We prove Malgrange’s conjecture on the absence of confluence phenomena for integrable meromorphic connections. More precisely,
if
Y→
X is a complex-analytic fibration by smooth curves,
Z a hypersurface of
Y finite over
X, and ∇ an integrable meromorphic connection on
Y with poles along
Z, then the function which attaches to
x∈
X the sum of the irregularities of the fiber ∇
(x) at the points of
Z
x
is lower semicontinuous.
The proof relies upon a study of the formal structure of integrable meromorphic connections in several variables.