arXiv subjects

Alfonso Zamora

Publications and source records attributed to Alfonso Zamora.

16 recordsLinked to original sources

arXiv2026

$e$-polynomials of character varieties

These lecture notes contain the material presented at one of the mini-courses of the Workshop on Character Varieties and Higgs Bundles held in Liberia, Guanacaste, Costa Rica, in August 2025. They also contain some exercises for the students attending the conference. This manuscript contains the basic ideas and constructions about $e$-polynomials in character varieties and the state of the art of certain research in the field, plus some new further directions. We introduce mixed Hodge structures and $e$-polynomials, together with a series of arithmetic (counting points over finite fields) and geometric (stratification into parabolic types) techniques to compute them. We include a complete example of the calculation of the $e$-polynomial for the ${\rm GL}_3$-character variety of the free group. Finally, we extend the geometric stratification into parabolic types to a general reductive group $G$ to obtain explicit motivic expressions for the $G$-character varieties, and reduce certain topological mirror symmetry conjectures for these moduli spaces.

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arXiv2026

Segre invariants of principal bundles over a curve

For a vector bundle $V$ over a curve $X$, the Segre invariant $s_n (V)$ encodes the maximal degree attained by rank $n$ subbundles of $V$. The functions $s_n$ define stratifications on moduli of $V$ which are well studied. Let $G$ be a connected reductive algebraic group, and $E \to X$ a principal $G$-bundle. For each parabolic subgroup $P \subset G$ there is a Segre number $s_P (E)$, generalising $s_n (V)$. We show that $s_P$ is semicontinuous in families of $G$-bundles, and thus defines stratifications on moduli spaces of $G$-bundles over $X$. We study the invariance properties of $s_P$, relating the behaviour of $s_P$ and $s_{\phi(P)}$ for a surjective homomorphism $\phi \colon G \to H$ and allowing us to compare the Segre stratifications for $G$ and $H$. Finally, we analyse the stratification for the Borel subgroup $B$ of ${\rm GL}_3$, identifying patterns in the geometry and proving, in particular, a sharp Hirschowitz-type bound on $s_B (E)$ for certain topological types.

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arXiv2024

Root data in character varieties

Given $G$ an algebraic reductive group over an algebraically closed field of characteristic zero and $Γ$ a finitely generated group, we provide a stratification of the $G$-character variety of $Γ$ in terms of conjugacy classes of parabolic subgroups of $G$. Each stratum has the structure of a pseudo-quotient, which is a relaxed GIT notion capturing the topology of the quotient and, therefore, behaving well for motivic computations of invariants of the character varieties. These stratifications are constructed by analyzing the root datum of $G$ to encode parabolic classes. Finally, detailed and explicit motivic formulae are provided for cases with Dynkin diagram of types $A$, $B$, $C$ and $D$.

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arXiv2024

The moduli stack of principal $ρ$-sheaves and Gieseker-Harder-Narasimhan filtrations

Let X be a smooth projective variety and let G be a connected reductive group, both defined over a field of characteristic 0. Given a faithful representation $ρ$ of G into a product of general linear groups, we define a moduli stack of principal $ρ$-sheaves that compactifies the stack of G-bundles on X. We apply the theory developed by Alper, Halpern-Leistner and Heinloth to construct a moduli space of Gieseker semistable principal $ρ$-sheaves. This provides an intrinsic stack-theoretic construction of the moduli space of semistable singular principal bundles as constructed by Schmitt and Gómez-Langer-Schmitt-Sols. Our second main result is the definition of a schematic Gieseker-Harder-Narasimhan filtration for $ρ$-sheaves, which induces a stratification of the stack by locally closed substacks. This filtration for a general reductive group G is a refinement of the canonical slope parabolic reductions previously considered at the level of points by Anchouche-Azad-Biswas and as a stratification of the stack by Gurjar-Nitsure. In an appendix, we apply the same techniques to define Gieseker-Harder-Narasimhan filtrations in arbitrary characteristic and show that they induce a stratification of the stack by radicial morphisms.

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arXiv2023

A guide to moduli theory beyond GIT

In this survey we provide an overview of some recent developments in the construction of moduli spaces using stack-theoretic techniques. We will also explain the analogue of Harder-Narasimhan stratifications for general stacks, known as $Θ$-stratifications. As an application of the ideas exposed here, we address the moduli problem of principal bundles over higher dimensional projective varieties, as well as its different compactifications by the so-called principal $ρ$-sheaves. We construct a stratification by instability types whose lower strata admits a proper good moduli space of ``Gieseker semistable" objects and a new Gieseker-type Harder-Narasimhan filtration for these objects. Detailed proofs of the latter results will appear elsewhere.

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arXiv2020

Serre polynomials of $SL_n$- and $PGL_n$-character varieties of free groups

Let $G$ be a complex reductive group and $\mathcal{X}_{r}G$ denote the $G$-character variety of the free group of rank $r$. Using geometric methods, we prove that $E(\mathcal{X}_{r}SL_{n})=E(\mathcal{X}_{r}PGL_{n})$, for any $n,r\in\mathbb{N}$, where E(X) denotes the Serre (also known as E-) polynomial of the complex quasi-projective variety $X$, settling a conjecture of Lawton-Muñoz in [LM]. The proof involves the stratification by polystable type introduced in [FNZ], and shows moreover that the equality of E-polynomials holds for every stratum and, in particular, for the irreducible stratum of $\mathcal{X}_{r}SL_{n}$ and $\mathcal{X}_{r}PGL_{n}$. We also present explicit computations of these polynomials, and of the corresponding Euler characteristics, based on our previous results and on formulas of Mozgovoy-Reineke for $GL_{n}$-character varieties over finite fields.

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arXiv2020

On Hodge polynomials of Singular Character Varieties

Let $\mathcal{X}_ΓG:=\mathrm{Hom}(Γ,G)/\!/G$ be the $G$-character variety of $Γ$, where $G$ is a complex reductive group and $Γ$ a finitely presented group. We introduce new techniques for computing Hodge-Deligne and Serre polynomials of $\mathcal{X}_ΓG$, and present some applications, focusing on the cases when $Γ$ is a free or free abelian group. Detailed constructions and proofs of the main results will appear elsewhere.

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arXiv2020

Generating series for the $E$-polynomials of $GL(n,{\mathbb C})$-character varieties

With G=GL(n,C), let $\mathcal{X}_ΓG$ be the G-character variety of a given finitely presented group $Γ$, and let $\mathcal{X}^{irr}_ΓG \subset \mathcal{X}_ΓG$ be the locus of irreducible representation conjugacy classes. We provide a concrete relation, in terms of plethystic functions, between the generating series for E- polynomials of $\mathcal{X}_ΓG$ and the one for $\mathcal{X}^{irr}_ΓG$, generalizing a formula of Mozgovoy-Reineke [MR]. The proof uses a natural stratification of $\mathcal{X}_ΓG$ coming from affine GIT, the combinatorics of partitions, and the formula of MacDonald-Cheah for symmetric products; we also adapt it to the so-called Cartan brane in the moduli space of Higgs bundles. Combining our methods with arithmetic ones yields explicit expressions for the E-polynomials of the irreducible stratum of GL(n,C)-character varieties of some groups $Γ$, including surface groups, free groups, and torus knot groups, for low values of $n$.

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arXiv2019

Geometric Invariant Theory, holomorphic vector bundles and the Harder-Narasimhan filtration

This survey intends to present the basic notions of Geometric Invariant Theory (GIT) through its paradigmatic application in the construction of the moduli space of holomorphic vector bundles. Special attention is paid to the notion of stability from different points of view and to the concept of maximal unstability, represented by the Harder-Narasimhan filtration and, from which, correspondences with the GIT picture and results derived from stratifications on the moduli space are discussed.

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arXiv2019

Generating sets for Clifford Algebras

The aim of this paper is to find generating sets of commuting involutions and use them to explicitly construct minimal representations of Clifford algebras $Cl(n)_{p,q}$. By results of [HL] and [LW], we know the dimension of such minimal representations, which is linked to the maximal number of commuting involutions in the algebra, dependent only on $p$ and $q$. We provide an algorithm to construct these generating sets of involutions explicitly for all Clifford algebras $Cl(n)_{p,q}$ and provide some examples. Involutions yield mutually non-annihilating idempotents whose product gives a projection map $P^+$ with image $Cl(n)P^+$ being a minimal left ideal, which is a spinor space. Using the projections, we find minimal representations of Clifford algebras by combining matrices of left multiplication endomorphisms in the spinor spaces. Finally, we provide examples showing calculations of minimal representations.

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arXiv2015

Stability conditions and related filtrations for $(G,h)$-constellations

Given an infinite reductive algebraic group $G$, we consider $G$-equivariant coherent sheaves with prescribed multiplicities, called $(G,h)$-constellations, for which two stability notions arise. The first one is analogous to the $\theta$-stability defined for quiver representations by King and for $G$-constellations by Craw and Ishii, but depending on infinitely many parameters. The second one comes from Geometric Invariant Theory in the construction of a moduli space for $(G,h)$-constellations, and depends on some finite subset $D$ of the isomorphy classes of irreducible representations of $G$. We show that these two stability notions do not coincide, answering negatively a question raised in [BT15]. Also, we construct Harder-Narasimhan filtrations for $(G,h)$-constellations with respect to both stability notions (namely, the $\mu_{\theta}$-HN and $\mu_D$-HN filtrations). Even though these filtrations do not coincide in general, we prove that they are strongly related: the $\mu_{\theta}$-HN filtration is a subfiltration of the $\mu_D$-HN filtration, and the polygons of the $\mu_D$-HN filtrations converge to the polygon of the $\mu_{\theta}$-HN filtration when $D$ grows.

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arXiv2014

Harder-Narasimhan filtration for rank 2 tensors and stable coverings

We construct a Harder-Narasimhan filtration for rank $2$ tensors, where there does not exist any such notion a priori, as coming from a GIT notion of maximal unstability. The filtration associated to the 1-parameter subgroup of Kempf giving the maximal way to destabilize, in the GIT sense, a point in the parameter space of the construction of the moduli space of rank $2$ tensors over a smooth projective complex variety, does not depend on certain integer used in the construction of the moduli space, for large values of the integer. Hence, this filtration is unique and we define the Harder-Narasimhan filtration for rank $2$ tensors as this unique filtration coming from GIT. Symmetric rank $2$ tensors over smooth projective complex curves define curve coverings lying on a ruled surface, hence we can translate the stability condition to define stable coverings and characterize the Harder-Narasimhan filtration in terms of intersection theory.

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arXiv2014

On the Gieseker Harder-Narasimhan filtration for principal bundles

We give an example of an orthogonal bundle where the Harder-Narasimhan filtration, with respect to Gieseker semistability, of its underlying vector bundle does not correspond to any parabolic reduction of the orthogonal bundle. A similar example is given for the symplectic case.

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arXiv2014

GIT characterizations of Harder-Narasimhan filtrations

This Ph.D. thesis studies the relation between the Harder-Narasimhan filtration and a notion of GIT maximal unstability. When constructing a moduli space by using Geometric Invariant Theory (GIT), a notion of GIT stability appears, which is determined by 1-parameter subgroups. This thesis shows a correspondence between the 1-parameter subgroup giving maximal unstability from the GIT point of view and the Harder-Narasimhan filtration for different moduli problems: torsion free coherent sheaves, holomorphic pairs, Higgs sheaves, rank 2 tensors and quiver representations. The article [GSZ] contains the correspondence for torsion free coherent sheaves, whereas [Za1] and [Za2] are devoted to finite dimensional quiver representations and rank 2 tensors. In [HK], the authors explore this kind of correspondences while identifying the Hesselink stratification on conjugacy classes of 1-parameter subgroups with the one on Harder-Narasimhan types, showing that the Hesselink's adapted 1-parameter subgroup corresponds to the Harder-Narasimhan filtration (which is previously prescribed on each strata). Futher work of Hoskins (c.f. [Ho1, Ho2]) continues on the same direction for other moduli problems, comprising finite dimensional quiver representations as it is done in [Za1], and coherent sheaves using the functorial construction of Alvarez-Consul and King (c.f. [ACK]), as it appears in Section 3.2 of this thesis.

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arXiv2014

A GIT interpretration of the Harder-Narasimhan filtration

An unstable torsion free sheaf on a smooth projective variety gives a GIT unstable point in certain Quot scheme. To a GIT unstable point, Kempf associates a "maximally destabilizing" 1-parameter subgroup, and this induces a filtration of the torsion free sheaf. We show that this filtration coincides with the Harder-Narasimhan filtration.

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arXiv2013

On the Harder-Narasimhan filtration for finite dimensional representations of quivers

We prove that the Harder-Narasimhan filtration for an unstable finite dimensional representation of a finite quiver coincides with the filtration associated to the 1-parameter subgroup of Kempf, which gives maximal unstability in the sense of Geometric Invariant Theory for the corresponding point in the parameter space where these objects are parametrized in the construction of the moduli space.

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