Related papers: Derived category of projectivizations and flops
We introduce the concept of directed orbifold, namely triples (X, V, D) formed by a directed algebraic or analytic variety (X, V), and a ramification divisor D, where V is a coherent subsheaf of the tangent bundle TX. In this context, we…
We give two proofs to the following theorem and its generalization: if a finite dimensional algebra $A$ is derived equivalent to a smooth projective scheme, then any derived equivalence between $A$ and another algebra $B$ is standard, that…
A counterexample to the Jordan-H\"older property for semiorthogonal decompositions of derived categories of smooth projective varieties was constructed by B\"ohning, Graf von Bothmer and Sosna. In this short note we present a simpler…
In this note we use results of Minamoto and Amiot, Iyama, Reiten to construct an embedding of the graded singularity category of certain graded Gorenstein algebras into the derived categories of coherent sheaves over its projective scheme.…
Suppose $f,g$ are homogeneous polynomials of degree $d$ defining smooth hypersurfaces $X_f = V(f)\subset \mathbb{P}^{m-1}$ and $X_g = V(g)\subset\mathbb{P}^{n-1}$. Then the sum $f(x)+g(y)$ defines a smooth hypersurface…
We present a formula for computing proper pushforwards of classes in the Chow ring of a projective bundle under the projection $\pi:\Pbb(\Escr)\rightarrow B$, for $B$ a non-singular compact complex algebraic variety of any dimension. Our…
In this paper, we investigate Keller's deformed Calabi--Yau completion of the derived category of coherent sheaves on a smooth variety. In particular, for an $n$-dimensional smooth variety $Y$, we describe the derived category of the total…
We leverage the results of the prequel in combination with a theorem of D. Orlov to yield some results in Hodge theory of derived categories of factorizations and derived categories of coherent sheaves on varieties. In particular, we…
We prove that integral projective curves with Cohen-Macaulay singularities over an algebraically closed field of characteristic zero are determined by their triangulated category of perfect complexes.This partially extends a theorem of…
We prove a generalized vanishing theorem for certain quasi-coherent sheaves along the derived blow-ups of quasi-smooth derived Artin stacks. We give four applications of the generalized vanishing theorem: we prove a $K$-theoretic version of…
If Y,Z are three-dimensional smooth varieties related by a flop, then Bondal and Orlov conjectured that the derived categories of coherent sheaves on Y and Z are equivalent. This conjecture was recently proved by Bridgeland. Our aim in this…
The aim of this paper is twofold: First we give an explicit construction of the infinitesimal deformations of the category Coh(X) of coherent sheaves on a smooth projective variety X. Secondly we show that any Fourier-Mukai transform…
By resolving an arbitrary perfect derived object over a Deligne-Mumford stack, we define its Euler class. We then apply it to define the Euler numbers for a smooth Calabi-Yau threefold in the 4-dimensional projective space. These numbers…
We study the derived category of coherent sheaves on various versions of moduli space of vector bundles on curves by the Borel-Weil-Bott theory for loop groups and $\Theta$-stratification, and construct a semiorthogonal decomposition with…
We extend the classical formula of Porteous for blowing-up Chern classes to the case of blow-ups of possibly singular varieties along regularly embedded centers. The proof of this generalization is perhaps conceptually simpler than the…
We study matrix factorizations of locally free coherent sheaves on a scheme. For a scheme that is projective over an affine scheme, we show that homomorphisms in the homotopy category of matrix factorizations may be computed as the…
A conjecture of Orlov predicts that derived equivalent smooth projective varieties over a field have isomorphic Chow motives. The conjecture is known for curves, and was recently observed for surfaces by Huybrechts. In this paper we focus…
In earlier papers arXiv:0802.3120, arXiv:0806.0463 of this series we constructed a sequence of intermediate moduli spaces $\bM^m(c)$ connecting a moduli space $M(c)$ of stable torsion free sheaves on a nonsingular complex projective surface…
We develop an approach that allows to construct semiorthogonal decompositions of derived categories of surfaces with cyclic quotient singularities whose components are equivalent to derived categories of local finite dimensional algebras.…
Kuznetsov showed that for a flat quadric fibration $\mathcal{Q}$ over a smooth base $S$, $\mathrm{D}^b(\mathcal{Q})$ admits a semiorthogonal decomposition where one of the components is the derived category of the sheaf of even parts of a…