LAUSR

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 ≤ h 1 , 1 ≤ 491. We show that the Kähler cone of X is very narrow at large h 1 , 1 , and as a consequence, control of the α′ expansion in string compactifications on X is correlated with the presence of ultralight axions. If every effective curve has volume ≥ 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 ≲ p ≲ 7 depending on the type of cycle in question. Instantons from branes wrapping these cycles are thus highly suppressed.