How does geometry organize its possibilities?

12 papers · Suggested reading sequence

The manuscripts.

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 blowups and locally trivial toric bundles.

   Preprint · 2026 · Abstract, PDF & citation ↗

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   We study anticanonical slope stability of tangent bundles for two classes of smooth toric Fano varieties. For codimension-two blowups of products of projective spaces indexed by a tree, 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. A specified degree-four neighbourhood forces instability; other degree-four and degree-five arrangements are stable. For 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

   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.

5. 05

   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.

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

   Nonsimplicial local stability chambers of toric tangent bundles

   Geometric realization: nonsimplicial chambers and full slope accessibility on genuine weak-Fano toric tangent bundles, proved independently of the spectral analysis.

   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.

8. 08

   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.

9. 09

   The central-charge threshold for wall-crossing in two-variable open FJRW theory

   When primary invariants become independent of symmetric canonical boundary conditions in two-variable Fermat theory.

   Preprint · 2026 · Abstract, PDF & citation ↗

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   For the genus-zero open FJRW theory of Gross, Kelly and Tessler attached to a two-variable Fermat polynomial, we prove, as they predicted, that the primary invariants do not depend on the choice of symmetric canonical boundary conditions exactly for the simple singularities, of types $A$, $D_4$, $E_6$ and $E_8$, that is, if and only if the central charge is less than one. The proof bounds the critical graphs, which carry the jumps, by the central charge and passes to the invariants through the wall-crossing group. Whenever a critical graph has one boundary marking of each type, this group acts simply transitively only after the vector field of the anti-diagonal torus action is adjoined. Beyond the simple range, dependence first appears with two interior insertions when both exponents are at least four and with three otherwise, while one insertion of descendent degree one produces wall-crossing for every pair. At central charge one every primary wall generator is a multiple of the torus vector field. A B-model analogue, for open Saito potentials, is due to Maher.

10. 10

    Slope-ordered factorization of wall-crossing in two-variable open FJRW theory

    A slope-ordered factorization of wall crossing, constructed in finite truncations.

    Preprint · 2026 · Abstract, PDF & citation ↗

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    For the genus-zero open FJRW invariants of Gross, Kelly and Tessler attached to a two-variable Fermat polynomial, we prove that the wall-crossing transformation between the two extremal systems of invariants factors uniquely, in every finite truncation, into automorphisms supported on rays in the quadrant of boundary-marking numbers, ordered by slope. In these two systems, within each set of invariants mixed by wall-crossing, all vanish except the one with the fewest, respectively most, boundary markings of the first type. Each critical graph contributes a generator on the ray of its boundary-marking numbers, the only ray whose monomials that generator fixes. The proof combines the Kontsevich–Soibelman ordered-product lemma, run with a filtration in place of a grading, with the simply transitive action of the wall-crossing group enlarged by a torus direction. The factorization is realized ray by ray by concatenated homotopies of boundary conditions, though in general not by a single homotopy from the extension lemma of Gross, Kelly and Tessler. Finally, in each truncation the potentials between consecutive rays are transported from one set of initial coefficients, analogous to theta-function coefficients, which can be nonzero on terms with several internal markings.

11. 11

    Annular scattering in quartic open FJRW theory

    An integral-affine annulus and its monodromy in the quartic open FJRW comparison.

    Preprint · 2026 · Abstract, PDF & citation ↗

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    For the open FJRW theory of Gross, Kelly and Tessler attached to the quartic Fermat polynomial in two variables, we prove that, with its two monomials matched to toric divisors of $\mathbb{P}(1,1,2)$, the smallest cover of the tropical base of Carl, Pumperla and Siebert on which the affine monodromy preserves the lattice spanned by the exponents of the base and of the first-descendent wall-crossing generators of twists $(1,0)$ and $(0,1)$ is the double cover branched at two of its three singular points, an integral-affine annulus. Its core holonomy is hyperbolic, and lifting its transitions requires roots of the slab functions that generate an infinite Kummer extension under transport. The annulus carries a wall structure, consistent at its two joints in a sense weaker than that of Gross and Siebert, whose wall-crossing agrees with that of Gross, Kelly and Tessler between the extremal systems of invariants for a single marking but not for two; the difference does not change periods, and its removal from the annular structure is left open. The homotopies of boundary conditions realizing this wall-crossing ray by ray can be chosen compatibly for all markings.

12. 12

    Inverse Moduli Theory: Realizability of Degeneration Data by Compactified Moduli Spaces

    The inverse problem for moduli-space structures, including an obstruction to extending stabilizing bundles.

    Preprint · 2026 · Abstract, PDF & citation ↗

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    Donaldson, Seiberg–Witten, Floer, and Gromov–Witten theories extract invariants or algebraic structures from compactified moduli spaces. We study the inverse problem: which prescribed gluing data and algebraic identities can be realized by compactified moduli spaces, and when can such a realization arise from Fredholm geometry? The formulation uses decorated Feynman categories, marked realization categories, and smoothing profiles that distinguish boundary, real-normal, and complex-normal degenerations.

    The principal result is an obstruction to extending stabilizing bundles across the associahedron $K_4$. A compatible family over $K_2$ and $K_3$ extends if and only if its glued boundary bundle extends over $K_4$; reflecting one endpoint identification produces a Möbius summand on $\partial K_4$ and hence a nonextendable datum. We also prove that the equations $d^2=0$ and the $A_{\infty }$ relations are precisely oriented filling conditions in dimension one, with higher extensions obstructed by ordinary smooth $\theta$-bordism classes. Realization categories form fibers of a cartesian fibration over gluing data, and rigid realizations satisfy descent over overlapping color sets.

    Family-index, compactness, and determinant-line conditions yield further obstructions to Fredholm realization; we examine them in flow, planar, stable-curve, and stable-disk examples. The resulting framework is a sequence of realization and obstruction problems, not a universal existence theorem.