Small Hodge eigenvalues on nodal degenerations of threefolds

We determine the small positive Hodge eigenvalues on two-forms in a proper holomorphic family of compact threefolds over a disc, with smooth Kähler total space, smooth fibres away from the origin, and a reduced central fibre whose only singularities are ordinary double points. The fibre metrics are induced from the total space. Assume that a small resolution Y of the central fibre has vanishing first Betti number and second real cohomology spanned by first Chern classes of holomorphic line bundles. The number of small eigenvalues is the drop in the second Betti number from Y to a smooth fibre. Each is asymptotic to 2πμ/log(1/|t|), where t is the family parameter. The numbers μ are the relative eigenvalues of two quadratic forms on H²(Y;ℝ) modulo the classes that extend to the smooth fibres: the sum of squared degrees on the exceptional curves, and the minimum squared L² norm of a harmonic representative on the regular part of the central fibre. We express the latter form through divisor intersections and the Kähler class. The remaining positive spectra in degrees one and two have a uniform gap, and the small eigenforms are coexact, primitive and of type (1,1). The proof combines the norm calculation for a harmonic two-form of unit period on the node link, a sharp leading energy bound on the affine quadric smoothing, and the compactness theorem of Anné and Takahashi.