arXiv subjects

Carlos Florentino

Publications and source records attributed to Carlos Florentino.

At least 19 recordsLinked to original sources

arXiv2026

Stringy invariants for abelian character varieties

We compute all stringy invariants, encoding orbifold cohomology numbers, of (the normalization of) the identity component of $G$-character varieties of free abelian groups and of moduli spaces of $G$-Higgs bundles on abelian varieties, for all complex connected reductive groups $G$. As an application, we provide a direct proof of a topological mirror symmetry statement: the equality of the stringy invariants for Langlands dual groups. In the framework of Ginzburg--Kaledin local resolutions, we discuss the meaning of these stringy invariants, showing that symplectic resolutions of these moduli spaces can only exist in the cases of groups of Dynkin type $A$, $B$, or $C$. When such resolutions exist, the computed stringy Hodge numbers agree with the actual Hodge numbers of the resolution.

math.AG↗

arXiv2025

Character Varieties of Generalized Torus Knot Groups

Given $\mathbf{n}=(n_{1},\ldots,n_{r})\in\mathbb{N}^r$, let $Γ_{\mathbf{n}}$ be a group presentable as $$\left\langle γ_{1},\ldots,γ_{r}\:|\:γ_{1}^{n_{1}}=γ_{2}^{n_{2}}=\cdots=γ_{r}^{n_{r}}\right\rangle. $$ If $\gcd(n_i,n_j)=1$ for all $i\not=j$, we say $Γ_{\mathbf{n}}$ is a {\it generalized torus knot group} and otherwise say it is a {\it generalized torus link group}. This definition includes torus knot and link groups ($r=2$), that is, fundamental groups of the complement of a torus knot or link in $S^{3}$. Let $G$ be a connected complex reductive affine algebraic group. We show that the $G$-character varieties of generalized torus knot groups are path-connected. We then count the number of irreducible components of the $\mathrm{SL}(2,\mathbb{C})$-character varieties of $Γ_{\mathbf{n}}$ when $n_i$ is odd for all $i$.

math.GT↗

arXiv2024

Mixed Hodge structures on character varieties of nilpotent groups

Let R be the connected component of the identity of the variety of representations of a finitely generated nilpotent group N into a connected reductive complex affine algebraic group G. We determine the mixed Hodge structure on the representation variety R and on the character variety R//G. We obtain explicit formulae (both closed and recursive) for the mixed Hodge polynomial of these representation and character varieties.

math.AG↗

arXiv2023

Flawed groups and the topology of character varieties

A finitely presented group F is called flawed if Hom(F,G)//G deformation retracts onto its subspace Hom(F,K)/K for reductive affine algebraic groups G and maximal compact subgroups K in G. After discussing generalities concerning flawed groups, we show that all finitely generated groups isomorphic to a free product of nilpotent groups are flawed. This unifies and generalizes all previously known classes of flawed groups. We also provide further evidence for the authors' conjecture that RAAGs (with torsion) are flawed. Lastly, we show direct products between finite groups and some flawed group are also flawed. These latter two theorems enlarge the known class of flawed groups.

math.GR↗

arXiv2023

Characterizing maximal varieties via Bredon cohomology

We obtain a characterization of Maximal and Galois-Maximal $C_2$-spaces (including real algebraic varieties) in terms of $\operatorname{RO}(C_2)$-graded cohomology with coefficients in the constant Mackey functor $\underline{\mathbf{F}}_2$, using the structure theorem of \cite{clover_may:structure_theorem}. Other known characterizations, for instance in terms of equivariant Borel cohomology, are also rederived from this. For the particular case of a smooth projective real variety $V$, equivariant Poincaré duality from \cite{pedro&paulo:quaternionic_algebraic_cycles} is used to deduce further symmetry restrictions for the decomposition of the $\operatorname{RO}(C_2)$-graded cohomology of the complex locus $V(\mathbf{C})$ given by the same structure theorem. We illustrate this result with some computations, including the $\operatorname{RO}(C_2)$-graded cohomology with $\underline{\mathbf{F}}_2$ coefficients of real $K3$ surfaces.

math.AG↗

arXiv2023

Topology of the moduli spaces of Higgs bundles over abelian varieties

Abstract. Let G be a complex reductive group and A be an Abelian variety of dimension d over $\mathbb{C}$. We determine the Poincaré polynomials and also the mixed Hodge polynomials of the moduli space $\mathcal{M}_{A}^{H}(G)$ of G-Higgs bundles over A. We show that these are normal varieties with symplectic singularities, when G is a classical semisimple group. For $G=GL_{n}(\mathbb{C})$, we also compute Poincaré polynomials of natural desingularizations of $\mathcal{M}_{A}^{H}(G)$ and of G-character varieties of free abelian groups, in some cases. In particular, explicit formulas are obtained when dim A=d=1, and also for rank 2 and 3 Higgs bundles, for arbitrary d>1.

math.AG↗

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.

math.AG↗

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.

math.AG↗

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 $n$.

math.AG↗

arXiv2018

Homotopy Groups of Free Group Character Varieties

Let G be a connected, complex reductive Lie group with maximal compact subgroup K, and let X denote the moduli space of G- or K-valued representations of a rank r free group. In this article, we develop methods for studying the low-dimensional homotopy groups of these spaces and of their subspaces Y of irreducible representations. Our main result is that when G = GL(n,C) or SL(n,C), the second homotopy group of X is trivial. The proof depends on a new general position-type result in a singular setting. This result is proven in the Appendix and may be of independent interest. We also obtain new information regarding the homotopy groups of the subspaces Y. Recent work of Biswas and Lawton determined the fundamental group of X for general G, and we describe the fundamental group of Y. Specializing to the case G = GL(n,C), we explicitly compute the homotopy groups of the smooth locus of X in a large range of dimensions, finding that they exhibit Bott Periodicity. As a further application of our methods (and in particular our general position result) we obtain new results regarding centralizers of subgroups of G and K, motivated by a question of Sikora. Additionally, we use work of Richardson to solve a conjecture of Florentino-Lawton about the singular locus of X, and we give a topological proof that for G= GL(n,C) or SL(n,C), the space X is not a rational Poincaré Duality Space for r>3 and n=2.

math.AT↗

arXiv2015

Higgs bundles and representation spaces associated to morphisms

Let $G$ be a connected reductive affine algebraic group defined over the complex numbers, and $K\subset G$ be a maximal compact subgroup. Let $X , Y$ be irreducible smooth complex projective varieties and $f: X \rightarrow Y$ an algebraic morphism, such that $π_1(Y)$ is virtually nilpotent and the homomorphism $f_* : π_1(X) \rightarrowπ_1(Y)$ is surjective. Define $$ {\mathcal R }^f(π_1(X),\, G)\,=\, \{ρ\, \in\, \text{Hom}(π_1(X),\, G)\, \mid\, A\circρ\ \text{ factors through }~ f_*\}\, , $$ $$ {\mathcal R }^f(π_1(X),\, K)\,=\, \{ρ\, \in\, \text{Hom}(π_1(X),\, K)\, \mid\, A\circρ\ \text{ factors through }~ f_*\}\, , $$ where $A: G \rightarrow \text{GL}(\text{Lie}(G))$ is the adjoint action. We prove that the geometric invariant theoretic quotient ${\mathcal R }^f(π_1(X, x_0), G)/\!\!/G$ admits a deformation retraction to ${\mathcal R }^f(π_1(X, x_0),\, K)/K$. We also show that the space of conjugacy classes of $n$ almost commuting elements in $G$ admits a deformation retraction to the space of conjugacy classes of $n$ almost commuting elements in $K$.

math.AG↗

arXiv2015

Character varieties of virtually nilpotent Kähler groups and G-Higgs bundles

Let G be a connected complex reductive affine algebraic group, and let K be a maximal compact subgroup. Let X be a compact connected Kähler manifold whose fundamental group Gamma is virtually nilpotent. We prove that the character variety Hom(Gamma, G)/G admits a natural strong deformation retraction to the subset Hom(Gamma, K)/K. The natural action of C^* on the moduli space of G-Higgs bundles over X extends to an action of C. This produces the deformation retraction.

math.AG↗

arXiv2014

Homotopy type of free group character varieties

Let G be a real reductive algebraic group with maximal compact subgroup K, and let F be a rank r free group. Here, we summarize the construction of a natural strong deformation retraction from the space of closed orbits in Hom(F,G)/G to the orbit space Hom(F,K)/K. In particular, these spaces have the same homotopy type.

math.AT↗

arXiv2014

Topology of Moduli Spaces of Free Group Representations in Real Reductive Groups

Let $G$ be a real reductive algebraic group with maximal compact subgroup $K$, and let $F_r$ be a rank $r$ free group. We show that the space of closed orbits in $\mathrm{Hom}(F_r,G)/G$ admits a strong deformation retraction to the orbit space $\mathrm{Hom}(F_r,K)/K$. In particular, all such spaces have the same homotopy type. We compute the Poincaré polynomials of these spaces for some low rank groups $G$, such as $\mathrm{Sp}(4,\mathbb{R})$ and $\mathrm{U}(2,2)$. We also compare these real moduli spaces to the real points of the corresponding complex moduli spaces, and describe the geometry of many examples.

math.AT↗

arXiv2013

Symplectic form on hyperpolygon spaces

In [GM], a family of parabolic Higgs bundles on $CP^1$ has been constructed and identified with a moduli space of hyperpolygons. Our aim here is to give a canonical alternative construction of this family. This enables us to compute the Higgs symplectic form for this family and show that the isomorphism of [GM] is a symplectomorphism.

math.SG↗

arXiv2013

Commuting elements in reductive groups and Higgs bundles on abelian varieties

Let G be a connected real reductive algebraic group, and let K be a maximal compact subgroup of G. We prove that the conjugation orbit space Hom(Z^{2d},K)/K is a strong deformation retract of the space Hom(Z^{2d},G)/G of equivalence classes of representations of Z^{2d} into G. This is proved by showing that the homotopy type of the moduli space of principal G-Higgs bundles of vanishing rational characteristic classes on a complex abelian variety of dimension d depends only on K.

math.AG↗

arXiv2013

Character Varieties and the Moduli of Quiver Representations

Let G be a Lie group and Q a quiver with relations. In this paper, we define G-valued representations of Q which directly generalize G-valued representations of finitely generated groups. Although as G-spaces, the G-valued quiver representations are more general than G-valued representations of finitely generated groups, we show by collapsing arrows that their quotient spaces are equivalent. We then establish a general criterion for the moduli of G-valued quiver representations with relations to admit a strong deformation retraction to a compact quotient by pinching vertices on the quiver. This provides two different generalizations of main results in our previous work. Lastly, we establish quiver theoretic conditions for the moduli spaces of GL(n,C) and SL(n,C)-valued quiver representations to embed into traditional moduli spaces of quiver representations having constant dimension vector.

math.GT↗

arXiv2012

The Topology of Parabolic Character Varieties of Free Groups

Let G be a complex affine algebraic reductive group, and let K be a maximal compact subgroup of G. Fix elements h_1,...,h_m in K. For n greater than or equal to 0, let X (respectively, Y) be the space of equivalence classes of representations of the free group of m+n generators in G (respectively, K) such that for each i between 1 and m, the image of the i-th free generator is conjugate to h_i. These spaces are parabolic analogues of character varieties of free groups. We prove that Y is a strong deformation retraction of X. In particular, X and Y are homotopy equivalent. We also describe explicit examples relating X to relative character varieties.

math.AG↗