Related papers: All Quiet on the Exceptional Locus

We show various properties of smooth projective D-affine varieties. In particular, any smooth projective D-affine variety is algebraically simply connected and its image under a fibration is D-affine. In characteristic zero such D-affine…

For a geometrically rational surface X over an arbitrary field of characteristic different from 2 and 3 that contains all roots of 1, we show that either X is birational to a product of a projective line and a conic, or the group of…

To a smooth and proper morphism $\mathcal{X}\to U$ with quasicompact semiseparated target we associate a sheaf in the \'etale topology, which takes an affine $U$-scheme $V$ to the set of $V$-linear semiorthogonal decompositions (of fixed…

For a projective K3 surface X we introduce the dense triangulated subcategory S^* of the bounded derived category D^b(Coh(X)) of coherent sheaves on X that is generated by spherical objects. For a K3 surface X over \bar Q it is shown that…

Given a nontrivial semi-orthogonal decomposition $\Perf(\X)=\langle \mathcal{A},\mathcal{B}\rangle$, and assume that the base locus of $\omega_{\X}$ is a proper closed subset, it was proved by Kotaro Kawatani and Shinnosuke Okawa that all…

Bondal and Kapranov describe how to assign to a full exceptional collection on a variety X a DG category C such that the bounded derived category of coherent sheaves on X is equivalent to the bounded derived category of C. In this paper we…

In this paper we study families of projective manifolds with good minimal models. After constructing a suitable moduli functor for polarized varieties with canonical singularities, we show that, if not birationally isotrivial, the base…

Let f : X -> Y be a morphism between normal complex varieties, and assume that Y is Kawamata log terminal. Given any differential form, defined on the smooth locus of Y, we construct a "pull-back form" on X. The pull-back map obtained by…

Let $X$ be a smooth projective variety over an algebraically closed field $k$ of characteristic $p>0$ of $\dim X\geq 4$ and Picard number $\rho(X)=1$. Suppose that $X$ satisfies $H^i(X,F^{m*}_X(\Omg^j_X)\otimes\Ls^{-1})=0$ for any ample…

We introduce and study adhesive spaces. Using this concept we obtain a characterization of stable Baire maps $f:X\to Y$ of the class $\alpha$ for wide classes of topological spaces. In particular, we prove that for a topological space $X$…

Using an obstruction based on Donaldson's theorem, we derive strong restrictions on when a Seifert fibered space $Y = F(e; \frac{p_1}{q_1}, \ldots, \frac{p_k}{q_k})$ over an orientable base surface $F$ can smoothly embed in $S^4$. This…

We generalize a result of Orlov and Van den Bergh on the representability of a cohomological functor from the bounded derived category of a smooth projective variety over a field to the category of L-modules, to the case where L is a field…

We prove a relative version of the fact that semiorthogonal decompositions of the bounded derived category of coherent sheaves are strongly constrained by the base locus of the canonical linear system. As an application we prove that the…

The classical Beauville-Bogomolov Decomposition Theorem asserts that any compact K\"ahler manifold with numerically trivial canonical bundle admits an \'etale cover that decomposes into a product of a torus, and irreducible,…

Let $f: X \to Y$ be a dominant morphism of smooth, proper and geometrically integral varieties over a number field $k$, with geometrically integral generic fibre. We give a necessary and sufficient geometric criterion for the induced map…

A classical result of Bondal-Orlov states that a standard flip in birational geometry gives rise to a fully faithful functor between derived categories of coherent sheaves. We complete their embedding into a semiorthogonal decomposition by…

A smooth hypersurface over a finite field $\mathbb{F}_q$ is called Frobenius nonclassical if the image of every geometric point under the $q$-th Frobenius endomorphism remains in the unique hyperplane tangent to the point. In this paper, we…

We discuss the problem of classifying birational extremal contractions of smooth threefolds where the canonical bundle is trivial along the curves contracted, in the case when a divisor is contracted to a point. We prove the analytic…

We show that for an arbitrarily given closed Riemannian manifold $M$ admitting a point $p \in M$ with a single cut point, every closed Riemannian manifold $N$ admitting a point $q \in N$ with a single cut point is diffeomorphic to $M$ if…

For any smooth complex projective surface $S$, we construct semistable refined Vafa-Witten invariants of $S$ which prove the main conjecture of arXiv:1810.00078. This is done by extending part of Joyce's universal wall-crossing formalism to…