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.Comment: Final version. To appear in Indagationes Mathematica

Model categories in deformation theory

The aim is the formalization of Deformation Theory in an abstract model category, in order to study several geometric deformation problems from a unified point of view. The main geometric application is the description of the DG-Lie algebra controlling infinitesimal deformations of a separated scheme over a field of characteristic 0

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

Hyper-holomorphic connections on vector bundles on hyper-K\"ahler manifolds

We study infinitesimal deformations of autodual and hyper-holomorphic connections on complex vector bundles on hyper-K\"ahler manifolds of arbitrary dimension. We describe the DG Lie algebra controlling this deformation problem, and prove that it is formal when the connection is hyper-holomorphic. Moreover, we prove associative formality for derived endomorphisms of a holomorphic vector bundle admitting a projectively hyper-holomorphic connection.Comment: 26 pages. Comments are very welcome

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.Comment: Post-print version; accepted for publication in Compositio Mathematic

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.Comment: 20 pages, comments welcome

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

We consider a finite acyclic quiver \mathcalQ and a quasi-Frobenius ring $R$. We then characterise Gorenstein-projective modules over the path algebra R\mathcalQ in terms of the corresponding quiver representations over $R$, generalizing the work of X.-H. Luo and P. Zhang to the case of not necessarily finitely generated R\mathcalQ-modules. The proofs are based on Model Category Theory. In particular we endow the category \mathbfRep(\mathcalQ, R) of quiver representations over $R$ with a cofibrantly generated model structure, and we recover the stable category of Gorenstein-projective R\mathcalQ-modules as the homotopy category \mathbfHo(\mathbfRep(\mathcalQ, R))

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

We consider a finite acyclic quiver \mathcalQ and a quasi-Frobenius ring $R$. We then characterise Gorenstein-projective modules over the path algebra R\mathcalQ in terms of the corresponding quiver representations over $R$, generalizing the work of X.-H. Luo and P. Zhang to the case of not necessarily finitely generated R\mathcalQ-modules. The proofs are based on Model Category Theory. In particular we endow the category \mathbfRep(\mathcalQ, R) of quiver representations over $R$ with a cofibrantly generated model structure, and we recover the stable category of Gorenstein-projective R\mathcalQ-modules as the homotopy category \mathbfHo(\mathbfRep(\mathcalQ, R))