Diego), Amihay Hanany (Imperial), Alexander Kuznetsov(Steklov Institute) and Balázs Szendroi (Oxford) Additional lectures by Pawel Sosna (Universität Hamburg), John Calabrese(Oxford), Richard Thomas (Imperial), Paul Hacking (UMass/Amherst), Timothy Logvinenko (Warwick), Agnieszka Bodzenta-Skibinska (Warsaw), Matthew Ballard (Wien), Hokuto Uehara (Tokyo Metropolitan University), Miles Reid (University of Warwick), Program http://www2.warwick.ac.uk/fac/sci/maths/research/events/2011-2012/agtp/ |
Year(s) Of Engagement Activity | 2012 |
URL | http://www2.warwick.ac.uk/fac/sci/maths/research/events/2011-2012/agtp/ |
Description | Geometry and Algebra of Orbifolds and the McKay Correspondence |
Form Of Engagement Activity | Participation in an activity, workshop or similar |
Part Of Official Scheme? | No |
Geographic Reach | International |
Primary Audience | Professional Practitioners |
Results and Impact | Geometry and Algebra of Orbifolds and the McKay Correspondence, Aug 9th-14th 2010 Mon 9th - Sat 14th Aug 2010 Organisers: Timothy Logvinenko (Warwick) and Miles Reid (Warwick) in collaboration with Yukari Ito (Nagoya) and Ali Craw (Glasgow), and continues the Nagoya workshop 8th-10th Mar 2010. The topics include McKay correspondence, explicit construction of resolutions as variants on G-Hilb, relations with noncommutative algebra and representations of quivers and orbifold RR. The workshop is the final follow-up event of EPSRC 2007-08 Warwick symposium on algebraic geometry. However, the WAG 07-08 money is now long gone, and at present Warwick does not have much funds to support this activity. Confirmed Participants include: Alastair Craw (Glasgow), Alastair King (Bath), Alessio Corti (Imperial), Alexander Quintero (Glasgow), Alvaro Nolla (Nagoya), Anita Buckley (Ljubljana), Balázs Szendröi (Oxford), Barbara Fantechi (SISSA, Trieste), Dai Evans (Cardiff), David Ploog (Hannover), Diane Maclagan (Warwick), Elena Andreini (SISSA), Fabio Nironi (Columbia), HAYASHI Toshihiro (Nagoya), ISHII Akira (Hiroshima), ITO Yukari (Nagoya), Mark Blume (Münster), Markus Perling (Bochum), Miles Reid (Warwick), Nathan Broomhead (Hannover), Oskar Kedzierski (Warsaw), Raf Bocklandt (Newcastle), Ryo Ohkawa (TITech), SEKIYA Yuhi (Nagoya), Stéphane Lamy (Warwick), Stephen Coughlan (Bayreuth), Tarig Abdel Gadir (Glasgow), Timothy Logvinenko (Warwick), UEHARA Hokuto (Tokyo Metropolitan), Graduate Students: Fabio Tonini (Scuola Normale, Pisa) HAYASHI Toshihiro (Nagoya), SEKIYA Yuhi (Nagoya), TAKAHASHI Keisuke (Nagoya), Tarig Abdel Gadir (Glasgow), Tom Sutherland (Sheffield), Warwick Graduate Students: Andrew Chan, Eduardo Dias, JUNG Seung-Jo, Martha Bernal, Michael Selig, Michele Torielli, Piotr Zwiernik, Sam Derbyshire, Sarah Davis, Sohail Iqbal, Umar Hayat, ZHOU Shengtian, For more informations, see http://www2.warwick.ac.uk/fac/sci/maths/research/events/2009_2010/workshops/orb Poster http://www2.warwick.ac.uk/fac/sci/maths/research/events/2009_2010/workshops/orb/orbifolds_2.pdf Program http://www2.warwick.ac.uk/fac/sci/maths/research/events/2009_2010/workshops/orb/orbifolds_programme.pdf |
Year(s) Of Engagement Activity | 2010 |
URL | http://www2.warwick.ac.uk/fac/sci/maths/research/events/2009_2010/workshops/orb |
Description | Homological Projective Duality and Non-commutative Geometry |
Form Of Engagement Activity | Participation in an activity, workshop or similar |
Part Of Official Scheme? | No |
Geographic Reach | International |
Primary Audience | Professional Practitioners |
Results and Impact | Mon 8th-Sat 14th October 2012 Homological Projective Duality and Non-commutative Geometry Organiser: Timothy Logvinenko Lecture courses by Alexander Kuznetsov, Dmitri Kaledin and Alexander Efimov. Additional lectures by Hwayoung Lee (KIAS), Timothy Logvinenko (Warwick), Igor Netay (HSE, Moscow), Andrew MacPherson (Imperial), Andrey Trepalin (HSE, Moscow), Tom Bridgeland (Oxford), Artan Sheshmani (Max Planck) The workshop is centered around three minicourses on homological methods in algebraic geometry delivered by Alexander Kuznetsov, Dmitri Kaledin and Alexander Efimov of the Steklov Mathematical Institute (Moscow). The lectures should, in principle, be accessible to PhD students and postdocs with appropriate background. There will also be several stand-alone talks on current research problems. The workshop incorporates a CALF day on Wed Oct 10th and a COW day (in Oxford) on Thu Oct 11th, and traditional events such as a curry evening and Saturday pub lunch. Abstracts http://www2.warwick.ac.uk/fac/sci/maths/research/events/2012-2013/nonsymp/hpd/abstracts/ Program http://www2.warwick.ac.uk/fac/sci/maths/research/events/2012-2013/nonsymp/hpd |
Year(s) Of Engagement Activity | 2012 |
URL | http://www2.warwick.ac.uk/fac/sci/maths/research/events/2012-2013/nonsymp/hpd |
Description | Russian-British Winter School on the McKay Correspondence |
Form Of Engagement Activity | Participation in an activity, workshop or similar |
Part Of Official Scheme? | No |
Geographic Reach | International |
Primary Audience | Professional Practitioners |
Results and Impact | Russian-British Winter School on the McKay Correspondence, Mon 20th-Sat 25th Feb 2012 This school was partly supported by the Royal Society International Joint Project "Varieties, orbifolds and derived categories" Organisers Miles Reid (Warwick), Timothy Logvinenko (Warwick), Costya Shramov (Steklov Institute RAS) Speakers: Artem Avilov, Chris Brav, Kuzma Khrabrov, Timothy Logvinenko, Alvaro Nolla, Miles Reid, Taro Sano, Constantin Shramov, Evgeny Smirnov, Michael Wemyss Lectures on Hilbert schemes, G-Hilbert Schemes, the McKay correspondence, Quiver Reps and Stability, McKay correspondence in the style of Auslander and related topics. Preschool reading: http://www2.warwick.ac.uk/fac/sci/maths/research/events/2011-2012/rbws/abstracts/ http://www2.warwick.ac.uk/fac/sci/maths/research/events/2011-2012/rbws/literature/ Program http://www2.warwick.ac.uk/fac/sci/maths/research/events/2011-2012/rbws/ |
Year(s) Of Engagement Activity | 2012 |
URL | http://www2.warwick.ac.uk/fac/sci/maths/research/events/2011-2012/rbws/ |