  We study the topological properties of Calabi-Yau threefold hypersurfaces at
large $h^{1,1}$. We obtain two million threefolds $X$ by triangulating
polytopes from the Kreuzer-Skarke list, including all polytopes with $240 \le
h^{1,1}\le 491$. We show that the K\"ahler cone of $X$ is very narrow at large
$h^{1,1}$, and as a consequence, control of the $\alpha^{\prime}$ expansion in
string compactifications on $X$ is correlated with the presence of ultralight
axions. If every effective curve has volume $\ge 1$ in string units, then the
typical volumes of irreducible effective curves and divisors, and of $X$
itself, scale as $(h^{1,1})^p$, with $3\lesssim p \lesssim 7$ depending on the
type of cycle in question. Instantons from branes wrapping these cycles are
thus highly suppressed.
