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Ana Peón-Nieto

Publications and source records attributed to Ana Peón-Nieto.

14 recordsLinked to original sources
arXiv2026

Spectral correspondence for cyclic Higgs bundles

In this paper, we describe the spectral correspondence for cyclic Higgs bundles from the viewpoint of quiver bundles. Under this framework, we establish a one-to-one correspondence between cyclic Higgs bundles on a curve and sheaves on a noncommutative surface whose noncommutative structure originates from the path algebra associated to the cyclic quiver. As applications, this correspondence generalizes the known spectral correspondence for $U(p,p)$-Higgs bundles and establish a connection between the spectral data for $U(p,q)$-Higgs bundles and modules over the sheaf of even Clifford algebras of a conic fibration.

math.AG↗
arXiv2026

Wobbly moduli of chains, equivariant multiplicities and $\mathrm{U}(n_0,n_1)$-Higgs bundles

We give a birational description of the reduced schemes underlying the irreducible components of the nilpotent cone and the $\CC^\times$-fixed point locus of length two in the moduli space of Higgs bundles. Using these results, we prove Drinfeld's conjecture for the sublocus of type $(n_0,n_1)$ fixed points. We introduce the notion of $\U(n_0,n_1)$-wobbliness (stronger than the one of wobbliness) and show that fixed point components of type $(n_0,n_1)$ are wobbly in rank higher than three, if and only if they are also $\U(n_0,n_1)$-wobbly. This yields a computable criterion to check wobbliness of fixed point components, that simplifies the existing ones. We analyse the virtual equivariant multiplicities of fixed points of type $(n_0,n_1)$ and their Euler pairings with downward flows for type $(1,\dots, 1)$ fixed points. We find that both invariants fail to fully detect all wobbly components for ordered partitions other than $(2,1)$.

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arXiv2023

Narasimhan--Ramanan branes and wobbly Higgs bundles

Narasimhan--Ramanan branes were introduced by the authors in a previous article. They consist of a family of $BBB$-branes inside the moduli space of Higgs bundles, and a family of complex Lagrangian subvarieties. It was conjectured that these complex Lagrangian subvarieties support the $BAA$-branes that are mirror dual to the Narasimhan--Ramanan $BBB$-branes. In this article we show that the support of these branes intersects non-trivially the locus of wobbly Higgs bundles.

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arXiv2023

Equality of the wobbly and shaky loci

Let $X$ be a smooth complex projective curve of genus $g\geq 2$. We prove that a parabolic vector bundle $\mathcal{E}$ on $X$ on $X$ is (strongly) wobbly, i.e. $\mathcal{E}$ has a non-zero (strongly) parabolic nilpotent Higgs field, if and only if it is (strongly) shaky, i.e., it is in the image of the exceptional divisor of a suitable resolution of the rational map from the (strongly) parabolic Higgs moduli to the parabolic bundle moduli space, both assumed to be smooth. This solves a conjecture by Donagi-Pantev [DP1] in the parabolic and the vector bundle context. To this end, we prove the stability of strongly very stable parabolic bundles, and criteria for very stability of parabolic bundles.

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arXiv2020

Unramified covers and branes on the Hitchin system

We study the locus of the moduli space of Higgs bundles on a curve given by those Higgs bundles obtained by pushforward under an unramified cover. We equip these loci with a hyperholomorphic bundle so that they can be viewed as BBB-branes, and we introduce corresponding BAA-branes which can be described via Hecke modifications. We then show how these branes are naturally dual via explicit Fourier--Mukai transform, where we recall that the structure group $\mathrm{GL}(n,\mathbb{C})$ is Langlands self dual. It is noteworthy that these branes lie over the singular locus of the Hitchin fibration. As a particular case, our construction describes the behaviour under mirror symmetry of the fixed loci for the action of tensorization by a line bundle of order $n$. These loci play a key role in the work of Hausel and Thaddeus on topological mirror symmetry for Higgs moduli spaces.

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arXiv2020

Branes on the singular locus of the Hitchin system via Borel and other parabolic subgroups

We study mirror symmetry on the singular locus of the Hitchin system at two levels. Firstly, by covering it by (supports of) $(BBB)$-branes, corresponding to Higgs bundles reducing their structure group to the Levi subgroup of some parabolic subgroup $\mathrm{P}$, whose conjectural dual $(BAA)$-branes we describe. Heuristically speaking, the latter are given by Higgs bundles reducing their structure group to the unipotent radical of $\mathrm{P}$. Secondly, when $\mathrm{P}$ is a Borel subgroup, we are able to construct a family of hyperholomorphic bundles on the $(BBB)$-brane, and study the variation of the dual under this choice. We give evidence of both families of branes being dual under mirror symmetry via an integral functor induced by Fourier-Mukai in the moduli stack of Higgs bundles.

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arXiv2019

Moduli spaces of framed $G$--Higgs bundles and symplectic geometry

Let $X$ be a compact connected Riemann surface, $D\, \subset\, X$ a reduced effective divisor, $G$ a connected complex reductive affine algebraic group and $H_x\, \subsetneq\, G_x$ a Zariski closed subgroup for every $x\, \in\, D$. A framed principal $G$--bundle is a pair $(E_G,\, ϕ)$, where $E_G$ is a holomorphic principal $G$--bundle on $X$ and $ϕ$ assigns to each $x\, \in\, D$ a point of the quotient space $(E_G)_x/H_x$. A framed $G$--Higgs bundle is a framed principal $G$--bundle $(E_G,\, ϕ)$ together with a section $θ\, \in\, H^0(X,\, \text{ad}(E_G)\otimes K_X\otimes{\mathcal O}_X(D))$ such that $θ(x)$ is compatible with the framing $ϕ$ for every $x\, \in\, D$. We construct a holomorphic symplectic structure on the moduli space $\mathcal{M}_{FH}(G)$ of stable framed $G$--Higgs bundles. Moreover, we prove that the natural morphism from $\mathcal{M}_{FH}(G)$ to the moduli space $\mathcal{M}_{H}(G)$ of $D$-twisted $G$--Higgs bundles $(E_G,\, θ)$ that forgets the framing, is Poisson. These results generalize \cite{BLP} where $(G,\, \{H_x\}_{x\in D})$ is taken to be $(\text{GL}(r,{\mathbb C}),\, \{\text{I}_{r\times r}\}_{x\in D})$. We also investigate the Hitchin system for $\mathcal{M}_{FH}(G)$ and its relationship with that for $\mathcal{M}_{H}(G)$.

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arXiv2019

Higgs bundles, abelian gerbes and cameral data

We study the Hitchin map for $G_{\mathbb{R}}$-Higgs bundles on a smooth curve, where $G_{\mathbb{R}}$ is a quasi-split real form of a complex reductive algebraic group $G$. By looking at the moduli stack of regular $G_{\mathbb{R}}$-Higgs bundles, we prove it induces a banded gerbe structure on a slightly larger stack, whose band is given by sheaves of tori. This characterization yields a cocyclic description of the fibres of the corresponding Hitchin map by means of cameral data. According to this, fibres of the Hitchin map are categories of principal torus bundles on the cameral cover. The corresponding points inside the stack of $G$-Higgs bundles are contained in the substack of points fixed by an involution induced by the Cartan involution of $G_{\mathbb{R}}$. We determine this substack of fixed points and prove that stable points are in correspondence with stable $G_{\mathbb{R}}$-Higgs bundles.

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arXiv2019

Symplectic geometry of a moduli space of framed Higgs bundles

Let $X$ be a compact connected Riemann surface and $D$ an effective divisor on $X$. Let ${\mathcal N}_H(r,d)$ denote the moduli space of $D$-twisted stable Higgs bundles (a special class of Hitchin pairs) on $X$ of rank $r$ and degree $d$. It is known that ${\mathcal N}_H(r,d)$ has a natural holomorphic Poisson structure which is in fact symplectic if and only if $D$ is the zero divisor. We prove that ${\mathcal N}_H(r,d)$ admits a natural enhancement to a holomorphic symplectic manifold which is called here ${\mathcal M}_H(r,d)$. This ${\mathcal M}_H(r,d)$ is constructed by trivializing, over $D$, the restriction of the vector bundles underlying the $D$-twisted Higgs bundles; such objects are called here as framed Higgs bundles. We also investigate the symplectic structure on the moduli space ${\mathcal M}_H(r,d)$ of framed Higgs bundles as well as the Hitchin system associated to it.

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arXiv2017

Very stable bundles and properness of the Hitchin map

Let $X$ be a smooth complex projective curve of genus $g\geq 2$ and let $K$ be its canonical bundle. In this note we show that a stable vector bundle $E$ on $X$ is very stable, i.e. $E$ has no non-zero nilpotent Higgs field, if and only if the restriction of the Hitchin map to the vector space of Higgs fields $H^0(X, \mathrm{End}(E) \otimes K)$ is a proper map.

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arXiv2016

Higgs bundles for real groups and the Hitchin-Kostant-Rallis section

We consider the moduli space of polystable $L$-twisted $G$-Higgs bundles over a compact Riemann surface $X$, where $G$ is a real reductive Lie group, and $L$ is a holomorphic line bundle over $X$. Evaluating the Higgs field at a basis of the ring of polynomial invariants of the isotropy representation, one defines the Hitchin map. This is a map to an affine space, whose dimension is determined by $L$ and the degrees of the polynomials in the basis. Building up on the work of Kostant-Rallis and Hitchin, in this paper, as a first step in the study of the Hitchin map, we construct a section of this map. This generalizes the section constructed by Hitchin when $L$ is the canonical line bundle of $X$ and $G$ is complex. In this case the image of the section is related to the Hitchin-Teichmüller components of the moduli space of representations of the fundamental group of $X$ in $G_{\mathrm{Split}}$, a split real form of $G$. In fact, our construction is very natural in that we can start with the moduli space for $G_{\mathrm{Split}}$, instead of $G$, and construct the section for the Hitchin map for $G_{\mathrm{Split}}$ directly. The construction involves the notion of maximal split subgroup of a real reductive Lie group.

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arXiv2015

Cameral data for SU(p+1,p)-Higgs bundles

We study the cameral and spectral data for the moduli space of polystable $\mathrm{SU}(p+1,p)$-Higgs bundles and deduce the latter from the former. As an application, we obtain that the Toledo invariant classifies the connected components of the regular fibers of the Hitchin map.

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arXiv2009

On sigma-delta-Picard-Vessiot extensions

We study the differential Galois theory of difference equations under weaker hypothesis on the field of constants of the automorphism. This framework yields a new approach to results by C.Hardouin and M.Singer, which answers possitively a question by M.Singer: under the classical hypothesis, the known results are still valid. In particular, our Galois group is isomorphic to theirs over a suitable field. We also explicitly calculate the number of connected components of the Galois group.

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