## 所員 -Helmke, Stefan-

名前

**Helmke, Stefan**
職
助教

E-Mail
helmke （emailアドレスには＠kurims.kyoto-u.ac.jp をつけてください）

U R L

研究内容
Algebraic Geometry

紹 介

In my long term research project, a proof of the Fujita Conjecture,
I made overall progress but there where also some setbacks. In fact,
it turned out that in order to apply the local theory of graded rings
associated to valuations of Abhyankar type, their filtrations and
deformations I have developed in recent years, to the problem of
constructing global sections of algebraic line bundles, some small
but important modifications where needed. Unfortunately, this
consumed most of my time, so that the global theory is still not
quite completed, as I had expected last year. On the other hand,
the local theory is now much more coherent and it is certainly
quite interesting on its own. In particular, it includes a very
general version of the Restriction Theorem for multiplier ideals
and dual versions of Teissier's inequality on the multiplicity of
ideals and Demailly-Ein-Lazarsfeld's Subadditivity Theorem on the
multiplier ideal of an effective divisor. The deformation theory
includes generalizations of Galligo's Borel-invariance of generic
initial ideals, the existence of universal Gröbner bases and
many more new results. Those results are crucial to construct
good filtrations of certain linear systems on projective
varieties and to prove their rigidity. Earlier approaches [1-4] to
the problem always relied on very special filtrations which do not
have the appropriate strong properties necessary to proof the
Fujita Conjecture. But as a drawback with these new filtrations,
the minimal center of log-canonical singularities does not
anymore always decrease in this process; it can even increase!
This poses the new difficulty that the process may not terminate.
But rigidity of the filtrations allows one to prove that it
indeed terminates. The final result of this process will then be
either a filtration with an isolated log-canonical singularity,
or a filtration whose centers on the variety have very small
degrees. In the first case, it is already well-known that the
point under consideration is then not a base point. On the
other hand, in the second case, the degrees of the subvarieties
are so small, that it leads to an obvious contradiction to the
assumptions of the Fujita Conjecture. These exciting new
developments will appear in a hopefully forthcoming preprint [5].

- S. Helmke, On Fujita's conjecture, Duke Math. J. 88 (1997), 201--216.
- S. Helmke, On global generation of adjoint linear systems, Math. Ann. 313 (1999), 635--652.
- 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.
- S. Helmke, Multiplier ideals and basepoint freeness, Oberwolfach reports 1, 2004, 1137--1139.
- S. Helmke, New Combinatorial Methods in Algebraic Geometry, in preparation.