108
N.M. Katz:
(7.2.3)
Remark.
~,
depends
additively
upon the invertible sheaf with
integrable connection (~, V).
7.3.
When T is (the spectrum of) a field k of characteristic zero, the
group Hl(S, t2~/r) has a classical interpretation. Recall that a mero
morphic closed oneform co on S is said to be a
differential of the third
kind
if there exists an open covering {q/i} of S, and on each q/i a closed
holomorphic one form
(Diff_F(d~i,~l/T)
and a nonzero
meromorphic
function gi on q/i, such that
(7.3.0.1)
co=coiqdgi/gi
onq/i,
dcoi=O.
Now define f~j =
gj/gi,
a meromorphic function on q/i n q/j. In fact,
fiijEz F(O~Zi
("3 q/j,
(Q~
),
because on
q/i n q/j
(7.3.0.2)
dfii/fij = dgj/gj d gi/gi = coi  coj
so that
dfij/fij
is holomorphic on q/i c3 q/j. Because we are in characteristic
zero, this implies that fij and
1/f~j
are holomorphic on q/i ~ q/j.
Thus we have assosciated to a differential of the third kind co on S
a Cech 1cocycle
(coi,flj)
for the complex I2~/T. If, on the same covering q/i,
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 Fall '11
 NormanKatz
 Complex number, forms co'ion q/i, d.t.k., bove constructed mapping, smooth connected scheme

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