arXiv subjects

Tomás L. Gómez

Publications and source records attributed to Tomás L. Gómez.

11 recordsLinked to original sources

arXiv2025

Isomorphisms between moduli stacks of vector bundles with fixed determinant

We classify all isomorphisms between moduli stacks of vector bundles of fixed determinant on a smooth complex projective of genus at least 4. It is shown that each isomorphism between two different moduli stacks can be described as a composition of a pullback using an isomorphism of curves, dualization of vector bundles and tensoring with the pullback of a line bundle on the curve. We finally compare the 2-group of automorphisms of the moduli stack of vector bundles with the group of automorphisms of the moduli space of semistable vector bundles.

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arXiv2024

Torelli theorem for moduli stacks of vector bundles and principal G-bundles

Given any irreducible smooth complex projective curve $X$, of genus at least $2$, consider the moduli stack of vector bundles on $X$ of fixed rank and determinant. It is proved that the isomorphism class of the stack uniquely determines the isomorphism class of the curve $X$ and the rank of the vector bundles. The case of trivial determinant, rank $2$ and genus $2$ is specially interesting: the curve can be recovered from the moduli stack, but not from the moduli space (since this moduli space is $\mathbb{P}^3$ thus independently of the curve). We also prove a Torelli theorem for moduli stacks of principal $G$-bundles on a curve of genus at least $3$, where $G$ is any non-abelian reductive group.

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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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arXiv2021

Comparison of Poisson structures on moduli spaces

Let $X$ be a complex irreducible smooth projective curve, and let ${\mathbb L}$ be an algebraic line bundle on $X$ with a nonzero section $σ_0$. Let $\mathcal{M}$ denote the moduli space of stable Hitchin pairs $(E,\, θ)$, where $E$ is an algebraic vector bundle on $X$ of fixed rank $r$ and degree $δ$, and $θ\, \in\, H^0(X,\, End(E)\otimes K_X\otimes{\mathbb L})$. Associating to every stable Hitchin pair its spectral data, an isomorphism of $\mathcal{M}$ with a moduli space $\mathcal{P}$ of stable sheaves of pure dimension one on the total space of $K_X\otimes{\mathbb L}$ is obtained. Both the moduli spaces $\mathcal{P}$ and $\mathcal{M}$ are equipped with algebraic Poisson structures, which are constructed using $σ_0$. Here we prove that the above isomorphism between $\mathcal{P}$ and $\mathcal{M}$ preserves the Poisson structures.

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arXiv2020

Stability of the Poincaré bundle

Let X be an irreducible smooth projective curve, of genus at least two, over an algebraically closed field k. Let $\mathcal{M}^d_G$ denote the moduli stack of principal G-bundles over X of fixed topological type $d \in π_1(G)$, where G is any almost simple affine algebraic group over k. We prove that the universal bundle over $X \times \mathcal{M}^d_G$ is stable with respect to any polarization on $X \times \mathcal{M}^d_G$. A similar result is proved for the Poincaré adjoint bundle over $X \times M_G^{d, rs}$, where $M_G^{d, rs}$ is the coarse moduli space of regularly stable principal G-bundles over X of fixed topological type d.

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arXiv2020

Complex Lagrangians in a hyperKaehler manifold and the relative Albanese

Let $M$ be the moduli space of complex Lagrangian submanifolds of a hyperKähler manifold $X$, and let $\varpi : \widehat{\mathcal{A}} \rightarrow M$ be the relative Albanese over $M$. We prove that $\widehat{\mathcal{A}}$ has a natural holomorphic symplectic structure. The projection $\varpi$ defines a completely integrable structure on the symplectic manifold $\widehat{\mathcal{A}}$. In particular, the fibers of $\varpi$ are complex Lagrangians with respect to the symplectic form on $\widehat{\mathcal{A}}$. We also prove analogous results for the relative Picard over $M$.

math.DG↗

arXiv2020

Motivic decompositions of moduli spaces of vector bundles on curves

Let $r \geq 2, d$ be two integers which are coprime to each other. Let $C$ be a smooth projective curve of genus $g \geq 2$ and $M(r,L)$ be the moduli space of rank $r$ stable vector bundles on $C$ whose determinants are isomorphic to a fixed line bundle $L$ of degree $d$ on $C.$ In this paper, we study motivic decomposition of $M(r,L)$ for $r=2, 3$ cases. We give a new proof of a version of the main result of arXiv:1806.11101. We also found a new motivic decomposition of $M(3,L).$

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arXiv2014

Integrable systems and Torelli theorems for the moduli spaces of parabolic bundles and parabolic Higgs bundles

We prove a Torelli theorem for the moduli space of semistable parabolic Higgs bundles over a smooth complex projective algebraic curve under the assumption that the parabolic weight system is generic. When the genus is at least two, using this result we also prove a Torelli theorem for the moduli space of semistable parabolic bundles of rank at least two with generic parabolic weights. The key input in the proofs is a method of J.C. Hurtubise, Integrable systems and algebraic surfaces, Duke Math. Jour. 83 (1996), 19--49.

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arXiv2014

Semistability of Principal Bundles on a Kähler Manifold with a Non-Connected Structure Group

We investigate principal $G$-bundles on a compact Kähler manifold, where $G$ is a complex algebraic group such that the connected component of it containing the identity element is reductive. Defining (semi)stability of such bundles, it is shown that a principal $G$-bundle $E_G$ admits an Einstein-Hermitian connection if and only if $E_G$ is polystable. We give an equivalent formulation of the (semi)stability condition. A question is to compare this definition with that of math.AG/0506511.

math.DG↗

arXiv2012

Torelli theorem for the Deligne--Hitchin moduli space, II

Let X and X' be compact Riemann surfaces of genus at least three. Let G and G' be nontrivial connected semisimple linear algebraic groups over C. If some components $M_{DH}^d(X,G)$ and $M_{DH}^{d'}(X',G')$ of the associated Deligne--Hitchin moduli spaces are biholomorphic, then X' is isomorphic to X or to the conjugate Riemann surface $\bar{X}$.

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