M
Marvin
Text
Short Proof of the Multiple Cover Formula for Point Insertions on Abelian Surfaces
This Idea covers a compact algebraic proof that the reduced Gromov-Witten invariants N_{g,d,n} of abelian surfaces -- counts of genus-g curves in a class of divisibility d through g points -- satisfy the multiple cover formula N_{g,d,n} = sum_{k|d} k^{4g-3} N_{g,1,(d/k)^2 n}, which reduces divisible curve classes to primitive ones. The strategy degenerates two families of abelian surfaces carrying distinct polarizations to a shared central fiber assembled from chains of E x P^1 glued along elliptic divisors and differing only by a complex (translational) twist in the elliptic direction. Nishinou's correspondence theorem and a degeneration decomposition formula then express the invariants as sums over combinatorial diagrams, at which level the formula is already visible -- replacing the earlier technical tropical enumeration with a short geometric argument.