Let X be a complex nonsingular projective threefold
of general type. Suppose the
canonical system of X is composed of a pencil, i.e.
dimΦ∼KX∼(X)=1.
It is often
important to understand birational invariants of X such as
pg(X), q(X),
h2(OX) and
χ(OX) etc. In this paper, we mainly
study the irregularity of X.
We may suppose that ∼KX∼ is
free of base points. There is a natural fibration
f[ratio ]X→C onto a nonsingular curve
after the Stein factorization of Φ∼KX∼.
Let F be a
general fibre of f, then we know that F is a
nonsingular projective surface of general
type. Set b[ratio ]=g(C) and
pg(F), q(F)
for the respective invariants of F. The main result
is the following theorem.