Can finite models predict how geometry degenerates?

16 manuscripts · Reading order

The manuscripts.

Chambers and geometric realizability · 4 manuscriptsThe section ↑

1. 01

   Slope stability of tangent bundles of smooth toric Fano varieties

   Constructions of toric Fano varieties with stable tangent bundles, compared with Kähler–Einstein conditions.

   Preprint · 2026 · Abstract, PDF & citation ↗

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   We construct smooth toric Fano $n$-folds of Picard number $n+2$ whose tangent bundles are slope-stable with respect to the anticanonical polarization, for every $n\ge4$. The construction gives toric fibrations with fibre the del Pezzo surface of degree six over products of projective lines, parametrized by multisets of roots of $A_2$. Every nonempty multiset of nonzero roots with vanishing sum yields a stable tangent bundle. Among these multisets, the resulting variety is Kähler–Einstein if and only if the multiset is invariant under negation or under the order-three rotation of the root hexagon. We also construct stable, non-Kähler–Einstein examples with nonzero twist sum in every even dimension at least four. The proofs reduce slope inequalities and barycenter computations to integrals over the moment hexagon.

   For a smooth toric Fano variety with strictly semistable tangent bundle, we prove that polystability is equivalent to decomposition as a nontrivial product of smooth toric Fano varieties with anticanonically stable tangent bundles. Exact evaluation of Klyachko’s criterion extends the stability classification to dimensions five and six, recovering the classifications of Steffens and Reynolds in dimensions three and four. Together with the product criterion, this determines polystability for all $8{,}630$ varieties in dimensions three through six, and hence the existence of Hermitian–Einstein metrics on their tangent bundles with respect to an anticanonical Kähler form.

2. 02

   Tangent stability on toric Fano blowups and bundles

   Stability criteria for specified Fano blowups and bundles, with a complete exact census of Picard-number-ten eightfolds.

   Preprint · 2026 · Abstract, PDF & citation ↗

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   For Fano blowups of products of projective spaces along invariant codimension-two centres indexed by a tree, using distinct coordinate hyperplanes at each factor, anticanonical tangent stability is equivalent to one divisor-degree inequality at each vertex. The exceptional contributions are mixed intersections on the blowup centres. Stability holds for every tree of maximum degree at most three whose nonleaf factor dimensions equal their valences, with arbitrary positive leaf dimensions. For smooth toric Fano varieties that are locally trivial toric bundles with fibre a product of degree-six del Pezzo surfaces and base a product of projective spaces, the largest Picard number compatible with tangent stability in dimension $n\ge 4$ is $\lfloor 5n/4\rfloor +1$, allowing arbitrary integral twists. Every fixed tree of maximum degree at most three with an edge yields bundles attaining this bound after sufficiently long independent edge subdivisions. Their number, up to isomorphism of toric fans, is unbounded as the dimension grows through multiples of four. The proofs use relative tangent bundles, ranks of subspaces spanned by rays, and integral comparisons. The bundle existence proofs also use finite exact rational calculations with uniform error estimates and induction.

3. 03

   Local stability chambers for toric tangent bundles

   Varying the polarization: an all-dimensional projective-bundle classification with arbitrarily many chamber boundaries, a geometric criterion, and local chambers for the 31 indecomposable semistable census cases.

   Preprint · 2026 · Abstract, PDF & citation ↗

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   We classify the anticanonical slope stability of the tangent bundles of $X_{p,q}={\mathbb{P}}_{({\mathbb{P}}^1{)}^{p+q}}(\mathcal{O}\oplus \mathcal{O}(1^p,-1^q))$, where $p,q$ are nonnegative integers and $p+q\ge 1$. The tangent bundle is stable when $p=q$, strictly semistable when $|p-q|=1$, and unstable otherwise. For $p=k$ and $q=k-1$, the germ of the semistable polarization locus at the anticanonical class is real analytically equivalent to $[0,\infty {)}^k\times {ℝ}^k$, with stable locus $(0,\infty {)}^k\times {ℝ}^k$. In particular, the number of independent local chamber boundaries is unbounded.

   For a smooth projective toric variety whose tangent bundle is semistable at a polarization $H_0$, we describe the local chamber at $H_0$ by the order ideals of a finite poset determined by the equal-slope ray subspaces. If each such subspace is spanned by the rays it contains in every maximal cone, nearby stable polarizations exist precisely when the variety is not a nontrivial toric product. Applied to the existing anticanonical census in dimensions three through six, this criterion gives nearby stability in all 31 indecomposable strictly semistable cases, whose local chambers have one, two, or three independent boundaries. We determine locally free, pairwise nonisomorphic Jordan–Hölder factors both for these cases and for the family $X_{k,k-1}$.

4. 04

   Nonsimplicial local stability chambers of toric tangent bundles

   Geometric realization: nonsimplicial tangent chambers, positive-spanning slope access without local freeness, and an explicit Calabi–Yau conifold transition.

   Preprint · 2026 · Abstract, PDF & citation ↗

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   We construct smooth projective toric varieties with nonsimplicial local slope-stability chambers for their tangent bundles. In every even dimension $2k+2\ge 4$, a fibre product of two projective-line bundles over ${\left({\mathbb{P}}^1\right)}^{2k}$ admits an ample real class at which the tangent bundle is strictly semistable. Near this class, the semistable locus is analytically isomorphic to a pointed cone over ${\Delta }^1\times {\Delta }^{2k-1}$, times a line, where ${\Delta }^j$ denotes the $j$-simplex. The torus-invariant saturated subsheaves of equal slope form the ideal lattice of the complete bipartite poset with two minimal and $2k$ maximal elements, and the stable line factors realize every centered slope direction. These varieties are weak Fano and not Fano. The fourfold has minimal dimension and Picard rank among toric varieties with such a chamber and nearby stable polarizations. We also prove a local normal form at any ample real class where the tangent bundle of a smooth projective toric variety is semistable. It retains precisely the slope directions realized by polarizations, requires no local-freeness assumption on equal-slope subsheaves, and localizes every nearby Harder–Narasimhan filtration to their equality lattice. If the fan rays in each equal-slope subspace positively span that subspace, every centered slope direction occurs. A semistable surface satisfies this condition although one equal-slope quotient is not locally free.

Stationary metrics and small spectra · 4 manuscriptsThe section ↑

5. 05

   Small Dolbeault eigenvalues near stability walls

   From stability chambers to the equilibrium spectrum: every small Hermitian–Einstein eigenvalue is determined to uniform relative error by ranks and slope gaps, even at arbitrarily separated scales.

   Preprint · 2026 · Abstract, PDF & citation ↗

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   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.

6. 06

   Uniform Hermitian–Einstein spectra for bipartite two-step extensions

   The spectral mechanism: relative estimates for every small eigenvalue on connected bipartite supports, uniformly through arbitrarily separated positive scales.

   Preprint · 2026 · Abstract, PDF & citation ↗

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   We determine every small eigenvalue of the Dolbeault Laplacian on trace-free endomorphisms at a Hermitian–Einstein metric near an intersection of slope-stability walls. The bundle is a fixed extension of one direct sum of stable bundles by another on a compact Kähler manifold, with pairwise nonisomorphic factors of equal slope at the reference form and a connected graph of nonzero extension components. In the stable chamber, the small spectrum is asymptotic to the nonzero spectrum of a weighted Laplacian on this graph. Its vertex masses are the factor ranks, and its edge weights balance the slope differences against the harmonic extension data. The relative error is of order the total edge weight, uniformly as positive stability margins vanish at arbitrarily different rates, and the remaining spectrum has a uniform gap. The proof uses curvature identities across graph cuts to control the nonlinear correction at each vanishing scale. As consequences, we obtain a relative approximation of the metric and a spanning-tree formula for the leading vanishing factor of the zeta-regularized determinant. For trees the leading spectrum is determined by ranks and slopes; cycles can retain dependence on the harmonic extension data.

7. 07

   Uniform Hermitian–Einstein spectra with repeated stable factors

   Repeated stable factors: actual HE metric comparison and every small matrix mode, uniformly at arbitrarily separated positive scales.

   Preprint · 2026 · Abstract, PDF & citation ↗

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   We prove uniform relative asymptotics for all small positive Dolbeault eigenvalues on trace-free endomorphism-valued zero-forms for Hermitian–Einstein metrics on simple two-step extensions near an equal-slope Kähler polarization. The stable factors may occur with arbitrary fixed multiplicities, with distinct isomorphism types in the two layers. The finite comparison operator is the commutator Laplacian on the full complex multiplicity algebra, evaluated at an independently balanced representative of the extension. Every small eigenvalue has relative error bounded by a fixed multiple of the total balanced extension energy, and the remaining spectrum has a uniform positive lower bound. No lower bound on positive eigenvalue ratios or stability margins is imposed. The main estimate bounds the curvature residual against every multiplicity projection by its crossing energy; a degree argument then compares the actual metric with the balanced metric without inverting a degenerating finite Hessian. A fixed rank-five bundle on a product of four projective lines exhibits six small modes and arbitrarily separated positive scales.

8. 08

   Hermitian–Einstein metrics and small spectra for three-step extensions

   Three-step extensions: actual metrics, every small spectral scale and intrinsic polarization response.

   Preprint · 2026 · Abstract, PDF & citation ↗

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   We prove uniform relative asymptotics for every small Dolbeault eigenvalue on trace-free endomorphism-valued zero-forms in a class of three-step Hermitian–Einstein extensions, allowing arbitrary separation of positive spectral scales. The graded factors are distinct line bundles of equal slope at a reference polarization on a compact Kähler manifold. Adjacent extension classes vary in fixed cohomology subspaces; the remaining terms are quadratic coexact Hodge primitives of their compositions, with no independent first-to-third-layer parameters. Assume integrability and a connected extension graph. At nearby polarizations, suppose diagonal rescaling balances the squared norms of the adjacent classes against the slope differences, with sufficiently small sum of rescaled squared norms. The rescaled split metric then approximates the Hermitian–Einstein metric with the same determinant, with logarithmic error bounded by a uniform constant times that sum. Every small eigenvalue has the same relative accuracy with respect to the corresponding eigenvalue of the graph Laplacian weighted by these squared norms; the complementary spectrum has a uniform positive gap. The proof controls curvature traces by the energy crossing each vertex cut. A family on the product of three projective lines has a spectral scale undetected by its polarization parameters. At each fixed bundle in this family, both singular values of the intrinsic metric response to the two polarization parameters agree, with the same relative accuracy, with those of the differentiated finite balancing equation.

Quotient metrics and repeated-factor moduli · 3 manuscriptsThe section ↑

9. 09

   Relative metric estimates near reducible Yang–Mills connections

   Relative surface gauge quotient-metric and small-spectrum estimates, uniform across changing stabilizers and arbitrarily separated scales.

   Preprint · 2026 · Abstract, PDF & citation ↗

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   We study the natural $L^2$ metric near a central Yang–Mills connection on a closed surface. In the classical quadratic symplectic model, we prove a relative metric estimate with error quadratic in the harmonic deformation parameter, uniformly across all stabilizer types. We also compute the leading metric correction using the Green operators of the reference connection. The proof first eliminates the gauge directions with a fixed positive spectral gap and then minimizes over the remaining finite-dimensional orbit; it requires no lower bound on a positive orbit eigenvalue. The same argument compares every small eigenvalue of the gauge Laplacian with the infinitesimal stabilizer-action operator, including at changes of kernel dimension. As a curved finite-dimensional comparison, quaternionic projective and hyperbolic reductions admit exact orbit-metric formulas on their flat moment-map zero cone. A four-cycle is realized in both settings with independently vanishing weights. The resulting estimates distinguish uniform control of the quotient metric from estimates that depend on the smallest orbit scale.

10. 10

    L² metrics on Hermitian–Yang–Mills moduli spaces near a stability wall

    Full local moduli near a wall: uniform relative harmonic L² metrics, chart transitions and intrinsic distance.

    Preprint · 2026 · Abstract, PDF & citation ↗

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    We prove a uniform relative comparison between the harmonic $L^2$ metric on a local Hermitian–Yang–Mills moduli space and the quotient metric of its finite-dimensional harmonic deformation model. The result concerns a rank-four family on the product of three projective lines, near a split sum of four line bundles at a stability wall. The model retains the full analytic obstruction and describes a neighborhood of the corresponding Hermitian–Einstein classes. On its smooth complex pieces, the relative metric error is quadratic in the coefficient radius. Both the comparison constant and the allowed radius are independent of the distance to the wall, even when the infinitesimal orbit map and the derivative of the obstruction lose rank. The proof estimates the ambient deformation norm before taking the quotient, avoiding inversion of either degenerating map. We also identify the transition from this wall model to the harmonic model at a reducible rank-three-plus-line endpoint, with quantitative domains of validity. These comparisons extend to intrinsic lengths and their completions; in an explicit family of stable extensions they give a joint asymptotic for the intrinsic distance to the reducible bundle.

11. 11

    Hermitian–Einstein metrics and moduli of three-step extensions with repeated factors

    Repeated factors in three steps: actual rank-five metrics, six matrix modes and local quotient geometry.

    Preprint · 2026 · Abstract, PDF & citation ↗

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    We construct Hermitian–Einstein metrics and obtain relative asymptotics for all six small eigenvalues of the Dolbeault Laplacian on trace-free endomorphism-valued zero-forms in a rank-five deformation problem with a repeated line factor on the product of three projective lines. The estimates are uniform as positive spectral scales separate or vanish. At nearby product polarizations, the metric construction applies to every sufficiently small simple integrable deformation satisfying the quadratic moment-map equation at the centered slope level. Its logarithmic error relative to the split metric is quadratic in the harmonic coefficients. The key estimate uses harmonic orthogonality to eliminate cubic curvature terms and bounds the remaining error against each multiplicity projection by the energy mixing its image and kernel. This gives relative spectral comparison with the matrix commutator form without a lower bound on its positive eigenvalues. On smooth pieces of the full local Hermitian–Einstein quotient, we likewise compare the harmonic $L^2$ metric with the coefficient quotient metric. At a reducible locus where one copy of the repeated line splits off, the induced metric agrees with that of the remaining stable rank-four summand.

Metric response and dynamics · 4 manuscriptsThe section ↑

12. 12

    Small Dolbeault eigenvalues along Hermitian Yang–Mills flow

    Small spectral modes along Hermitian Yang–Mills flow, described by evolving weighted graph Laplacians in specified bundle classes.

    Preprint · 2026 · Abstract, PDF & citation ↗

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    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.

13. 13

    Metric response and Hermitian Yang–Mills flow for two-step extensions

    Metrics and flows: intrinsic response, singular inverses and actual HYM comparisons, culminating in the integrated theorem on the same tangent family.

    Preprint · 2026 · Abstract, PDF & citation ↗

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    For two-step extensions of pairwise nonisomorphic stable bundles on a compact Kähler manifold, we quantify the degeneration of Hermitian–Einstein metrics and Hermitian Yang–Mills flow near intersecting slope-stability walls. We assume equal slopes at the reference form and connected support for the extension class. When the centered slopes provide full local parameters, every singular value of the trace-free part of $H^{-1}dH$, for the Hermitian–Einstein metric $H$, is relatively asymptotic to the reciprocal of a nonzero eigenvalue of a weighted extension-graph Laplacian. The comparison uses rank-weighted slope and normalized $L^2$ norms and remains uniform for arbitrarily separated positive stability margins. We also construct a finite-dimensional invariant manifold of the flow and compare its solutions with flows from any smooth family of initial metrics. For each tolerance, one late time and then one parameter neighborhood give multiplicative metric comparison, up to a constant scalar, and relative accuracy for every small endomorphism Dolbeault eigenvalue for all subsequent times. The comparison applies in the stable, semistable and unstable regimes. In the two-factor case, we determine a uniform asymptotic profile for the smallest eigenvalue across the wall, including exponential decay on the unstable side. An application to explicit weak Fano toric tangent bundles connects nonsimplicial stability chambers, metric response and flow on the same bundles.

14. 14

    Initial conditions and asymptotic phase for Hermitian–Yang–Mills flow

    Stable dynamics: differentiable initial-condition selection and uniform exponential tracking of a finite flow.

    Preprint · 2026 · Abstract, PDF & citation ↗

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    We prove that the initial Hermitian metric selects a finite-dimensional trajectory that tracks Hermitian–Yang–Mills flow with exponentially decaying relative metric error, uniformly as the positive relaxation rates separate or tend to zero. The result holds for a nearby stable family of deformations of a specified sum of four line bundles on the product of three projective lines, along a fixed polarization path. We construct an exact invariant graph over the full integrable harmonic deformation space. Relative to this chosen graph, the asymptotic phase is continuously differentiable in the initial metric at fixed bundle and polarization. The local theorem permits bounded diagonal metric coordinates and sufficiently small transverse displacement. A prescribed smooth determinant-normalized metric enters this regime after a common restart time, chosen before the deformation, polarization and nearby-metric neighborhoods. The proof uses a fixed complementary spectral gap and metric order, without inverting a small relaxation rate. For a three-layer family with two independent positive slope parameters, we also prove all-future comparison with a small, time-independent relative error.

15. 15

    Hermitian–Yang–Mills flow at and beyond a stability wall

    Wall and unstable dynamics: all-time coefficient confinement, decay and a limiting residual of initial conditions.

    Preprint · 2026 · Abstract, PDF & citation ↗

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    We prove all-time confinement and two-sided decay estimates for Hermitian–Yang–Mills flow whose initial metric is a sufficiently small transverse perturbation of a metric on a fixed finite-dimensional invariant graph. The bundles are small integrable deformations of a specified sum of four line bundles on the product of three projective lines. At the stability wall the directed extension graph is acyclic and connected; beyond it we treat a connected triangular class with locally free Harder–Narasimhan line quotients. In a moving diagonal presentation, the harmonic coefficients remain small although the diagonal metric factors become unbounded. Their energy decays reciprocally at the wall and crosses to exponential decay on the unstable side, uniformly in the ratio of wall distance to initial energy. This confinement yields an asymptotic phase relative to a chosen invariant graph: each such initial metric selects a unique reduced trajectory with exponentially decaying relative metric error. The phase is continuously differentiable in the initial metric at fixed bundle and polarization. At each fixed nearby unstable polarization, subtracting the linear diagonal drift leaves a limiting residual that is locally invertible in the three real diagonal initial coordinates.

Singular graded limits · 1 manuscriptThe section ↑

16. 16

    Singular graded limits and metric growth under Hermitian–Yang–Mills flow

    Singular graded sheaves: a toric surface obstruction to bounded-remainder weight-filtration metric asymptotics.

    Preprint · 2026 · Abstract, PDF & citation ↗

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    We construct a semistable toric tangent bundle for which the weight-filtration description of Hermitian–Yang–Mills metric growth cannot extend to singular graded sheaves with a uniformly bounded remainder. The bundle lies on a smooth projective surface, and its Jordan–Hölder graded sheaf is singular at one point. For every smooth initial metric with fixed Hermitian–Einstein determinant, the induced squared norm on its unique saturated equal-slope line subsheaf is $o\left(T^{-1/2}\right)$ as the flow time $T$ tends to infinity, uniformly on compact subsets away from that point. The rank-polarized iterated weight filtration of Haiden–Katzarkov–Kontsevich–Pandit instead assigns the scale $T^{-1/2}$, with no further nonzero weights. A splitting-independent local residue forces divergent normalized extension energy, which an exact dissipation identity converts into the strict decay estimate. Approaching the wall from the stable side, we also identify the Hermitian–Einstein limit with the reflexive graded bundle and locate its multiplicity-one curvature concentration at the same point.

The broader subject overview, including open FJRW theory and inverse moduli, is the Geometry & moduli guide.