Related papers: Comparison Isomorphisms for Smooth Formal Schemes

Let $p$ be a prime, and let $\mathrm{X}$ be a smooth $p$-adic formal scheme over $\mathrm{Spf} \mathcal{O}_K$ where $K/\mathbf{Q}_p$ is a finite extension. We show that reflexive sheaves on the stack $\mathrm{X}^{\mathrm{Syn}}$ are…

Let $\Lambda$ be a complete noetherian local ring with finite residue field of characteristic $p$ and $K/\mathbb{Q}_p$ a $p$-adic field. We show that, by deformation of the structure sheaf on the (transversal) prismatic site of a bounded…

We prove a general version of the crystalline equivalence principle which gives an equivalence of categories between a category of TQFTs defined on a generic space with $G$-symmetry, and a category of TQFTs with internal symmetry. We give a…

We present a pair of smooth fiber bundles over the circle with a common $4$-dimensional fiber with the following properties: (1) their total spaces are diffeomorphic to each other; (2) they are isomorphic to each other as topological fiber…

Let C be a smooth projective curve over a discretely valued field K, defined by an affine equation f(x,y)=0. We construct a model of C over the ring of integers of K using a toroidal embedding associated to the Newton polygon of f. We show…

We introduce the notion of integrable connections for a sheaf of differential graded algebras on a topological space. We then describe them in the finite locally projective setting, when the sheaf is either the de Rham complex of a formal…

We construct a canonical chain of formality quasiisomorphisms for the operad of chains on framed little disks and the operad of chains on little disks. The construction is done in terms of logarithmic algebraic geometry and is remarkable…

A theorem by Orlov states that any equivalence between the bounded derived categories of coherent sheaves of two smooth projective varieties, X and Y, is isomorphic to a Fourier-Mukai transform with kernel in the bounded derived category of…

Let X be a flexible variety of F be an isomorphism of closed one-dimensional subschemes of $X$. We develop criteria which guarantee that F extends to au automorphism of X.

We discuss the isomorphism problem of projective schemes; given two projective schemes, can we algorithmically decide whether they are isomorphic? We give affirmative answers in the case of one-dimensional projective schemes, the case of…

In this note, we introduce and study the Cartier--Witt stack $\mathrm{WCart}_X$ attached to a $p$-adic formal scheme $X$ as well as some variants. In particular, we reinterpret the notion of prismatic crystals on $X$ and their cohomology in…

For smooth affine varieties in positive characteristic, we identify a slope obstruction to the injectivity of the comparison morphism from rigid cohomology to rationalised crystalline cohomology. This yields a negative answer to a question…

We describe a natural isomorphism between the set of equivalence classes of pseudocycles and the integral homology groups of a smooth manifold. Our arguments generalize to settings well-suited for applications in enumerative algebraic…

We show that the natural morphism $\phi:\pi_1(X_{\eta},x_{\eta})\to \pi_1(X,x)_{\eta}$ between the fundamental group scheme of the generic fiber $X_{\eta}$ of a scheme $X$ over a connected Dedekind scheme and the generic fiber of the…

Let $X$ be a smooth proper scheme over an algebraically closed field $k$ in characteristic $p$. In this short note, by interpreting $\mathcal{D}_{X}$-modules as $F$-divided sheaves and establishing a cohomological boundedness property for…

We explicitly describe cohomology of the sheaf of differential forms with poles along a semiample divisor on a complete simplicial toric variety. As an application, we obtain a new vanishing theorem which is an analogue of the…

We introduce smooth L^\infty differential forms on a singular (semialgebraic) set X in R^n. Roughly speaking, a smooth L^\infty differential form is a certain class of equivalence of 'stratified forms', that is, a collection of smooth forms…

A different proof to a known criterion of derived equivalence implying birationality is given. Derived equivalent smooth projective curves over an algebraically closed field are proved to be isomorphic. A different proof of derived…

This a first step to develop a theory of smooth, etale and unramified morphisms between noetherian formal schemes. Our main tool is the complete module of differentials, that is a coherent sheaf whenever the map of formal schemes is of…

Quantum sheaf cohomology is a deformation of the cohomology ring of a sheaf. In recent years, this subject had an impetuous development in connection with the $(0; 2)$ non-linear sigma model from super-strings theory. The basic piece in…