Anomalous D-brane charge in F-theory compactifications

Tadpole cancellation in F-theory on an elliptic Calabi-Yau fourfold X --> B -3 demands some spacetime-filling three-branes (points in B-3). If moved to the discriminant surface, which supports the gauge group, and dissolved in to a finite size instanton, the second Chern class of the corresponding bun dle E is expected to give a compensating contribution. However the dependen ce of D-brane charge on the geometry of W and on the embedding i : W --> B- 3 gives a correction to c(2)(E). We show how this is reconciled by consider ing the torsion sheaf i(*)E and discuss some integrality issues related to global properties of X as well as the moduli space of this object.