Related papers: On the deformation theory of pair (X, E)

We prove a general version of the homological perturbation lemma which works in the presence of curvature, and without the restriction to strong deformation retracts, building on work of Markl. A key observation is that the notion of strong…

We develop an invariant deformation theory, in a form accessible to practice, for affine schemes $W$ equipped with an action of a reductive algebraic group $G$. Given the defining equations of a $G$-invariant subscheme $X \subset W$, we…

In this paper, we construct in characteristic zero a derived foliation on derived mapping stacks $\underline{\mathbf{Map}}_S(X,Y)$, for $S$ a base derived stack, $X$ a proper schematic, flat, and local complete intersection derived stack…

We develop square zero obstruction theory for modules over $\mathbb{E}_1$-algebras in an arbitrary stable (presentably) monoidal $\infty$-category. We explicitly describe the obstruction element as the homotopy class of a canonically…

For an $(\infty,n)$-category $\mathscr E$ we define an $(\infty,1)$ category $\mathrm{TwAr}(\mathscr E)$ and provide an isomorphism between the stabilization of the overcategory of $\mathscr E$ in $\mathrm{Cat}_{(\infty,n)}$ and the…

First, we show that a compact object $C$ in a triangulated category, which satisfies suitable conditions, induces a $t$-structure. Second, in an abelian category we show that a complex $P^{\centerdot}$ of small projective objects of term…

Let $R\subseteq E$ be two Lie conformal algebras and $Q$ be a given complement of $R$ in $E$. Classifying complements problem asks for describing and classifying all complements of $R$ in $E$ up to an isomorphism. It is known that $E$ is…

We prove that the automorphism group of a general complete intersection $X$ in a projective space is trivial with a few well-understood exceptions. We also prove that the automorphism group of a complete intersection $X$ acts on the…

For a pair (algebra, module) with equidimensional and isolated singularity we establish the existence of a versal henselian deformation. Obstruction theory in terms of an Andr\'e-Quillen cohomology for pairs is a central ingredient in the…

We investigate the old problem of determining the exact bulk moduli of generic $\mathrm{SU}(3)$-structure flux backgrounds of type II string theory. Using techniques from generalised geometry, we show that the infinitesimal deformations are…

Deformation theory is treated for locally notherian formal schemes (non necessarily smooth). The cotangent complex is defined in the derived category through the homology localization functor. The basic properties and results of a…

Let X be a projective complex K3 surface. Beauville and Voisin singled out a 0-cycle c_X on X of degree 1: it is represented by any point lying on a rational curve in X. Huybrechts proved that the second Chern class of a rigid simple…

Let $X$ be an arbitrary scheme. The category $\mathfrak{Qcoh}(X)$ of quasi--coherent sheaves on $X$ is known that admits arbitrary direct products. However their structure seems to be rather mysterious. In the present paper we will describe…

We classify compactly generated co-t-structures on the derived category of a commutative noetherian ring. In order to accomplish that, we develop a theory for compactly generated Hom-orthogonal pairs (also known as torsion pairs in the…

Let $X$ be any rational surface. We construct a tilting bundle $T$ on $X$. Moreover, we can choose $T$ in such way that its endomorphism algebra is quasi-hereditary. In particular, the bounded derived category of coherent sheaves on $X$ is…

Let $\pi: X \to Y$ be a morphism of projective varieties and suppose that $\alpha$ is a pseudo-effective numerical cycle class satisfying $\pi_*\alpha = 0$. A conjecture of Debarre, Jiang, and Voisin predicts that $\alpha$ is a limit of…

We study the interaction between geometric operations on stacks and algebraic operations on their categories of sheaves. We work in the general setting of derived algebraic geometry: our basic objects are derived stacks X and their…

We prove an unobstructedness result for deformations of subvarieties constrained by intersections with another, fixed subvariety. We deduce smoothness and expected-dimension results for multiple-point loci of generic projections, mainly…

Main topic of the paper is the determination, for a compact complex manifold $M$, of the class of manifolds $X$ which are deformation equivalent to it. If $M$ is a complex torus, then also $X$ is so. After describing the structure of…

Consider a family of integral complex locally planar curves. We show that under some assumptions on the basis, the relative nested Hilbert scheme is smooth. In this case, the decomposition theorem of Beilinson, Bernstein and Deligne asserts…