Related papers: Stably semiorthogonally indecomposable varieties
This paper introduces the notion of a stability condition on a triangulated category. The motivation comes from the study of Dirichlet branes in string theory, and especially from M.R. Douglas's notion of $\Pi$-stability. From a…
The nonexistence of semi-orthogonal decompositions in algebraic geometry is known to be governed by the base locus of the canonical bundle. We study another locus, namely the intersection of the base loci of line bundles that are isomorphic…
We study admissible subcategories of the derived categories of smooth noncommutative (nc) curves as classified by Reiten-van den Bergh. We prove that any admissible subcategory of the derived category of a smooth nc curve is again the…
Given a variety $Y$ with a rectangular Lefschetz decomposition of its derived category, we consider a degree $n$ cyclic cover $X \to Y$ ramified over a divisor $Z \subset Y$. We construct semiorthogonal decompositions of $\mathrm{D^b}(X)$…
Given a smooth variety $X$ with an action of a finite group $G$, and a semiorthogonal decomposition of the derived category, $\mathcal{D}([X/G])$, of $G$-equivariant coherent sheaves on $X$ into subcategories equivalent to derived…
This paper deals with properties of the algebraic variety defined as the set of zeros of a "typical" sequence of polynomials. We consider various types of "nice" varieties: set-theoretic and ideal-theoretic complete intersections,…
We define the functor $\textrm{ncDef}_{(Z_1,\ldots,Z_n)}$ of non-commutative deformations of an $n$-tuple of objects in an arbitrary $k$-linear abelian category $\mathcal{Z}$. In our categorified approach, we view the underlying spaces of…
A smooth variety is said to satisfy Condition (A) if every finite abelian subgroup of its automorphism group has a fixed point. We classify smooth Fano 3-folds that satisfy Condition (A).
In this paper we introduce a local-refinement procedure to investigate finite t-stabilities on a triangulated category, and show a direct sufficient condition for a finite t-stability to be finite finest. We classify all finite finest…
The deformation theory of singular varieties plays a central role in understanding the geometry and moduli of algebraic varieties. For a variety $X$ with possibly singular points, the space of first-order infinitesimal deformations is given…
In this paper we prove that every pure-state $\Psi ^{(N)}$ of N (N$\geqslant 3)$ continuous variables corresponds to a pair of convex rigid covers (CRCs) structures in the continuous-dimensional Hilbert-Schmidt space. Next we strictly…
We classify the thick subcategories of an algebraic triangulated standard category with finitely many indecomposable objects.
We show that the bounded derived category of coherent sheaves on a smooth projective curve except the projective line admits no non-trivial semi-orthogonal decompositions.
For a singular variety X, an essential step to determine its smoothability and study its deformations is the understanding of the tangent sheaf and of the sheaf T^1_X:=ext^1(Omega_X,O_X). A variety is semi-smooth if its singularities are…
We investigate the behavior of semi-orthogonal decompositions of bounded derived categories of singular varieties under flat deformations to smooth varieties. We consider a Q-Gorenstein smoothing of a surface with a quotient singularity,…
In the definition of irreducible holomorphic symplectic manifolds the condition of being simply connected can be replaced by vanishing irregularity. We discuss finite quotients X of complex tori where the space of reflexive 2-forms is…
This note aims to clarify the deep relationship between birational modifications of a variety and semiorthogonal decompositions of its derived category of coherent sheaves. The result is a conjecture on the existence and properties of…
We extend the criterion of Kawatani and Okawa for indecomposability of the derived category of a smooth projective variety to arbitrary schemes. For relative schemes, we also give a criterion for the nonexistence of semiorthogonal…
Using the moduli space of semiorthogonal decompositions in a smooth projective family, introduced by the second, the third and the fourth author, we propose a novel approach to indecomposability questions for derived categories. Modulo a…
In this paper we show that any smoothable complex projective variety, smooth in codimension two, with klt singularities and numerically trivial canonical class admits a finite cover, \'etale in codimension one, that decomposes as a product…