Small Dolbeault eigenvalues along Hermitian Yang–Mills flow

We study the small eigenvalues of the Dolbeault Laplacian on trace-free endomorphisms of a semistable holomorphic bundle along Hermitian Yang–Mills flow, normalized as \(h^{-1}\partial_\tau h=-2(K_h-\kappa)\) with \(K_h\) the mean curvature and \(\kappa\) its average. The bundles considered are simple bundles on a compact curve whose Jordan–Hölder factors are pairwise nonisomorphic, and extensions of one polystable bundle by another on compact Kähler manifolds, again with pairwise nonisomorphic stable factors. From every smooth initial metric, each small eigenvalue is asymptotic to an eigenvalue of a weighted graph Laplacian, whose conductances evolve by an explicit finite-dimensional flow; the relative error tends to zero. The graph records which stable factors are joined by nonzero extension classes. The asymptotics are governed by the weight grading of Haiden, Katzarkov, Kontsevich, and Pandit, the solution of a quadratic program on this graph. Suppose some Lagrange multiplier vector of this program is strictly positive on every active constraint. We then determine the decay exponent and the leading coefficient of every small eigenvalue, and the limits of the spectral subspaces. The eigenvalues of order \(\tau^{-1}\) have coefficients independent of the initial metric, and \(1/2\) is always one of them. At each faster rate the coefficients depend on the initial metric, and their sum is unbounded over metrics on one fixed bundle. A key step shows that the bounded remainder in the asymptotics of the finite-dimensional flow converges. Examples include an eigenvalue decaying like \(\tau^{-4/3}\) and, without the positivity condition, an eigenvalue of order \((\tau\log\tau)^{-1}\). We also treat families of harmonic extensions with fixed metrics, including repeated factors and recovery of the graded object, and we show that the constants in a slope inequality relating the lowest eigenvalue to the averaged trace-free mean curvature are sharp in every rank.