## 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

As I had planned for the preceding academic year and explained in
my last report, I worked on the final details of a possible proof
of Fujita's Conjecture on global generation of adjoint linear
systems [1--4]. I could solve all the remaining problems except the
following. In order to construct a global section of an algebraic
line bundle $\cal L$ on a projective variety X which is not vanishing
at a closed point $P\in X$, one only needs to find a global section
of the line bundle ${\cal L}^n\otimes\omega_X^{-n}$ which is singular
enough at P, but not too singular in $X\setminus\{P\}$,
where $\omega_X$ denotes the canonical sheaf and n is a positive
integer. This idea goes back to the Italian school of algebraic
geometry around 1940, where it was used in the case when X is
an algebraic surface of general type, ${\cal L}=\omega_X$ and n
is a very specific small integer. Then, 25 years later the techniques
where considerably improved with the help of cohomology theory and
in particular Kodaira's Vanishing Theorem. In the 1990's there
were strong efforts to generalize these results to higher dimensions.
But in contrast to the two dimensional case, the integer n could
now be arbitrary large. For the procedures used at that time, this
caused no essential problem since one could always increase n
as desired, but the results obtained in this way were not
satisfying. However, with my new techniques increasing n is impossible
since then the procedure may never come to an end. Therefore, one
needs an a priori estimate for n. But unfortunately, the
technique I designated to this problem and which was successful
for the corresponding local problem failed in the global situation.
However, during this rather time consuming course or research, I
developed another technique, which is more promising. In fact,
at least in low dimensions the argument already works and there
is good experimental evidence that this will also work in
all dimensions. So the new plan is now to spend some more time on
these new techniques and in case this would fail again, then to
publish the already obtained results[5] independently, without
the Fujita Conjecture.

- 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.