  We describe the dimensional reduction of the IIB B-fields in F-theory using a
conjectured description of normalizable B-fields in terms of perverse sheaves.
Computations are facilitated using the Decomposition Theorem. Many of our
descriptions are new, and all our results are all consistent with known results
in physics. We also conjecture a physical framework for normalizable B-fields
and show consistency with mathematics.
  We dedicate this paper to Herb Clemens, in admiration for his myriad
fundamental contributions to complex algebraic geometry, together with his more
recent interest in F-theory in physics. This paper deals with three of Herb's
interests: Hodge theory, topology of algebraic varieties, and F-theory, and so
is a fitting way for us to express our appreciation for his contributions over
a period of more than five decades.
