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Emma Lepri

Publications and source records attributed to Emma Lepri.

8 recordsLinked to original sources
arXiv2026

Deformations of morphisms of coherent sheaves

We generalise Hinich's Theorem of descent of Deligne groupoids to the case where the dgLas involved have no negative cohomology. We apply this result to study the infinitesimal deformations of a morphism $\alpha: {\mathcal F} \to {\mathcal G}$ of coherent sheaves, where both the sheaves $ {\mathcal F}$ and $ {\mathcal G}$ and the map $\alpha$ can be deformed, on a smooth variety over a field of characteristic zero. In particular, we provide an explicit dgLa that controls these deformations via the Deligne functor, applying the Thom-Whitney totalisation to a specific semicosimplicial dgLa, constructed from geometrical data.

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arXiv2024

Trivial extension DG-algebras, unitally positive $A_\infty$-algebras, and applications

To any periodic module over any algebra, this paper introduces an associated trivial extension DG-algebra T. After first passing to a strictly unital $A_\infty$-minimal model, it then constructs a particular $A_\infty$-algebra N, called the unitally positive $A_\infty$-algebra, which roughly speaking describes the identity in degree zero and all the positive cohomology. The object N is fundamental, and can be constructed for any DG-category satisfying very mild assumptions. The main application is to birational geometry. When applied to contraction algebras, the construction gives a simple and direct proof of the Donovan-Wemyss conjecture, namely that smooth irreducible 3-fold flops are classified by their contraction algebras, and thus by noncommutative data.

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arXiv2022

An elementary approach to the model structure on DG-Lie algebras

This paper contains an elementary proof of the existence of the classical model structure on the category of unbounded DG-Lie algebras over a field of characteristic zero, with an emphasis on the properties of free and semifree extensions, which are particularly nice cofibrations. The cobar construction of a locally conilpotent cocommutative coalgebra is shown to be an example of semifree DG-Lie algebra. We also give an example of a non-cofibrant DG-Lie algebra whose underlying graded Lie algebra is free; this cannot occur in the bounded above case, where DG-Lie algebras of this form are always cofibrant.

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arXiv2022

Semiregularity maps and deformations of modules over Lie algebroids

We determine a DG-Lie algebra controlling deformations of a locally free module over a Lie algebroid $\mathcal{A}$. Moreover, for every flat inclusion of Lie algebroids $\mathcal{A}\subset \mathcal{L}$ we introduce semiregularity maps and prove that they annihilate obstructions, provided that the Leray spectral sequence of the pair $(\mathcal{L},\mathcal{A})$ degenerates at $E_1$.

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arXiv2021

Cyclic forms on DG-Lie algebroids and semiregularity

Given a transitive DG-Lie algebroid $(\mathcal{A}, \rho)$ over a smooth separated scheme $X$ of finite type over a field $\mathbb{K}$ of characteristic $0$ we define a notion of connection $\nabla \colon \mathbf{R}\Gamma(X,\mathrm{Ker} \rho) \to \mathbf{R}\Gamma (X,\Omega_X^1[-1]\otimes \mathrm{Ker} \rho)$ and construct an $L_\infty$ morphism between DG-Lie algebras $f \colon \mathbf{R}\Gamma(X, \mathrm{Ker} \rho) \rightsquigarrow\mathbf{R}\Gamma(X, \Omega_X^{\leq 1} [2])$ associated to a connection and to a cyclic form on the DG-Lie algebroid. In this way, we obtain a lifting of the first component of the modified Buchweitz-Flenner semiregularity map in the algebraic context, which has an application to the deformation theory of coherent sheaves on $X$ admitting a finite locally free resolution. Another application is to the deformations of (Zariski) principal bundles on $X$.

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arXiv2021

Connections and $L_{\infty}$ liftings of semiregularity maps

Let $E^*$ be a finite complex of locally free sheaves on a complex manifold $X$. We prove that to every connection of type $(1,0)$ on $E^*$ it is canonically associated an $L_{\infty}$ morphism $g\colon A^{0, *}_X(\mathcal{H}om^*_{O_X}(E^*,E^*))\to \dfrac{A^{*,*}_X}{A^{\ge 2,*}_X}[2]$ that lifts the 1-component of Buchweitz-Flenner semiregularity map. An application to deformations of coherent sheaves on projective manifolds is given.

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arXiv2019

On deformations of diagrams of commutative algebras

In this paper we study classical deformations of diagrams of commutative algebras over a field of characteristic 0. In particular we determine several homotopy classes of DG-Lie algebras, each one of them controlling this above deformation problem: the first homotopy type is described in terms of the projective model structure on the category of diagrams of differential graded algebras, the others in terms of the Reedy model structure on truncated Bousfield-Kan approximations. The first half of the paper contains an elementary introduction to the projective model structure on the category of commutative differential graded algebras, while the second half is devoted to the main results.

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