Quantum Extremal Transitions and Special L-values

Abstract

A threefold extremal transition $Y \searrow X$ consists of a crepant extremal contraction $\phi \colon Y \to \bar Y$ with curve class $\ell \in \operatorname{NE}(Y)$, followed by a smoothing $\bar Y\rightsquigarrow X$. We consider the Type II case that $\phi$ contracts a divisor $E$ to a point and prove that the quantum cohomology $QH(X)$ is obtained from $QH(Y)$ via analytic continuation, regularization, and specialization in $Q^\ell$. Besides roots of unity, special $\mathrm{L}$-values appear in $\lim Q^\ell$ whenever $\bar Y$ admits more than one smoothings. Further techniques are employed and explored beyond known tools in Gromov--Witten theory including (i) the canonical local B model attached to $Y \searrow X$, (ii) existence of semistable reduction of double point type for the smoothing, (iii) the modularity of the extremal function $\mathbb{E} := E^3/\langle E, E, E\rangle^Y$, and (iv) periods integrals of Eisenstein series. Our study provides a geometric framework linking classifications of del Pezzo surfaces, Ramanujan's theta functions, and Zagier's special ODE list via Type II transitions.