送交者: tom_cat 于 2005-10-19, 10:29:57:
Dear Shing-Tung:
Thank you for your letter of November 6, 1986. In response to your lette
r I would like to bring up a few points.
1. In my talk in Columbia I presented my method of proving the existence
of Kahler-Einstein metrics on compact Kahler manifolds with positive an
ticanonical line bundle and a suitable finite group of symmetry. I menti
oned at the end that I was still checking some concrete examples. Right
after my return to Harvard I verified the case of the Fermat cubic surfa
ce. Within two weeks after the Columbia Conference a typed manuxxxx wa
s available. Morris Kalka and Pit-Mann Wong had copies at that time. Whe
n I looked over my manuxxxx to prepare my talk in Paris in June I adde
d the examples of blowing up three point in $\cal{P}_2$ and higher-dimen
sional Fermat hypersurfaces. All those examples were verified before las
t summer. In any case what is important is the method. Checking concrete
examples is only the simple task of counting the degrees of certain cur
ves and the number of points of the curves belonging to the same orbit.
2. Concerning your student's independent derivation of the equivalence o
f $sup\phi$ and $-inf\phi$, I would like to point out that the equivalen
ce of bounds for $sup_M\phi$, $-inf_M\phi$, $\int\phi$, $-\int\phi e^{-t
\phi+F}$, $log\int e^{-\alpha\phi-t\phi+F}$, and $log\int c^{\alpha\phi}
$ was already in the handwritten notes entitled "Reflection Methods" whi
ch I sent you about nine years ago. That was way before your student cam
e to the Unites States. Even a few year later you mentioned to me that y
ou were still keeping those notes. Unless in your discussions with your
students you were deliberately keeping back from them what you already k
new, I do not see how independent your student's derivation can be.
3. Since I have not seen your student's manuxxxx, it is impossible for
me to determine exactly how much your student's method is similar to or
different from the one I presented at the Columbia Conference. However,
from the inxxxxation that I have, it seems to me that his method is ess
entially a rexxxxulation using the equivalent bound of $\int e^{-\alpha\
phi}$.
It is against my