京都大学数理解析研究所

紹 介

　My work on Fujita's conjecture made some considerable progress recently. The process of constructing a valuation with very peculiar properties, as described in my last report, had some technical difficulties but I was finally able to overcome most of them. The conjecture was previously only solved in a few cases when the variety is smooth and of very low dimension[1,2,5]. In those cases, the valuation is always induced from blowing-up a point. But in general we need a much wider class of valuations, which can actually be optimized for this problem. The result of this process should be not only a proof of the Fujita Conjecture in all dimensions, but even the more precise conjecture formulated in[8]. The method also applies to the problem of effectively separating several points on a projective variety and leads to some unexpected results. If point separation fails, one can find a whole tree of subvarieties of low degree instead of just a flag as conjectured in[8]. This is in so far interesting, as in some known examples one can find such trees, but previous methods could only produce a single flag with much weaker properties. Unfortunately, this all took a long time, but it is now in the process of final clean-up and will be finished soon.
　My other research project, a continuation of a joint project with P.Slodowy, did still not make much progress, since I was concentrating on my work on the Fujita Conjecture. I am planning to continue this work only after completing this project. In the 1960's, E.Brieskorn discovered some remarkable similarities between simple singularities and simple algebraic groups. At the International Congress of Mathematicians in 1970 he gave a completely natural explanation of those similarities, based on some ideas of A. Grothendieck. Many mathematicians at that time were impressed by the beauty of this connection and therefore, it was only natural to ask for a generalization to a larger class of singularities. In 1981, P. Slodowy suggested a similar connection between simple elliptic singularities and affine Kac-Moody groups. But it turned out that there is no `subregular orbit' in affine Kac-Moody groups, which is required in order to generalize Brieskorn's connection. It was therefore necessary to find a completion of affine Kac-Moody groups which contains subregular orbits. Already in the finite dimensional case, the existence of a subregular orbit is a difficult problem and only solved by a partial classification of conjugacy classes, which was not known at all for those infinite dimensional groups. However, there is one completion, the so-called holomorphic loop group, for which a classification of their conjugacy classes is possible via the classification of principal bundles over elliptic curves[3,4]. Using our classification, we proved the existence of subregular orbits and were then able to generalize Brieskorn's connection to simple elliptic singularities and holomorphic loop groups [6,7]. Since there are much more holomorphic loop groups than simple elliptic singularities, we also had to deal with the other cases, which leads to some interesting geometry of non-isolated singularities. In particular, in a holomorphic loop group associated to the special linear group of rank 5, one would expect to find the cone over an elliptic curve of degree 5. Since this is not a complete intersection, it cannot be directly realized as in Brieskorn's construction. However, using geometric invariant theory on a slice which is slightly bigger than the transversal slice to a subregular orbit, I was able to construct this singularity together with its semi-universal deformation. Interestingly, the construction also works in some other cases, including the special linear group of rank 3, where we find a cone over an elliptic curve of degree 6 and its semi-universal deformation!

1. S. Helmke, On Fujita's conjecture, Duke Math. J. 88 (1997), 201--216.
2. S. Helmke, On global generation of adjoint linear systems, Math. Ann. 313 (1999), 635--652.
3. S. Helmke and P. Slodowy, Loop groups, principal bundles over elliptic curves and elliptic singularities, Annual Meeting of the Math. Soc. of Japan, Hiroshima, Sept. 1999, Abstracts, Section Infinite-dimensional Analysis, 67--77.
4. S. Helmke and P. Slodowy, On unstable principal bundles over elliptic curves, Publ. RIMS 37 (2001), 349--395.
5. S. Helmke, The base point free theorem and the Fujita conjecture, Vanishing theorems and effective results in algebraic geometry, ICTP Lecture Notes 6, Trieste, 2001, 215--248.
6. S. Helmke and P. Slodowy, Loop groups, elliptic singularities and principal bundles over elliptic curves, Geometry and Topology of Caustics -- Caustics '02, Banach Center Publ. 62, Warszawa, 2004, 87--99.
7. S. Helmke and P. Slodowy, Singular elements of affine Kac-Moody groups, European Congress of Mathematics. Proceedings of the 4th Congress (4ecm) held in Stockholm, June 29 -- July 2, 2004, Ed. A. Laptev, Stockholm, 2005, 155--172.
8. S. Helmke, Multiplier ideals and basepoint freeness, Oberwolfach reports 1, 2004, 1137--1139.