arXiv subjects

Andrés Ibáñez Núñez

Publications and source records attributed to Andrés Ibáñez Núñez.

6 recordsLinked to original sources

arXiv2026

Relative étale slices and cohomology of moduli spaces

We use techniques of Alper-Hall-Rydh to prove a local structure theorem for smooth morphisms between smooth stacks around points with linearly reductive stabilizers. This implies that the good moduli space of a smooth stack over a base has equisingular fibers. As an application, we show that any two fibers have isomorphic $\ell$-adic cohomology rings and intersection cohomology groups. If we work over the complex numbers, we show that the family is topologically locally trivial on the base, and that the intersection cohomology groups of the fibers fit into a polarizable variation of pure Hodge structures. We apply these results to derive some consequences for the moduli spaces of $G$-bundles on smooth projective curves, and for certain moduli spaces of sheaves on del Pezzo surfaces.

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arXiv2026

Stability and disconnected groups

We study the notion of semistability for principal bundles over curves with possibly disconnected reductive structure group. We establish a new characterization of the behavior of semistability under change of group, novel even in the connected case, and prove that all existing notions of semistability are equivalent, thus settling a question by Biswas-Gomez. The key ingredients for our results include a study of cocharacters and characters of disconnected linear algebraic groups, and an extension of the recursive description of Kirwan stratifications in Geometric Invariant Theory to the case of disconnected groups.

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arXiv2025

Intrinsic Donaldson-Thomas theory. I. Component lattices of stacks

This is the first paper in a series on intrinsic Donaldson-Thomas theory, where we develop a new framework for enumerative geometry that allows the generalization of constructions and results from linear moduli stacks to general non-linear algebraic stacks. In this paper, we introduce the component lattice of an algebraic stack. This is a key object in our theory, defined using the formalism of stacks of graded and filtered points. It provides the combinatorial data needed to formulate various results in enumerative geometry, such as decomposition-type theorems and wall-crossing formulae. Later papers in the series will focus on extending Donaldson-Thomas theory to the non-linear case, and we expect that our approach will be useful for extending many other flavours of enumerative invariants beyond the linear case as well. This paper proves several foundational results of our framework. The first is the constancy theorem, which states that the isomorphism types of connected components of the stacks of graded and filtered points stay constant within chambers in the component lattice. The second is the finiteness theorem, which provides a criterion for the finiteness of the number of possible isomorphism types of these components. The third is the associativity theorem, generalizing the structure of Hall algebras from linear stacks to general stacks. We also discuss some applications of these results outside Donaldson-Thomas theory, including a construction of stacks of real-weighted filtrations, and a generalization of the semistable reduction theorem to real-weighted filtrations.

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arXiv2025

Cohomology of symmetric stacks

We construct decompositions of: (1) the cohomology of smooth stacks, (2) the Borel--Moore homology of $0$-shifted symplectic stacks, and (3) the vanishing cycle cohomology of $(-1)$-shifted symplectic stacks, assuming a good moduli space exists and the tangent space has a pointwise orthogonal structure. These conditions are satisfied by many stacks of interest, including moduli stacks of semistable $G$-bundles and (twisted) $G$-Higgs bundles on curves, $G$-character stacks of oriented closed 2-manifolds and various 3-manifolds, and moduli stacks of semistable coherent sheaves on Calabi--Yau threefolds and K3 surfaces with generic polarization. As a special case, we prove a PBW-type theorem for cohomological Hall algebras of $3$-Calabi--Yau categories with commutative orientation data, a strong form of the cohomological integrality conjecture for such categories. We define the BPS cohomology as the primary summand of the decomposition. When the stack is smooth, the BPS cohomology coincides with the intersection cohomology of the good moduli space, generalizing a theorem of Meinhardt--Reineke. Using the BPS cohomology for singular spaces, we propose a formulation of the topological mirror symmetry conjecture for the stack of $G$-Higgs bundles generalizing the work of Hausel and Thaddeus for type A groups, and a version of Langlands duality for character stacks of compact oriented 3-manifolds, following Ben-Zvi--Gunningham--Jordan--Safronov.

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arXiv2025

Intrinsic Donaldson-Thomas theory. II. Stability measures and invariants

This is the second paper in a series on intrinsic Donaldson-Thomas theory, a framework for studying the enumerative geometry of general algebraic stacks. In this paper, we present the construction of Donaldson-Thomas invariants for general $(-1)$-shifted symplectic derived Artin stacks, generalizing the constructions of Joyce-Song and Kontsevich-Soibelman for moduli stacks of objects in $3$-Calabi-Yau abelian categories. Our invariants are defined using rings of motives, and depend intrinsically on the stack, together with a set of combinatorial data similar to a stability condition, called a stability measure on the component lattice of the stack. For our invariants to be well-defined, we prove a generalization of Joyce's no-pole theorem to general stacks, using a simpler and more conceptual argument than the original proof in the abelian category case. Further properties and applications of these invariants, such as wall-crossing formulae, will be discussed in a forthcoming paper.

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arXiv2023

Refined Harder-Narasimhan filtrations in moduli theory

We define canonical refinements of Harder-Narasimhan filtrations and stratifications in moduli theory, generalising and relating work of Haiden-Katzarkov-Kontsevich-Pandit and Kirwan. More precisely, we define a canonical stratification on any noetherian algebraic stack $\mathcal X$ with affine diagonal that admits a good moduli space and is endowed with a norm on graded points. The strata live in a newly defined stack of sequential filtrations of $\mathcal X$. Therefore the stratification gives a canonical sequential filtration, the iterated balanced filtration, for each point of $\mathcal X$. Examples of applicability include moduli of principal bundles on a curve, moduli of objects at the heart of a Bridgeland stability condition and moduli of K-semistable Fano varieties. We conjecture that the iterated balanced filtration describes the asymptotics of the Kempf-Ness flow in Geometric Invariant Theory, as part of a larger project aiming to describe the asymptotics of natural flows in moduli theory. In the case of quotient stacks by diagonalisable algebraic groups, we give an explicit description of the iterated balanced filtration in terms of convex geometry.

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