An approach towards the Kollár-Peskine problem via the Instanton Moduli Space
Abstract
We look at the following question raised by Kollár and Peskine. (Actually, it is a slightly weaker version of their question.)
Let
V
t
be a family of rank two vector bundles on
P
3
. Assume that the general member of the family is a trivial vector bundle. Then, is the special member
V
0
also a trivial vector bundle?
We show that this question is equivalent to the nonexistence of morphisms from
P
3
→X
, where
X
is the infinite Grassmannian associated to SL(2). We further reduce this question to the nonexistence of
C
∗
-equivariant morphisms from
C
3
∖{0}→
M
d
(for any
d>0
), where
M
d
is the Donaldson moduli space of isomorphism classes of rank two vector bundles
V
over
P
2
with trivial determinant and with second Chern class
d
together with a trivialization of
V
|
P
1
.