arXiv subjects

Francesco Meazzini

Publications and source records attributed to Francesco Meazzini.

11 recordsLinked to original sources

arXiv2026

Higher cotangent cohomology for Stanley-Reisner rings

Inspired by work of Altmann and Christophersen, we study the graded pieces of the cotangent cohomology $T^i_{S_{\mathcal{K}}}$, $i\geq 3$ of the Stanley-Reisner ring $S_{\mathcal{K}}$ associated to a simplicial complex $\mathcal{K}$. We prove a localization formula allowing one to reduce to the case of negative weights. Our results give a complete description of $T^3$ and $T^4$ in terms of the topology of $\mathcal{K}$ whenever $\mathcal{K}$ is a flag complex. As an application, we give a sufficient criterion for the vanishing of $T^3$ for simplicial spheres, classify two-spheres that have vanishing $T^3$, and show that the boundary complex of the dual associahedron has vanishing $T^3$. Our results make use of the arborescent resolutions considered by Hancharuk, Laurent-Gengoux, and Strobl. We give an alternative and self-contained treatment of these resolutions that may be of independent interest.

math.AG↗

arXiv2025

Local Equations for Hilbert Schemes of Points

We compute the completion of the local ring of the Hilbert scheme of degree $n+1$ subschemes of $\mathbb{A}^n$ at the point corresponding to the ideal $\langle x_1,\ldots,x_n\rangle^2$, and describe the completion of the universal family. For the purposes of comparison, we do this computation with both classical and DGLA methods. We use our explicit equations to produce high dimensional linear subspaces of the Hilbert scheme, and compare our equations with those coming from deformations of based algebras.

math.AG↗

arXiv2025

Deformations of twisted sheaves and formality results

We show that infinitesimal deformations of twisted sheaves are controlled by the DG Lie algebra of their derived automorphisms. We prove that such DG Lie algebra is formal for polystable twisted sheaves on minimal surfaces of Kodaira dimension 0 and for projectively hyper-holomorphic locally free twisted sheaves on hyper-K\"ahler manifolds.

math.AG↗

arXiv2022

Hyper-holomorphic connections on vector bundles on hyper-Kähler manifolds

We study infinitesimal deformations of autodual and hyper-holomorphic connections on complex vector bundles on hyper-Kähler manifolds of arbitrary dimension. In particular, we describe the DG Lie algebra controlling this deformation problem. Moreover, we prove associative formality for derived endomorphisms of a holomorphic vector bundle admitting a projectively hyper-holomorphic connection.

math.AG↗

arXiv2022

Hilbert squares of degeneracy loci

Let $S$ be the first degeneracy locus of a morphism of vector bundles corresponding to a general matrix of linear forms in $\mathbb{P}^s$. We prove that, under certain positivity conditions, its Hilbert square $\mathrm{Hilb}^2(S)$ is isomorphic to the zero locus of a global section of an irreducible homogeneous vector bundle on a product of Grassmannians. Our construction involves a naturally associated Fano variety, and an explicit description of the isomorphism.

math.AG↗

arXiv2020

Quiver representations over a quasi-Frobenius ring and Gorenstein-projective modules

We consider a finite acyclic quiver $\mathcal{Q}$ and a quasi-Frobenius ring $R$. We endow the category of quiver representations over $R$ with a model structure, whose homotopy category is equivalent to the stable category of Gorenstein-projective modules over the path algebra $R\mathcal{Q}$. As an application, we then characterize Gorenstein-projective $R\mathcal{Q}$-modules in terms of the corresponding quiver $R$-representations; this generalizes a result obtained by Luo-Zhang to the case of not necessarily finitely generated $R\mathcal{Q}$-modules, and partially recover results due to Enochs-Estrada-García Rozas, and to Eshraghi-Hafezi-Salarian. Our approach to the problem is completely different since the proofs mainly rely on model category theory.

math.RT↗

arXiv2020

Formality conjecture for minimal surfaces of Kodaira dimension 0

Let F be a polystable sheaf on a smooth minimal projective surface of Kodaira dimension 0. Then the DG-Lie algebra RHom(F,F) of derived endomorphisms of F is formal. The proof is based on the study of equivariant $L_{\infty}$ minimal models of DG-Lie algebras equipped with a cyclic structure of degree 2 which is non-degenerate in cohomology, and does not rely (even for K3 surfaces) on previous results on the same subject.

math.AG↗

arXiv2020

Deformations of polystable sheaves on surfaces: quadraticity implies formality

We study relations between the quadraticity of the Kuranishi family of a coherent sheaf on a complex projective scheme and the formality of the DG-Lie algebra of its derived endomorphisms. In particular, we prove that for a polystable coherent sheaf on a smooth complex projective surface the DG-Lie algebra of derived endomorphisms is formal if and only if the Kuranishi family is quadratic.

math.AG↗

arXiv2019

Deformations of algebraic schemes via Reedy-Palamodov cofibrant resolutions

Let $X$ be a Noetherian separated and finite dimensional scheme over a field $\mathbb{K}$ of characteristic zero. The goal of this paper is to study deformations of $X$ over a differential graded local Artin $\mathbb{K}$-algebra by using local Tate-Quillen resolutions, i.e., the algebraic analog of the Palamodov's resolvent of a complex space. The above goal is achieved by describing the DG-Lie algebra controlling deformation theory of a diagram of differential graded commutative algebras, indexed by a direct Reedy category.

math.CT↗

arXiv2018

A DG-enhancement of D(QCoh(X)) with applications in deformation theory

It is well-known that DG-enhancements of D(QCoh(X)) are all equivalent to each other, see [23]. Here we present an explicit model which leads to applications in deformation theory. In particular, we shall describe three models for derived endomorphisms of a quasi-coherent sheaf F on a finite-dimensional Noetherian separated scheme (even if F does not admit a locally free resolution). Moreover, these complexes are endowed with DG-Lie algebra structures, which we prove to control infinitesimal deformations of F.

math.AG↗

arXiv2018

Formal deformation theory in left-proper model categories

We develop the notion of deformation of a morphism in a left-proper model category. As an application we provide a geometric/homotopic description of deformations of commutative (non-positively) graded differential algebras over a local DG-Artin ring.

math.CT↗