Using a recent computation of the rational minus part of $SH(k)$ by Ananyevskiy, Levine and Panin, a theorem of Cisinski and Déglise and a version of the Röndigs and Østvær theorem, rational stable motivic homotopy theory over an infinite perfect field of characteristic different from 2 is recovered in this paper from finite Milnor–Witt correspondences in the sense of Calmès and Fasel.

We study the derived categories of coherent sheaves on Gushel–Mukai varieties. In the derived category of such a variety, we isolate a special semiorthogonal component, which is a K3 or Enriques category according to whether the dimension of the variety is even or odd. We analyze the basic properties of this category using Hochschild homology, Hochschild cohomology, and the Grothendieck group. We study the K3 category of a Gushel–Mukai fourfold in more detail. Namely, we show this category is equivalent to the derived category of a K3 surface for a certain codimension 1 family of rational Gushel–Mukai fourfolds, and to the K3 category of a birational cubic fourfold for a certain codimension 3 family. The first of these results verifies a special case of a duality conjecture which we formulate. We discuss our results in the context of the rationality problem for Gushel–Mukai varieties, which was one of the main motivations for this work.

We develop a theory of unbounded derived categories of quasi-coherent sheaves on algebraic stacks. In particular, we show that these categories are compactly generated by perfect complexes for stacks that either have finite stabilizers or are local quotient stacks. We also extend Toën and Antieau–Gepner’s results on derived Azumaya algebras and compact generation of sheaves on linear categories from derived schemes to derived Deligne–Mumford stacks. These are all consequences of our main theorem: compact generation of a presheaf of triangulated categories on an algebraic stack is local for the quasi-finite flat topology.

A result of André Weil allows one to describe rank $n$ vector bundles on a smooth complete algebraic curve up to isomorphism via a double quotient of the set $\text{GL}_{n}(\mathbb{A})$ of regular matrices over the ring of adèles (over algebraically closed fields, this result is also known to extend to $G$-torsors for a reductive algebraic group $G$). In the present paper we develop analogous adelic descriptions for vector and principal bundles on arbitrary Noetherian schemes, by proving an adelic descent theorem for perfect complexes. We show that for Beilinson’s co-simplicial ring of adèles $\mathbb{A}_{X}^{\bullet }$, we have an equivalence $\mathsf{Perf}(X)\simeq |\mathsf{Perf}(\mathbb{A}_{X}^{\bullet })|$ between perfect complexes on $X$ and cartesian perfect complexes for $\mathbb{A}_{X}^{\bullet }$. Using the Tannakian formalism for symmetric monoidal $\infty$-categories, we conclude that a Noetherian scheme can be reconstructed from the co-simplicial ring of adèles. We view this statement as a scheme-theoretic analogue of Gelfand–Naimark’s reconstruction theorem for locally compact topological spaces from their ring of continuous functions. Several results for categories of perfect complexes over (a strong form of) flasque sheaves of algebras are established, which might be of independent interest.

We prove a Knörrer-periodicity-type equivalence between derived factorization categories of gauged Landau–Ginzburg models, which is an analogy of a theorem proved by Shipman and Isik independently. As an application, we obtain a gauged Landau–Ginzburg version of Orlov’s theorem describing a relationship between categories of graded matrix factorizations and derived categories of hypersurfaces in projective spaces, by combining the above Knörrer periodicity type equivalence and the theory of variations of geometric invariant theory quotients due to Ballard, Favero and Katzarkov.

Smooth cubic hypersurfaces $X\subset \mathbb{P}^{5}$ (over $\mathbb{C}$) are linked to K3 surfaces via their Hodge structures, due to the work of Hassett, and via a subcategory ${\mathcal{A}}_{X}\subset \text{D}^{\text{b}}(X)$, due to the work of Kuznetsov. The relation between these two viewpoints has recently been elucidated by Addington and Thomas. In this paper, both aspects are studied further and extended to twisted K3 surfaces, which in particular allows us to determine the group of autoequivalences of ${\mathcal{A}}_{X}$ for the general cubic fourfold. Furthermore, we prove finiteness results for cubics with equivalent K3 categories and study periods of cubics in terms of generalized K3 surfaces.

Inspired by symplectic geometry and a microlocal characterizations of perverse (constructible) sheaves we consider an alternative definition of perverse coherent sheaves. We show that a coherent sheaf is perverse if and only if $R{\rm\Gamma}_{Z}{\mathcal{F}}$ is concentrated in degree $0$ for special subvarieties $Z$ of $X$. These subvarieties $Z$ are analogs of Lagrangians in the symplectic case.

We consider the natural $A_{\infty }$-structure on the $\mathrm{Ext}$-algebra $\mathrm{Ext}^*(G,G)$ associated with the coherent sheaf $G=\mathcal{O}_C\oplus \mathcal{O}_{p_1}\oplus \cdots \oplus \mathcal{O}_{p_n}$ on a smooth projective curve $C$, where $p_1,\ldots,p_n\in C$ are distinct points. We study the homotopy class of the product $m_3$. Assuming that $h^0(p_1+\cdots +p_n)=1$, we prove that $m_3$ is homotopic to zero if and only if $C$ is hyperelliptic and the points $p_i$ are Weierstrass points. In the latter case we show that $m_4$ is not homotopic to zero, provided the genus of $C$ is greater than $1$. In the case $n=g$ we prove that the $A_{\infty }$-structure is determined uniquely (up to homotopy) by the products $m_i$ with $i\le 6$. Also, in this case we study the rational map $\mathcal{M}_{g,g}\to \mathbb{A}^{g^2-2g}$ associated with the homotopy class of $m_3$. We prove that for $g\ge 6$ it is birational onto its image, while for $g\le 5$ it is dominant. We also give an interpretation of this map in terms of tangents to $C$ in the canonical embedding and in the projective embedding given by the linear series $|2(p_1+\cdots +p_g)|$.

We give a necessary and sufficient condition in order for a hyperplane arrangement to be of Torelli type, namely that it is recovered as the set of unstable hyperplanes of its Dolgachev sheaf of logarithmic differentials. Decompositions and semistability of non-Torelli arrangements are investigated.

A famous theorem of D. Orlov describes the derived bounded category of coherent sheaves on projective hypersurfaces in terms of an algebraic construction called graded matrix factorizations. In this article, I implement a proposal of E. Segal to prove Orlov’s theorem in the Calabi–Yau setting using a globalization of the category of graded matrix factorizations (graded D-branes). Let X⊂ℙ be a projective hypersurface. Segal has already established an equivalence between Orlov’s category of graded matrix factorizations and the category of graded D-branes on the canonical bundle Kℙ to ℙ. To complete the picture, I give an equivalence between the homotopy category of graded D-branes on Kℙ and Dbcoh(X). This can be achieved directly, as well as by deforming Kℙ to the normal bundle of X⊂Kℙ and invoking a global version of Knörrer periodicity. We also discuss an equivalence between graded D-branes on a general smooth quasiprojective variety and on the formal neighborhood of the singular locus of the zero fiber of the potential.

We introduce the idea of a geometric categorical Lie algebra action on derived categories of coherent sheaves. The main result is that such an action induces an action of the braid group associated to the Lie algebra. The same proof shows that strong categorical actions in the sense of Khovanov–Lauda and Rouquier also lead to braid group actions. As an example, we construct an action of Artin’s braid group on derived categories of coherent sheaves on cotangent bundles to partial flag varieties.

We prove that on separated algebraic surfaces every coherent sheaf is a quotient of a locally free sheaf. This class contains many schemes that are neither normal, reduced, quasiprojective nor embeddable into toric varieties. Our methods extend to arbitrary two-dimensional schemes that are proper over an excellent ring.

Using the middle convolution functor MCχ introduced by N. Katz, we prove the existence of rigid local systems whose monodromy is dense in the simple algebraic group G2. We derive the existence of motives for motivated cycles which have a motivic Galois group of type G2. Granting Grothendieck’s standard conjectures, the existence of motives with motivic Galois group of type G2 can be deduced, giving a partial answer to a question of Serre.

Let $f:X\to Y$ be a Cohen–Macaulay map of finite type between Noetherian schemes, and $g:Y'\to Y$ a map, with Y′ Noetherian. Let $f':X'\to Y'$ be the base change of f under g and $g':X'\to X$ the base change of g under f. We show that there is a canonical isomorphism $\theta_g^f: {g'}^*\omega_f \simeq \omega_{f'}$, where $\omega_f$ and $\omega_{f'}$ are the relative dualizing sheaves. The map $\theta_g^f$ is easily described when f is proper, and has a more subtle description when f is not. If f is smooth we show that $\theta_g^f$ corresponds to the canonical identification $g'^*\Omega_f^r= \Omega_{f'}^r$ of differential forms, where r is the relative dimension of f. This work is closely related to B. Conrad's work on base change. However, our approach to the problems and our viewpoint are very different from Conrad's: dualizing complexes and their Cousin versions, residual complexes, do not appear in this paper.