Rank 3 curve semistable Higgs bundles: the nilpotent case

Ugo Bruzzo; Beatriz Graña Otero

Home/Archive/arXiv

arXiv · 2610.32900

Rank 3 curve semistable Higgs bundles: the nilpotent case

Ugo Bruzzo·Beatriz Graña Otero

View original source ↗Read source document ↗Export metadata ↓

Abstract

A Higgs bundle on a smooth complex projective variety is curve semistable if its pullback to every smooth projective curve is semistable. A 2008 conjecture asserts that this is equivalent to slope semistability for some polarization plus vanishing discriminant. One direction is known, the converse holds in rank two, and Huang recently gave a rank-four counterexample. We prove the conjecture for rank three nilpotent Higgs bundles in any dimension. We treat separately the two possible cases: if the Higgs field $ϕ$ is nilpotent of order two, there is a divisorial component of a Higgs Grassmannian where nefness of the normalized tautological class forces the discriminant to vanish; otherwise, $ϕ$ is regular nilpotent, and then the canonical kernel flag, commutativity and Bogomolov's lemma on line subbundles of the cotangent bundle complete the proof.

Explore related subjects

math.AG

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Ugo Bruzzo, Beatriz Graña Otero. 2026-10-23. Rank 3 curve semistable Higgs bundles: the nilpotent case. https://arxiv.org/abs/2610.32900

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Atiyah duality for motivic spectra

We prove that Atiyah duality holds in the $\infty$-category of non-$\mathbb A^1$-invariant motivic spectra over arbitrary derived schemes: every smooth projective scheme is dualizable with dual given by the Thom spectrum of its negative tangent bundle. The Gysin maps recently constructed by L. Tang are a key ingredient in the proof. We then present several applications. First, we study $\mathbb A^1$-colocalization, which transforms any module over the $\mathbb A^1$-invariant sphere into an $\mathbb A^1$-invariant motivic spectrum without changing its values on smooth projective schemes. This can be applied to all known $p$-adic cohomology theories and gives a new elementary approach to "logarithmic" or "tame" cohomology theories; it recovers for instance the logarithmic crystalline cohomology of strict normal crossings compactifications over perfect fields and shows that the latter is independent of the choice of compactification. Second, we prove a motivic Landweber exact functor theorem, associating a motivic spectrum to any graded formal group law classified by a flat map to the moduli stack of formal groups. Using this theorem, we compute the ring of $\mathbb P^1$-stable cohomology operations on the algebraic K-theory of qcqs derived schemes, and we prove that rational motivic cohomology is an idempotent motivic spectrum.

math.AG↗

arXiv2026

Künneth and extension theorems for Morita invariants of non-commutative schemes

In this article, we apply the derived Morita theory of dg-categories to show how to extend the domain of validity of many identities relating Morita invariants from associative dg-algebras toward non-commutative scheme. Doing so, we obtain that the dg-category of associative algebras can be used to test the exactness of any sequence and the commutativity of any diagram involving Morita invariants depending multilinearly on their arguments, under a mild condition of stability by cofibrant replacement. This gives a simple picture of how the underlying algebra of a non-commutative scheme captures its Morita invariant properties. As an application, we use a Künneth formula on non-commutative schemes to factorise the Hochschild cohomology of a product of quasi-phantom categories such as built by Orlov and Gorshinsky in [GO13]. This is an occasion to explore how the derived Morita theory interacts with the Čech dg-enhancement of a projective scheme and to shed an unifying light upon the landscape of dg-modules over dg-categories.

math.AG↗

arXiv2026

Intersections of Schubert varieties and smooth $T$-stable subvarieties of flag varieties

A smooth projective variety equipped with a suitable torus action admits a cell decomposition, called the Białynicki-Birula decomposition. The singularities of the closures of these cells are not well understood in general. Schubert varieties in a flag variety~$G/B$ are classical examples of such closures, and several criteria for their smoothness are known. In this paper, we study the closures of Białynicki-Birula cells in regular semisimple Hessenberg varieties~$\mathrm{Hess}(s,h)$, called Hessenberg Schubert varieties. We first investigate the smoothness of intersections of Schubert varieties with $\mathrm{Hess}(s,h)$, obtaining a sufficient condition for a Hessenberg Schubert variety to be smooth. We also establish several combinatorial sufficient conditions for the smoothness of Hessenberg Schubert varieties in type $A$.

math.AG↗