  We show that for a polynomial map, the size of the Jordan blocks for the
eigenvalue 1 of the monodromy at infinity is bounded by the multiplicity of the
reduced divisor at infinity of a good compactification of a general fiber. The
existence of such Jordan blocks is related to global invariant cycles of the
graded pieces of the weight filtration. These imply some applications to period
integrals. We also show that such a Jordan block of size greater than 1 for the
graded pieces of the weight filtration is the restriction of a strictly larger
Jordan block for the total cohomology group. If there are no singularities at
infinity, we have a more precise statement on the monodromy.
