Integral topology and small eigenvalues in Calabi–Yau conifold transitions

We determine the leading small spectrum of the Hodge Laplacian on real two-forms for Ricci-flat metrics in polarized Calabi–Yau conifold transitions. We assume a simply connected projective small resolution and a fixed smoothing that smooths every node to first order, approached along a fixed ray. Exactly as many positive eigenvalues tend to zero as there are divisor directions lost under smoothing, and their decay is inverse-logarithmic in the smoothing parameter. The leading coefficients are 2π times the generalized eigenvalues of the squared exceptional-period form relative to the harmonic norm after projection off all surviving cohomology. We compute this norm from global intersections. The divisor-period restriction map also determines the integral relations among cohomological vanishing cycles. The small eigenspaces are coexact and primitive of real type (1,1), with a uniform gap in the remaining positive two-form spectrum. Applications to four-node and eight-node toric transitions give three and two distinct coefficients, respectively. For the sixteen-node split quintic with Kähler class c₁(O(1)), the coefficient is 10π; we also calculate its integral rings, attachment and specialization maps, primitive vanishing lattice and monodromy image of index sixteen. The resolution and smoothing branches converge to a common Ricci-flat metric completion, and their tangent Hermitian–Einstein connections converge smoothly to the central tangent connection on compact subsets of the regular locus.