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Donatella Iacono

Publications and source records attributed to Donatella Iacono.

At least 19 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 $α: {\mathcal F} \to {\mathcal G}$ of coherent sheaves, where both the sheaves $ {\mathcal F}$ and $ {\mathcal G}$ and the map $α$ 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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arXiv2026

Joint Deformations of Manifolds, Coherent Sheaves and Sections

We describe a differential graded Lie algebra controlling infinitesimal deformations of triples $(X,\mathcal{F},σ)$, where $\mathcal{F}$ is a coherent sheaf on a smooth variety $X$ over a field of characteristic 0 and $σ\in H^0(X,\mathcal{F})$. Then, we apply this result to investigate deformations of pairs (variety, divisor).

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arXiv2025

Deformations of morphisms of sheaves

We analyse infinitesimal deformations of morphisms of locally free sheaves on a smooth projective variety $X$ over an algebraically closed field of characteristic zero. In particular, we describe a differential graded Lie algebra controlling the deformation problem. As an application, we study infinitesimal deformations of pairs given by a locally free sheaf and a subspace of it sections with a view towards Brill-Noether theory.

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arXiv2023

On the local structure of the Brill-Noether locus of locally free sheaves on a smooth variety

We study the functor $\operatorname{Def}_E^k$ of infinitesimal deformations of a locally free sheaf $E$ of $\mathcal{O}_X$-modules on a smooth variety $X$, such that at least $k$ independent sections lift to the deformed sheaf, where $h^0(E) \geq k$. We deduce some information on the $k$-th Brill-Noether locus of $E$, such as the description of the tangent cone at some singular points, of the tangent space at some smooth ones and some links between the smoothness of the functor $\operatorname{Def}_E^k$ and the smoothness of some well know deformations functors and their associated moduli spaces. As a tool for the investigation of $\operatorname{Def}_E^k$, we study infinitesimal deformations of the pairs $(E,U)$, where $U$ is a linear subspace of sections of $E$. We generalise to the case where $E$ has any rank and $X$ any dimension many classical results concerning the moduli space of coherent systems, like the description of its tangent space and the link between its smoothness and the injectivity of the Petri map.

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arXiv2020

Deformations of Calabi-Yau manifolds in Fano toric varieties

In this article, we investigate deformations of a Calabi-Yau manifold $Z$ in a toric variety $F$, possibly not smooth. In particular, we prove that the forgetful morphism from the Hilbert functor $H^F_Z$ of infinitesimal deformations of $Z$ in $F$ to the functor of infinitesimal deformations of $Z$ is smooth. This implies the smoothness of $H^F_Z $ at the corresponding point in the Hilbert scheme. Moreover, we give some examples and include some computations on the Hodge numbers of Calabi-Yau manifolds in Fano toric varieties.

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arXiv2020

Homotopy abelianity of the DG-Lie algebra controlling deformations of pairs (variety with trivial canonical bundle, line bundle)

We investigate the deformations of pairs $(X,L)$, where $L$ is a line bundle on a smooth projective variety $X$, defined over an algebraically closed field $\mathbb{K}$ of characteristic 0. In particular, we prove that the DG-Lie algebra controlling the deformations of the pair $(X,L)$ is homotopy abelian whenever $X$ has trivial canonical bundle, and so these deformations are unobstructed.

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arXiv2018

On deformations of pairs (manifold, coherent sheaf)

We analyse infinitesimal deformations of pairs $(X,\mathcal{F})$ with $\mathcal{F}$ a coherent sheaf on a smooth projective manifold $X$ over an algebraic closed field of characteristic $0$. We describe a differential graded Lie algebra controlling the deformation problem, and we prove an analog of a Mukai-Artamkin Theorem about the trace map.

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arXiv2017

On the abstract Bogomolov-Tian-Todorov Theorem

We describe an abstract version of the Theorem of Bogomolov-Tian-Todorov, whose underlying idea is already contained in various papers by Bandiera, Fiorenza, Iacono, Manetti. More explicitly, we prove an algebraic criterion for a differential graded Lie algebras to be homotopy abelian. Then, we collect together many examples and applications in deformation theory and other settings.

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arXiv2016

Diffeomorphism classes of Calabi-Yau varieties

In this article we investigate diffeomorphism classes of Calabi-Yau threefolds. In particular, we focus on those embedded in toric Fano manifolds. Along the way, we give various examples and conclude with a curious remark regarding mirror symmetry.

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arXiv2013

Deformations and obstructions of pairs (X,D)

We study deformations of pairs (X,D), with X smooth projective variety and D a smooth or a normal crossing divisor, defined over an algebraically closed field of characteristic 0. Using the differential graded Lie algebras theory and the Cartan homotopy construction, we are able to prove in a completely algebraic way the unobstructedness of the deformations of the pair (X,D) in many cases, e.g., whenever (X,D) is a log Calabi-Yau pair, in the case of a smooth divisor D in a Calabi Yau variety X and when D is a smooth divisor in |-m K_X|, for some positive integer m.

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arXiv2012

Semiregularity and obstructions of complete intersections

We prove that, on a smooth projective variety over an algebraically closed field of characteristic 0, the semiregularity map annihilates every obstruction to embedded deformations of a local complete intersection subvariety with extendable normal bundle. The proof is based on the theory of L-infinity algebras and Tamarkin-Tsigan calculus on the de Rham complex of DG-schemes.

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arXiv2012

The Tutte polynomial of the Sierpinski and Hanoi graphs

We study the Tutte polynomial of two infinite families of finite graphs: the Sierpiński graphs, which are finite approximations of the well-known Sierpiński gasket, and the Schreier graphs of the Hanoi Towers group $H^{(3)}$ acting on the rooted ternary tree. For both of them, we recursively describe the Tutte polynomial and we compute several special evaluations of it, giving interesting results about the combinatorial structure of these graphs.

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arXiv2010

The Tutte Polynomial of the Schreier graphs of the Grigorchuk group and the Basilica group

We study the Tutte polynomial of two infinite families of finite graphs. These are the Schreier graphs associated with the action of two well-known self-similar groups acting on the binary rooted tree by automorphisms: the first Grigorchuk group of intermediate growth, and the iterated monodromy group of the complex polynomial $z^2-1$ known as the Basilica group. For both of them, we describe the Tutte polynomial and we compute several special evaluations of it, giving further information about the combinatorial structure of these graphs.

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arXiv2010

Deformations of algebraic subvarieties

In this paper, we use (bi)semicosimplicial language to study the classical problem of infinitesimal deformations of a closed subscheme in a fixed smooth variety, defined over an algebraically closed field of characteristic 0. In particular, we give an explicit description of the differential graded Lie algebra controlling this problem.

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arXiv2009

Differential graded Lie algebras controlling infinitesimal deformations of coherent sheaves

We use the Thom-Whitney construction to show that infinitesimal deformations of a coherent sheaf F are controlled by the differential graded Lie algebra of global sections of an acyclic resolution of the sheaf End(E), where E is any locally free resolution of F. In particular, one recovers the well known fact that the tangent space to deformations of F is Ext^1(F,F), and obstructions are contained in Ext^2(F,F). The main tool is the identification of the deformation functor associated with the Thom-Whitney DGLA of a semicosimplicial DGLA whose cohomology is concentrated in nonnegative degrees with a noncommutative Cech cohomology-type functor.

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arXiv2009

An algebraic proof of Bogomolov-Tian-Todorov theorem

We give a completely algebraic proof of the Bogomolov-Tian-Todorov theorem. More precisely, we shall prove that if X is a smooth projective variety with trivial canonical bundle defined over an algebraically closed field of characteristic 0, then the L-infinity algebra governing infinitesimal deformations of X is quasi-isomorphic to an abelian differential graded Lie algebra.

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