  Motivated by the Weak Gravity Conjecture, we uncover an intricate interplay
between black holes, BPS particle counting, and Calabi-Yau geometry in five
dimensions. In particular, we point out that extremal BPS black holes exist
only in certain directions in the charge lattice, and we argue that these
directions fill out a cone that is dual to the cone of effective divisors of
the Calabi-Yau threefold. The tower and sublattice versions of the Weak Gravity
Conjecture require an infinite tower of BPS particles in these directions, and
therefore imply purely geometric conjectures requiring the existence of
infinite towers towers of holomorphic curves in every direction within the dual
of the cone of effective divisors. We verify these geometric conjectures in a
number of examples by computing Gopakumar-Vafa invariants.
