  We generalize the super period matrix of a super Riemann surface to the case
that Ramond punctures are present. For a super Riemann surface of genus g with
2r Ramond punctures, we define, modulo certain choices that generalize those in
the classical theory (and assuming a certain generic condition is satisfied), a
g|r x g|r period matrix that is symmetric in the Z_2-graded sense. As an
application, we analyze the genus 2 vacuum amplitude in string theory
compactifications to four dimensions that are supersymmetric at tree level. We
find an explanation for a result that has been found in orbifold examples in
explicit computations by D'Hoker and Phong: with their integration procedure,
the genus 2 vacuum amplitude always vanishes "pointwise" after summing over
spin structures, and hence is given entirely by a boundary contribution.
