Small Dolbeault eigenvalues near stability walls

Abstract

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 ${⨁}_{j=1}^k{F}_j$ 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 $O({\sum }_j{ϵ}_j)$, with the eigenvalues of $diag({ϵ}_1,\dots ,{ϵ}_k)+r_0^{-1}v{v}^{*}$, where $v_j=\sqrt{r_j{ϵ}_j}$. 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 ${\lambda }_1=\delta -D{\delta }^2+o({\delta }^2)$, where $\delta >0$ is the quotient Einstein constant minus the subbundle Einstein constant and $D\ge 0$ is given by the Green operator of the graded connection. After rescaling the extension and making unitary identifications with the graded limit, we determine the normalized first eigensection through order $\delta$ in every spatial $C^m$ norm. Toric tangent bundles provide examples with arbitrarily many small eigenvalues whose orders of vanishing can be prescribed independently.