Quantum Extremal Transitions and Special L-values
Shuang-Yen Lee, Chin-Lung Wang, Sz-Sheng Wang
TL;DR
The paper develops a comprehensive framework for how big quantum cohomology behaves under Type II extremal transitions Y ⟶ X in smooth threefolds. By constructing canonical local B-models, performing semistable degenerations and applying GW degeneration formulas, the authors relate QH(Y) to QH(X) through analytic continuation, regularization, and specialization in Q^ℓ, uncovering root-of-unity and exceptional L-value phenomena. A rich structure emerges: the local models for del Pezzo transitions are tied to Picard–Fuchs equations, modular forms, Ramanujan theta functions, and Eisenstein-series periods, which together explain how topological changes encode arithmetic data. Globally, the approach reduces the global GW problem to local computations, yielding a precise subquotient relation QH(X) ⊂ QH(Y) and revealing deep connections between del Pezzo classifications, modular curves, and automorphic phenomena in GW theory. The results illuminate new arithmetic facets of GW theory in geometric transitions and provide a robust toolkit for future investigations of higher-dimensional analogs and modular-limit behaviors.
Abstract
A threefold extremal transition $Y \searrow X$ consists of a crepant extremal contraction $φ\colon Y \to \bar Y$ with curve class $\ell \in \operatorname{NE}(Y)$, followed by a smoothing $\bar Y\rightsquigarrow X$. We consider the Type II case that $φ$ 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.