Maria Sk\lodowska-Curie graduate school Threefolds in algebraic geometry (3-fAG) reference number MCFH-1999 00687 Research to be supported: Classification of surfaces and 3-folds. Group actions, singularities and McKay correspondence. Calabi-Yau 3-folds and mirror symmetry, related algebra, Kaehler and symplectic geometry and string theory in physics. Keywords: 1. Classification of varieties 2. Resolution of singularities 3. Mirror symmetry 4. String theory Fellows requested Number of researcher-months: 72 Indicative number of fellows: 10--12 Legal conditions: A candidate must be a citizen of the EU or an associated state, and must be registered for a Ph.D. in an EU country other than the UK. Studentships can only be awarded for a period between 3 months and 1 year. The geometry group at Warwick invites applications for postgraduate fellowships to be held in the EU Framework 5 Maria Sklodowska-Curie training site "Threefolds in algebraic geometry". The research topics covered by this program include the following: 3-folds, minimal models and classification, the birational geometry of Mori fibre spaces; Calabi-Yau 3-folds and mirror symmetry; group actions, quotient singularities and McKay correspondence; related areas of the classification of varieties, symplectic geometry, Kaehler geometry, string theory in theoretical physics, singularity theory and commutative algebra. These topics here will be interpreted broadly. The Warwick geometry group currently includes Mark Gross, John Jones, Mario Micallef, David Mond, John Moody, John Rawnsley, Miles Reid, Dmitry Rumynin, Balazs Szendroi, in addition to Marie Curie fellows Adrian Langer and Nikos Tziolas, and about 12 graduate students. The Maria Sklodowska-Curie training site program will cover two or three postgraduate fellowships each year (lasting for 3 to 12 months, with preference given to longer stays), from a starting date in mid 2000. Those interested should contact Prof Miles Reid (e-mail Miles@Maths.Warwick.Ac.UK). For further details, see www.maths.warwick.ac.uk/research. Recruitment plan: 1st year 2000 3 appointments 2nd year 2001 3--4 appointments 3rd year 2002 3--4 appointments 4th year 2003 1 appointments Description of research area of the doctoral training: The geometry group at Warwick offers training to postgraduate students in a broad area of algebraic geometry centred around three topics in 3-folds: (1) Mori theory of minimal models and classification of 3-folds, including the birational geometry of Mori fibre spaces; (2) Calabi--Yau 3-folds and mirror symmetry, including the Strominger-- Yau--Zaslow approach to mirror symmetry based on special Lagrangian fibrations and more recent geometric approaches to string dualities in theoretical physics; (3) group actions, quotient singularities, Hilbert schemes and the McKay correspondence. Mori theory generalises and extends the classification of curves and surfaces, using a whole range of ideas from other areas such as singularity theory, commutative algebra, Abelian varieties, toric geometry derived categories and Hodge theory, in addition to related areas of symplectic geometry, Kaehler geometry, cohomology theories and string theory. Graduate students working primarily in these areas will be encouraged to continue their direction of research in the context of the wider view of the subject afforded by 3-folds. Expected benefit to the fellows: As a deep and well-developed mathematical discipline having fruitful overlap with many other branches of science, algebraic geometry is a crucial area for training young mathematicians. The last 20 years have seen extraordinary developments in threefolds, both internal work such as Mori theory that build on extensive experience with curves and surfaces, and new applications to other areas of math and physics, some quite unexpected. Our program aims to provide European graduates with an opportunity to learn the methods and results of 3-folds, but just as important, to view these in the wider context of the rich interplay between algebraic geometry and other areas of math and physics. This will add to the intellectual value of their research, and certainly improve their prospects for a successful research career. Detailed proposal information Part I. Description of the doctoral training 1. The research area The foundations of minimal models of 3-folds (Mori theory) and the classification of 3-folds were laid during the 1980s by Mori, Reid, Kawamata, Miyaoka, Koll\'ar and others, the subject obtaining recognition in 1990 with Mori's Fields medal. This theory builds on work on algebraic curves and the classification of surfaces going back to the 19th century, but the generalisations to higher dimension involve a sophisticated new conceptual framework and many technical difficulties. Mori theory offers rewarding new insights into classical results on algebraic surfaces, and has been an essential ingredient in advances such as mirror symmetry and the McKay correspondence. Unfortunately, partly on account of the difficulty of the subject, the study of 3-folds has been localised to a few centres, mainly in Japan or the USA. The geometry group at Warwick is unique in Europe in being among the world leaders in 3-folds and several related areas of geometry; the training site ``Threefolds in algebraic geometry'' aims to share this expertise with graduate students from many European centres. While higher dimensional geometry and mirror symmetry certainly contain problem areas that vary from extremely difficult to completely intractable, there are also many worthwhile accessible research problems in the subject. For example, there are still challenging problems in reinterpretting classical results on algebraic curves and surfaces (say, Cremona transformations) from the standpoint of Mori theory, and to develop these further; these types of problems may be particularly appropriate for graduate students coming from European centres with interests in traditional areas of algebraic geometry. Members of the Warwick geometry group have extensive experience of finding good problems that are do-able while appropriate to the needs of graduate students. We mention here some areas each containing many problems for which a successful outcome can be predicted: i. Graded rings: Classical construction of algebraic surfaces can in many cases be viewed in terms of the commutative algebra of canonical rings. We can now frequently go beyond what was known classically, to find new treatments of algebraic varieties and their moduli spaces in terms of algebra. This links to methods of projective geometry and to research problems in commutative algebra. Algebraic methods based on Gorenstein graded rings are also applicable to the classification of the flips and divisorial contractions of the 3-fold minimal model program. ii. Biregular and birational geometry of Fano 3-folds and Mori fibre spaces: Kawamata has proved that Fano 3-folds form a bounded family, but there is still a long way to go before we can turn this result of principle into a concrete finite list; however, the conjectural outlines of such a classification are slowly coming into focus: in small codimension they are given by algebraic constructions involving graded rings; at the other end (in big codimension), they should be obtainable by projections in the style of Fano and by graded version of the Mukai symmetric spaces. iii. Singularity theory: One of the new conceptual tools of Mori theory is the realisation that nature is singular: Mori theory works with singular 3-folds, and only resolves their singularities as a technical device. Following Mirel Caib{\u a}r's 1999 Warwick Ph.D. thesis, we expect the study of 3-fold singularities to become a significant growth area in the next 5 years; the crepant resolution of a 3-fold canonical singularities behaves like a local piece of a Calabi--Yau 3-fold, and Caib{\u a}r showed how to calculate its invariants in singularity theory terms. With his results, many traditional tricks of singularity theory (such as deformation theory) become applicable to these models of Calabi--Yaus. iv. Group actions, quotient singularities and McKay correspondence: The McKay correspondence relates the geometry of the resolution of singularities of a quotient singularity to the representation theory of the group. The two dimensional case is the ``classical'' McKay correspondence, and there has been considerable progress in understanding the higher dimensional case, following conjectures by Reid in 1992. However, the known theorems are rather abstract statements (about derived categories or motivic integration) giving little information about how the McKay correspondence works in concrete cases. Since the groups are listed in the 3-fold case, we obtain a large number of concrete problems that involve toric geometry, rather easy group theory and representation theory, and several different categories of geometry. At the same time, this area contains problems that are subtle and likely to be deep, for example, on the interpretation of the resolution of singularities in terms of moduli, and the multiplicative structure of the cohomology. While developments on the algebraic geometry side of 3-folds continue apace, one of the substantial current application of Mori theory is to Calabi--Yau manifolds and mirror symmetry. This idea arose about 10 years ago in the work of string theorists in theoretical physics, and came as a big surprise to mathematicians, giving us a whole range of new problems at many levels. This theory is now in part mathematical, and Mark Gross has given several lecture courses on it as part of the Warwick M.Sc. and at a number of summer schools. The special Lagrangian approach to mirror symmetry of Strominger, Yau and Zaslow present many exciting problems. Gross' work gives new approaches to special Lagrangian fibrations around a singular fibre, and many problems in the style of toric geometry can be derived from this. With Gross and Szendr{\H o}i, the Warwick geometry group is in a strong position to teach this subject at the research level. Mirror symmetry is not an isolated case in recent interactions between geometry and physics. What happens typically is that ``string dualities'' in physics predict cross-over between different categories of geometry, for example, the algebraic geometry of a Calabi--Yau 3-fold and the special Lagrangian geometry of its mirror. The insights of the physicists teach us the importance of the broader view of the mathematical subject, including surprising new relations between algebraic 3-folds and singularity theory, PDEs and geometry, gauge theory, symplectic geometry, geometric quantisation, elliptic cohomology, Lie theory etc. The Warwick geometry group has specialists in all these disciplines (David Mond, Mario Micallef, Peter Topping, John Rawnsley, John Jones, John Moody, Dmitry Rumynin, etc.) 2. Benefits to the fellows Marie Curie postgraduate fellows visiting the Warwick Math Institute under 3-fAG will be allocated a shared office, and the same library, computer, and other facilities as other postgraduate students. They will be assigned a research tutor chosen from the Warwick geometry group, and will get expert advice on which of our regular M.Sc. and M.Math. lecture courses and seminars to attend; they will be encouraged to discuss math problems with our existing graduate students. It should be mentioned that in recent years we have had several visiting international students each year (for example, in addition to many short term visitors, Hirokazu Nasu from Nagoya visited for 6 months in 1997--98, and Thomas Fangel from Aarhus for 6 months in 1998); our visitors quickly fit into a lively and friendly group of fellow students with related research interests. Incoming graduate students are welcome to take part in all our other research activities, including our symposia, conferences and workshops. They may also visit Cambridge, Oxford, London and other British universities for meetings of the COW (Cambridge Oxford Warwick algebraic geometry symposium) and Calf (Junior COW, organised by graduate students). The members of the Warwick geometry group will provide a broad range of advice on research problems. Depending on the individual postgraduate, this may take the form of help with an existing problem, or a completely new project, or some combination of the two; a completely new research problem will mainly only be appropriate for those making longer term visits. We should again mention that, in recent years, several postgraduate students who have visited us obtained substantial help in starting up on a research problem, or in developing their ideas into a thesis: as examples, Yukari Ito (Tokyo University), Adrian Langer (Warszawa) and Massimiliano Mella (Trento) wrote their theses in part on ideas suggested during visits to Warwick. It is likely that visiting students will in some cases also indulge in joint projects with Warwick students, leading to joint publications and reciprocal visits. In addition to study and research, Warwick offers visiting students many facilities for self-improvement. Visitors will be persuaded to give seminars at a basic and a research level. Most of our graduate students have experience of mathematical presentation, classroom teaching, computer typesetting, preparing material for publication, use of the web, etc., and are happy to give advice on these things. Our students regularly comment on each others' presentation at seminars, and proofread each others' papers to improve the English and the math presentation. International students spending an entire 10 week term at Warwick will be given the opportunity to take part in undergraduate teaching for a few hours a week. International students may use the facilities of the Language Lab to brush up their English or to embark on an introductory course in Japanese. Finally, our collective experience and contacts in the world of research mathematics puts the Warwick geometry group in a strong position to provide visiting students with advice on further study, the opportunities for postdoc placing, or for permanent employment. Part II. Description of the training site 1. The Warwick Math Institute Warwick is one of the UK's most successful universities. It obtains high ratings in the various periodic assessments of teaching and research, and is commonly ranked 4th for quality of research among the UK's 100 universities. The University attracts a large number of international visitors at the undergraduate, M.Sc., postgraduate and postdoc level, and has excellent facilities for handling them. The Warwick Math Institute, founded by Professor Sir Christopher Zeeman in 1965, is one of the world's leading centres of teaching and research in math. It was one of only 3 UK Math departments awarded the 5* rating in the UK Funding Council's 1996 Research Assessment Exercise. The Institute has 39 permanent academic staff -- all active research mathematicians of international stature -- plus research fellows and junior staff. Our visitor program attracts an average of 200 visits per year, including leading mathematicians from all over the world. The Math Institute runs an annual research symposium, usually funded by the British Engineering and Physical Science Research Council (EPSRC), and runs high level workshops and conferences on a regular basis (listed on our website www.maths.warwick.ac.uk/research.) The geometry group has been particularly active in this respect, running an algebraic geometry symposium in 1995--96 funded by British EPSRC, featuring 4 or 5 workshops and conferences, including a highly successful concluding EuroConference. The current 1999--2000 EPSRC symposium in geometry and topology looks like being equally strong. Algebraic geometers at Warwick takes an active part in European research activities: we have hosted many meetings as part of the EU Science, HCM and TMR programs, and expects to be a major component of the EU research training network EAGER (currently under application, reference number RTN1-1999-00202). There are currently 10 graduate students in algebraic geometry, 5 of whom are cross-border Europeans, and another 10 in related areas of geometry. The Math Institute's graduate program trains around 70 Ph.D. students at any time, plus 20 M.Sc. students. Just over half these are cross-border European or overseas students. In the last 5 years, we have awarded 75 Ph.D. degrees, of which 36 to international Ph.D. students. The Math Institute is a popular destination for international visiting students at all levels: we take 12 undergraduate students each year under the Erasmus/Socrates program, most of whom take M.Sc. courses and participate actively in graduate seminars; European graduate students frequently visit conferences, workshops and seminars, or make short stays to collaborate with our research staff. 2. The training site ``Threefolds in algebraic geometry'' Current research staff Permanent appointments: Mark Gross, John Jones, Mario Micallef, David Mond, John Moody, John Rawnsley, Miles Reid, Peter Topping Warwick Zeeman lecturers (on 3-year contracts): Dmitry Rumynin, Bal\'azs Szendr{\H o}i Marie Curie fellows: Adrian Langer, Nikos Tziolas Several new appointments in geometry are expected during the lifetime of the proposed training site, one this year to replace Victor Pidstrigatch who moves to Goettingen. The training site will function against the background of a comprehensive M.Math. and M.Sc. teaching program, involving many specialists not listed here in algebra, topology, hyperbolic geometry, computation, dynamical systems, etc. Current doctoral students: Rui de Albuquerque (Portugal), Ricardo Castano Bernard (Mexico), Alastair Craw, Rita Gaio (Portugal), Rebecca Leng, Diego Matessi (Italy), Anita Mo{\v c}nik (Slovenia), Jonathan Munn, Elisabeta Nedita (Romania), Jorge Neves (Portugal), Stavros Papadakis (Greece), Simone Pavanelli (Italy), Daniel Ryder, Marc Sommers (Germany), Andrew Stacey, Andrei Caldar{\u a}ru (USA, visiting from Cornell) Approximately 6 M.Sc. students each year; in addition, visiting postgraduate students (for example, Hokuto Uehara from Tokyo University will visit for 12 months from April 2000). 3. Research quality of the training site The main researchers in the geometry group at Warwick and their research interest are as follows: Mark Gross: algebraic and differential geometry, Calabi--Yau manifolds, mirror symmetry, geometry and string theory, classification of algebraic varieties John Jones: geometry, topology, gauge theory, K-theory, cyclic homology, index theorems, elliptic genus Mario Micallef: partial differential equations, differential geometry, K\"ahler geometry David Mond: algebraic geometry, singularity theory John Moody: algebra, algebraic geometry, resolution of singularities John Rawnsley: symplectic geometry, quantisation, differential geometry Miles Reid: algebraic geometry and commutative algebra, classification of surfaces and 3-folds, Mori theory, canonical singularities, flips, McKay correspondence Peter Topping: analysis, partial differential equations and geometry, K\"ahler geometry Bal{\'a}zs Szendr{\H o}i: algebraic geometry, Calabi--Yau 3-folds, Torelli problems, toric geometry and mirror symmetry, string theory Of these, Reid, Gross and Szendr{\H o}i work directly in 3-folds, and the others in areas of geometry having important relations with algebraic geometry. We give a small number of references directly relevant to the work of the training site. Paul Aspinwall and Mark Gross, Heterotic--heterotic string duality and multiple K3 fibrations, Phys. Lett. B 382 (1996), 81--88 Paul Aspinwall and Mark Gross, The SO(32) heterotic string on a K3 surface, Phys. Lett. B 387 (1996), 735--742 Bal{\'a}zs Szendr{\H o}i, Calabi--Yau threefolds with a curve of singularities and counterexamples to the Torelli problem. I, to appear in Int. J. Math. II, to appear in Math. Proc. Camb. Phil. Soc. Bal{\'a}zs Szendr{\H o}i, Some finiteness results for Calabi--Yau threefolds, to appear in J. London Math. Soc. Tom Bridgeland, Alastair King and M. Reid, Mukai implies McKay, preprint: math/9908027, 17~pp. A. Corti, A. Pukhlikov and M. Reid, Birationally rigid Fano hypersurfaces (May 1999 draft pp. 84), to appear in book ``3-folds at Warwick'', A. Corti and M. Reid (eds.), CUP 2000 Alastair Craw and M. Reid, How to calculate A-Hilb C^3, preprint: math/9909085, 29~pp. Mark Gross, Topological Mirror Symmetry, preprint: math/9909015, 61~pp. Mark Gross and Sorin Popescu, The moduli space of (1,11)-polarized Abelian surfaces is unirational, preprint: math/9902017, 27~pp. Mark Gross, Special Lagrangian Fibrations, I. Topology, Integrable systems and algebraic geometry (Kobe/Kyoto, 1997), 156--193, World Sci. Publishing, River Edge, NJ, 1998. II. Geometry, preprint: math/9809072, 72~pp. Mark Gross and P.M.H. Wilson, Mirror symmetry via 3-tori for a class of Calabi--Yau threefolds, Math. Ann. 309 (1997), 505--531. Mark Gross and Sorin Popescu, Equations of (1,d)-polarized Abelian surfaces, Math. Ann. 310 (1998), 333--377 J.D.S. Jones and John Rawnsley, Hamiltonian circle actions on symplectic manifolds and the signature, J. Geom. Phys. 23 (1997), 301--307 John Moody, On resolving vector fields. J. Algebra 189 (1997), 90--100 Miles Reid, Young person's guide to canonical singularities, in Algebraic geometry (Bowdoin, 1985), Part 1, 345--414, Proc. Sympos. Pure Math., 46, A.M.S., 1987 Miles Reid, Canonical 3-folds. Journ\'ees de G\'eometrie Alg\'ebrique d'Angers, 1979, 273--310, Sijthoff \& Noordhoff, Alphen aan den Rijn 1980 M. Reid and Y. Ito, The McKay correspondence for finite subgroups of SL(3,C), in Higher Dimensional Complex Varieties (Trento, Jun 1994), M. Andreatta and others Eds., de Gruyter, Mar 1996, 221--240 M. Reid Chapters on algebraic surfaces, in Complex algebraic varieties, J. Koll\'ar (Ed.), IAS/Park City lecture notes series (1993 volume), AMS, Providence R.I., 1997, 1--154. Warwick preprint 9/1996, alg-geom/9602006 M. Reid, McKay correspondence, in Proc. of algebraic geometry symposium (Kinosaki, Nov 1996), T. Katsura (Ed.), 14--41, alg-geom 9702016, 30 pp. 4. Facilities See under Part~1, ``Benefit to the fellows'' above. 5. Evidence of past success Warwick has awarded 75 Ph.D.s in math since 1994, of which 36 to EU and overseas students. Succesful Ph.D.s trained by the geometry group include the following Selma Alt{\i}nok (Turkey), Santos Asin Lares (Spain), Claudio Arezzo (Italy), Mohan Bhupal (India), Mirel Caib{\u a}r (Romania), Jose Cisneros-Molina (Mexico), Marcello Felisatti (Italy), Peter Gothen (Denmark), Laurent Lazzarini (France), Andrea Loi (Italy), Sergio Santa Cruz (Spain), Ioannis Sardis (Greece), Yorgos Terizakis (Greece) Over the same period, a similar number of international postgraduate students visited the Warwick geometry group for extended periods as part of their studies towards a Ph.D. at their home university. P.S. The EU commission scheme to which we are applying is named after the Polish lady scientist Maria Sk{\l}odowska-Curie. In discussing and advertising our grant, we prefer to use her full name, which seems better to reflect the internationalist and non-discriminatory spirit of the times.