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Paolo Stellari

Publications and source records attributed to Paolo Stellari.

At least 19 recordsLinked to original sources
arXiv2026

Stability conditions and moduli spaces on projective families

We extend the construction of stability conditions on projective schemes over a field to projective families over an arbitrary base, and prove that they admit proper relative moduli spaces of semistable objects. We also prove a number of complementary results: the existence of mass-Hom bounds for these stability conditions, as conjectured by Halpern-Leistner and Robotis; a comparison with tilt-stability on surfaces and threefolds; a construction of stability conditions on the supported derived category of total spaces of certain vector bundles, including all local Calabi--Yau varieties; and a simple new proof of Bondal and Orlov's reconstruction theorem.

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arXiv2026

Metrics on triangulated categories and their enhancements

In this paper we investigate the uniqueness of enhancements of the natural subcategories of weakly approximable triangulated categories. The main idea is to enhance at the level of $\infty$-categories the recently developed theory of excellent metrics. The applications of our results include a vast generalization of the known results about the (strong) uniqueness of enhancements in the linear and nonlinear setting, providing positive answers to some open questions. In addition we prove that, under some natural assumptions, the equivalences between such subcategories can be lifted through their natural inclusions. This completes the picture started in our previous paper arXiv:2505.10374 and extends the known results about Margolis Uniqueness Conjecture for the homotopy category of spectra.

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arXiv2026

Hyper-Kähler varieties: Lagrangian fibrations, atomic sheaves, and categories

We review recent developments in the theory of compact hyper-Kähler varieties, from the viewpoint of Lagrangian fibrations, moduli spaces of stable sheaves, and derived categories. These notes originated from the lecture by the second named author at the 2025 Summer Institute in Algebraic Geometry, Colorado State University, Fort Collins (USA), July 14 - August 1, 2025.

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arXiv2025

Stability conditions on products of curves and Hilbert schemes of surfaces

We prove that stability conditions on the derived category of a product of curves of positive genus are uniquely determined by their central charge and the phase of skyscraper sheaves. As an application, we construct stability conditions on Hilbert schemes of points on certain surfaces, including some K3 surfaces of Kummer type.

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arXiv2025

Strong uniqueness of enhancements for the dual numbers: a case study

We prove that the bounded and bounded below derived categories of (all) modules over the dual numbers have strongly unique (dg) enhancements. To this end we relate those categories to the category of sequences of vector spaces, which allows a complete classification of indecomposable objects. Along the way we also prove that all the derived categories of any hereditary category have strongly unique enhancements.

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arXiv2024

Weakly approximable triangulated categories and enhancements: a survey

This paper surveys some recent results, concerning the intrinsicness of natural subcategories of weakly approximable triangulated categories. We also review the results about uniqueness of enhancements of triangulated categories, with the aim of showing the fruitful interplay. In particular, we show how this leads to a vast generalization of a result by Rickard about derived invariance for schemes and rings.

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arXiv2024

Localizations of the categories of $A_\infty$ categories and internal Homs over a ring

We show that, over an arbitrary commutative ring, the localizations of the categories of dg categories, of cohomologically unital, of unital and of strictly unital $A_\infty$ categories with respect to the corresponding classes of quasi-equivalences are all equivalent. The result is proven at the $\infty$-categorical level by considering the natural $\infty$-categorical models of the categories above. As an application of the techniques we develop to compare the localizations mentioned above, we provide a new proof of the existence of internal Homs for the homotopy category of dg categories in terms of the category of (strictly) unital $A_\infty$ functors. This yields a complete proof of a claim by Kontsevich and Keller.

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arXiv2024

The passage among the subcategories of weakly approximable triangulated categories

In this article we prove that all the inclusions between the 'classical' and naturally defined full triangulated subcategories of a weakly approximable triangulated category are intrinsic (in one case under a technical condition). This extends all the existing results about subcategories of weakly approximable triangulated categories. Together with a forthcoming paper about uniqueness of enhancements, our result allows us to generalize a celebrated theorem by Rickard which asserts that if $R$ and $S$ are left coherent rings, then a derived equivalence of $R$ and $S$ is "independent of the decorations". That is, if $D^?(R\text{-}\square)$ and $D^?(S\text{-}\square)$ are equivalent as triangulated categories for some choice of decorations $?$ and $\square$, then they are equivalent for every choice of decorations. But our theorem is much more general, and applies also to quasi-compact and quasi-separated schemes -- even to the relative version, in which the derived categories consist of complexes with cohomology supported on a given closed subscheme with quasi-compact complement.

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arXiv2022

Internal Homs via extensions of dg functors

We provide a simple proof of the existence of internal Homs in the localization of the category of dg categories with respect to all quasi-equivalences and of some of their main properties such as the so-called derived Morita theory. This was originally proved in a seminal paper by Toën.

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arXiv2022

Categorical Torelli theorems: results and open problems

We survey some recent results concerning the so called Categorical Torelli problem. This is to say how one can reconstruct a smooth projective variety up to isomorphism, by using the homological properties of special admissible subcategories of the bounded derived category of coherent sheaves of such a variety. The focus is on Enriques surfaces, prime Fano threefolds and cubic fourfolds.

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arXiv2022

Stability conditions in families

We develop a theory of Bridgeland stability conditions and moduli spaces of semistable objects for a family of varieties. Our approach is based on and generalizes previous work by Abramovich-Polishchuk, Kuznetsov, Lieblich, and Piyaratne-Toda. Our notion includes openness of stability, semistable reduction, a support property uniformly across the family, and boundedness of semistable objects. We show that such a structure exists whenever stability conditions are known to exist on the fibers. Our main application is the generalization of Mukai's theory for moduli spaces of semistable sheaves on K3 surfaces to moduli spaces of Bridgeland semistable objects in the Kuznetsov component associated to a cubic fourfold. This leads to the extension of theorems by Addington-Thomas and Huybrechts on the derived category of special cubic fourfolds, to a new proof of the integral Hodge conjecture, and to the construction of an infinite series of unirational locally complete families of polarized hyperkähler manifolds of K3 type. Other applications include the deformation-invariance of Donaldson-Thomas invariants counting Bridgeland stable objects on Calabi-Yau threefolds, and a method for constructing stability conditions on threefolds via degeneration.

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arXiv2022

Stability conditions on Kuznetsov components

We introduce a general method to induce Bridgeland stability conditions on semiorthogonal components of triangulated categories. In particular, we prove the existence of Bridgeland stability conditions on the Kuznetsov component of the derived category of Fano threefolds and of cubic fourfolds. As an application, in the appendix, written jointly with Xiaolei Zhao, we give a variant of the proof of the Torelli theorem for cubic fourfolds by Huybrechts and Rennemo.

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arXiv2021

Uniqueness of enhancements for derived and geometric categories

We prove that the derived categories of abelian categories have unique enhancements -- all of them, the unbounded, bounded, bounded above and bounded below derived categories. The unseparated and left completed derived categories of a Grothendieck abelian category are also shown to have unique enhancements. Finally we show that the derived category of complexes with quasi-coherent cohomology and the category of perfect complexes have unique enhancements for quasi-compact and quasi-separated schemes.

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arXiv2020

A refined Derived Torelli Theorem for Enriques surfaces

We prove that two general Enriques surfaces defined over an algebraically closed field of characteristic different from $2$ are isomorphic if their Kuznetsov components are equivalent. We apply the same techniques to give a new simple proof of a conjecture by Ingalls and Kuznetsov relating the derived categories of the blow-up of general Artin\textendash Mumford quartic double solids and of the associated Enriques surfaces. This paper originated from one of the problem sections at the workshop \emph{Semiorthogonal decompositions, stability conditions and sheaves of categories}, Université de Toulouse, May 2--5, 2018.

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arXiv2020

Localizations of the category of $A_\infty$ categories and internal Homs

We prove that the localizations of the categories of dg categories, of cohomologically unital and strictly unital $A_\infty$ categories with respect to the corresponding classes of quasi-equivalences are all equivalent. Moreover we show that the last two localizations are equivalent to the corresponding quotients by the relation of being isomorphic in the cohomology of the $A_\infty$ category of $A_\infty$ functors. As an application we give a complete proof of a claim by Kontsevich stating that the category of internal Homs for two dg categories can be described as the category of strictly unital $A_\infty$ functors between them.

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arXiv2019

A tour about existence and uniqueness of dg enhancements and lifts

This paper surveys the recent advances concerning the relations between triangulated (or derived) categories and their dg enhancements. We explain when some interesting triangulated categories arising in algebraic geometry have a unique dg enhancement. This is the case, for example, for the unbounded derived category of quasi-coherent sheaves on an algebraic stack or for its full triangulated subcategory of perfect complexes. Moreover we give an account of the recent results about the possibility to lift exact functors between the bounded derived categories of coherent sheaves on smooth schemes to dg (quasi-)functors.

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arXiv2019

Lectures on non-commutative K3 surfaces, Bridgeland stability, and moduli spaces

We survey the basic theory of non-commutative K3 surfaces, with a particular emphasis to the ones arising from cubic fourfolds. We focus on the problem of constructing Bridgeland stability conditions on these categories and we then investigate the geometry of the corresponding moduli spaces of stable objects. We discuss a number of consequences related to cubic fourfolds including new proofs of the Torelli theorem and of the integral Hodge conjecture, the extension of a result of Addington and Thomas and various applications to hyperkähler manifolds. These notes originated from the lecture series by the first author at the school on "Birational Geometry of Hypersurfaces", Palazzo Feltrinelli - Gargnano del Garda (Italy), March 19-23, 2018.

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