Small Dolbeault eigenvalues near stability walls

We determine the small spectrum of the Dolbeault Laplacian on trace-free endomorphisms of Hermitian–Einstein bundles near intersections of slope-stability walls. On a compact Kähler manifold of dimension at least two, consider an extension of \(ⱼ₌₁^kF_j\) by \(F_0\), where the factors are pairwise nonisomorphic and stable of equal slope at the limiting Kähler class, and every extension component is nonzero. Write \(r_i=rk F_i\) and let \(ε_j\) be the Einstein constant of \(F_j\) minus that of the extension. In the chamber \(ε_j>0\), exactly \(k\) eigenvalues tend to zero. They agree, to relative error \(O(∑_jε_j)\), with the eigenvalues of \(diag(ε_1,…,ε_k)+r_0⁻¹vv^*\), where \(v_j=√r_jε_j\). The estimate is uniform in all positive ratios of the slope gaps. Thus the ranks and slopes determine the leading spectrum even when the gaps vanish at arbitrarily different rates. For two factors we obtain \(λ_1=δ-Dδ^2+o(δ^2)\), where \(δ>0\) is the quotient Einstein constant minus the subbundle Einstein constant and \(D≥0\) is given by the Green operator of the graded connection. We also determine the first eigensection through order \(δ\) in every spatial \(C^m\) norm. Toric tangent bundles provide examples with arbitrarily many independently varying small eigenvalues.