arXiv subjects
Publications and source records attributed to Martin Olsson.
We describe the prismatic F-gauge associated to a finite flat height one group scheme over a smooth variety of positive characteristic. As applications, we derive the description of the crystalline Dieudonn\'e module of Berthelot-Breen-Messing in this case and recover results of Bragg-Olsson describing flat cohomology using a Hoobler-type sequence.
We prove that smooth, separated Deligne--Mumford stacks in mixed characteristic with quasi-projective coarse moduli space are global quotient stacks and satisfy the resolution property. This builds on work of Kresch and Vistoli and of Bragg, Hall, and Matthur which proves the case when the stack is over a base field, as well as work of Gabber and de Jong which proves the same holds for a $\mu_n$-gerbe over a scheme with an ample line bundle. The key technical input is a result that gerbes banded by so-called trigonalizable group schemes admit faithful vector bundles and are quotient stacks.
We study moduli spaces of stable maps from pointed curves, where the points are allowed to coincide, with target a tame Deligne-Mumford stack. This generalizes the Abramovich-Vistoli theory of twisted stable maps as well as work of Hassett, Alexeev and Guy, and Bayer and Manin, who studied stable maps to projective varieties from curves with weighted marked points.
We study the algebraic structure of the automorphism group of the derived category of coherent sheaves on a smooth projective variety twisted by a Brauer class. Our main results generalize results of Rouquier in the untwisted case.
We introduce a new notion of generalized log twisted curves, which are marked nodal curves with additional data at the marked points. In the case when the markings are distinct this notion agrees with the notion of twisted curve introduced by Abramovich and Vistoli. In addition to developing the basic notions and results, we study in this article the moduli of such curves as well as contraction maps between them. This is motivated, in part, by applications to twisted stable maps which will be studied in a subsequent article.
We explain how to define an embedding of a tame stack over a noetherian ring into a certain generalization of a weighted projective stack using a notion of ample vector bundle on the stack. As applications we construct algebraic moduli stacks of tame stacks equipped with an ample vector bundle and algebraic stacks of tame orbicurves.
We extend the notions of Hochschild and cyclic homology to morphisms from algebraic spaces to algebraic stacks. Using this, we obtain generalizations to log schemes in the sense of Fontaine and Illusie of these homology theories.
We classify the point objects in the derived category $D(X)$ of a torsor under an abelian variety over a field of characteristic $0$.
Let $X$ and $Y$ be smooth projective varieties over a field $k$ admitting morphisms $f:X \to T$ and $g:Y \to T$ to a third variety $T$. We formulate conditions on a derived equivalence $\Phi:D(X) \to D(Y)$ ensuring that $\Phi$ is induced by a complex $P \in D(X \times_T Y )$, defining derived equivalences between the fibers of $f$ and $g$. We apply our results to the canonical fibration and albanese fibration.
We prove various finiteness and representability results for cohomology of finite flat abelian group schemes. In particular, we show that if $f\colon X\rightarrow \mathrm{Spec}(k)$ is a projective scheme over a field $k$ and $G$ is a finite flat abelian group scheme over $X$ then $R^nf_*G$ is representable for all $n$. More generally, we study the derived pushforwards $R^nf_*G$ for $f\colon X\rightarrow S$ a projective morphism and $G$ a finite flat abelian group scheme over $X$. We also define compactly supported cohomology for finite flat abelian group schemes, describe cohomology in terms of the cotangent complex for group schemes of height $1$, and prove higher categorical versions of our main representability results.
We show that the deformation theory of a perfect complex and that of its determinant are related by the trace map, in a general setting of sheaves on a site. The key technical step, in passing from the setting of modules over a ring where one has global resolutions to the general setting, is achieved using $K$-theory and higher category theory.
We study Torelli-type theorems in the Zariski topology for varieties of dimension at least 2, over arbitrary fields. In place of the Hodge structure, we use the linear equivalence relation on Weil divisors. Using this setup, we prove a universal Torelli theorem in the sense of Bogomolov and Tschinkel. The proofs rely heavily on new variants of the classical Fundamental Theorem of Projective Geometry of Veblen and Young. For proper normal varieties over uncountable algebraically closed fields of characteristic 0, we show that the Zariski topological space can be used to recover the linear equivalence relation on divisors. As a consequence, we show that the underlying scheme of any such variety is uniquely determined by its Zariski topological space. We use this to prove a topological version of Gabriel's theorem, stating that a proper normal variety over an uncountable algebraically closed field of characteristic 0 is determined by its category of constructible abelian \'etale sheaves. We also discuss a conjecture in arbitrary characteristic, relating the Zariski topological space to the perfection of a proper normal variety.
We discuss the question of finding conditions on a derived equivalence between two smooth projective varieties $X$ and $Y$ that imply that $X$ and $Y$ are birational. The types of conditions we consider are in the spirit of finding categorical analogous of classical Torelli theorems. We study, in particular, a notion of strongly filtered derived equivalence and study cases where strongly filtered derived equivalence implies birationality. We also consider an open variant of our main question.
We show that varieties of dimension at least 2 over infinite fields are determined as abstract schemes by their Zariski topological spaces together with the rational equivalence relation on the set of effective divisors. This gives a universal Torelli theorem in the sense of Bogomolov and Tschinkel. The proof relies heavily on a rational version of the classical Fundamental Theorem of Projective Geometry.
We introduce a spreading out technique to deduce finiteness results for étale fundamental groups of complex varieties by characteristic $p$ methods, and apply this to recover a finiteness result proven recently for local fundamental groups in characteristic $0$ using birational geometry.
In an earlier paper the notion of a filtered derived equivalence was introduced, and it was shown that if two K3 surfaces admit such an equivalence then they are isomorphic. In this paper we study more refined aspects of filtered derived equivalences related to the action on the cohomological realizations of the Mukai motive. It is shown that if a filtered derived equivalence between K3 surfaces also preserves ample cones then one can find an isomorphism that induces the same map as the equivalence on the cohomological realizations.
We study Fourier-Mukai equivalence of K3 surfaces in positive characteristic and show that the classical results over the complex numbers all generalize. The key result is a positive-characteristic version of the Torelli theorem that uses the derived category in place of the Hodge structure on singular cohomology; this is proven by algebraizing formal lifts of Fourier-Mukai kernels to characteristic zero. As a consequence, any Shioda-supersingular K3 surface is uniquely determined up to isomorphism by its derived category of coherent sheaves. We also study different realizations of Mukai's Hodge structure in algebraic cohomology theories (etale, crystalline, de Rham) and use these to prove: 1) the zeta function of a K3 surface is a derived invariant (discovered independently by Huybrechts); 2) the variational crystalline Hodge conjecture holds for correspondences arising from Fourier-Mukai kernels on products of two K3 surfaces.
We prove the Nagata compactification theorem for any separated map of finite type between quasi-compact and quasi-separated algebraic spaces, generalizing earlier results of Raoult. Along the way we also prove (and use) absolute noetherian approximation for such algebraic spaces, generalizing earlier results in the case of schemes.