arXiv subjects

Tommaso Scognamiglio

Publications and source records attributed to Tommaso Scognamiglio.

6 recordsLinked to original sources

arXiv2026

Connected components of real loci in moduli spaces of vector and Higgs bundles over a Klein surface

Let $X$ be a Riemann surface of genus $g \geqslant 2$ and let $σ: X \to X$ be an antiholomorphic involution on $X$. Let $\mathcal{N}(r,d)$ be the moduli space of semistable vector bundles of rank $r$ and degree $d$ on $X$, with the induced real structure. Using a gauge-theoretic approach, we determine the number of connected components of the real locus of $\mathcal{N}(r,d)$ for general $r$ and $d$. We show in particular that, when the base curve has real points, quaternionic vector bundles can exist for even rank and degree but that the number of connected components of $\mathbb{R}\mathcal{N}(r,d)$ is still equal to that of $\mathbb{R}\mathrm{Pic}_d$. In contrast, when the base curve has empty real locus and $r$ and $d$ are not coprime, the number of connected components of $\mathbb{R}\mathcal{N}(r,d)$ can be smaller than that of $\mathbb{R}\mathrm{Pic}_d$. We then generalize these results to real loci of moduli spaces of Higgs bundles and apply them to the study of the topology of certain $(A,A,A)$ and $(A,B,A)$ branes in the associated hyperkähler quotient.

math.AG↗

arXiv2026

$\mathrm{PGL}_n(\mathbb{C})$-character stacks and Langlands duality over finite fields

In this paper we study the mixed Poincaré polynomial of generic $\mathrm{PGL}_n(\mathbb{C})$-character stacks with coefficients in some local systems arising from the conjugacy classes of $\mathrm{PGL}_n(\mathbb{C})$ which have non-connected stabiliser. We give a conjectural formula that we prove to be true under the Euler specialisation. We then prove that this conjectured formula interpolates the structure coefficients of the two based rings$ \left(\mathcal{C}(\mathrm{PGL}_n(\mathbb{F}_q)),Loc(\mathrm{PGL}_n),*\right)$ and $\left(\mathcal{C}(\mathrm{SL}_n(\mathbb{F}_q)), CS(\mathrm{SL}_n),\cdot\right) $ where for a group $H$, $\mathcal{C}(H)$ denotes the space of complex valued class functions on $H$, $Loc(\mathrm{PGL}_n)$ denotes the basis of characteristic functions of intermediate extensions of equivariant local systems on conjugacy classes of $\mathrm{PGL}_n$ and $CS(\mathrm{SL}_n)$ the basis of characteristic functions of Lusztig's character-sheaves on $\mathrm{SL}_n$. Our result reminds us of a non-abelian Fourier transform.

math.RT↗

arXiv2025

Real Bialynicki-Birula flows in moduli spaces of Higgs bundles

Let $X$ be a compact Riemann surface $X$ of genus $\geqslant 2$ and let $σ:X \to X$ be an anti-holomorphic involution. Using real and quaternionic systems of Hodge bundles, we study the topology of the real locus $\mathbb{R} \mathbf{M}_{\mathrm{Dol}}(r,d)$ of the moduli space of semistable Higgs bundles of rank $r$ and degree $d$ on $X$, for the induced real structure $(E,ϕ) \to (σ^*(\overline{E}),σ^*(\overlineϕ))$. We show in particular that, when $\mathrm{gcd}(r,d)=1$, the number of connected components of $\mathbb{R} \mathbf{M}_{\mathrm{Dol}}(r,d)$ coincides with that of $\mathbb{R} \mathrm{Pic}_d(X)$, which is well-known.

math.AG↗

arXiv2024

A generalization of Kac polynomials and tensor product of representations of $GL_n(\mathbb{F}_q)$

We study the multiplicities of semisimple split characters in tensor product of semisimple split characters of $GL_n(\mathbb{F}_q)$. We prove that these multiplicities are polynomial in q with non-negative integer coefficients and we obtain a criterion for their non-vanishing. We give moreover an interpretation of these polynomials in terms of the counting of the representations of star-shaped quivers, generalizing a previous result of Hausel, Letellier and Rodriguez-Villegas, who linked multiplicities for generic $k$-tuples of semisimple split characters and Kac polynomials.

math.RT↗

arXiv2024

Cohomology of non-generic character stacks

We study (compactly supported) cohomology of character stacks of punctured Riemann surfaces with prescribed semisimple local monodromies at punctures. In the case of generic local monodromies, the cohomology of these character stacks has already been studied by Hausel, Letellier and Rodriguez-Villegas and by Mellit. In this paper, we extend the results of Hausel, Letellier and Rodriguez-Villegas to the non-generic case. In particular, we compute the E-series and we give a conjectural formula for the mixed Poincaré series. Moreover, we verify our conjecture in the case of the projective line with 4 punctures and a certain choice of a non-generic quadruple.

math.AG↗

arXiv2023

On the cohomology of character stacks for non-orientable surfaces

We give a counterexample to a formula suggested by the work of Letellier and Rodriguez-Villegas (arXiv:2008.13435) for the mixed Poincaré series of character stacks for non-orientable surfaces. The counterexample is obtained by an explicit description of these character stacks for (real) elliptic curves.

math.AG↗