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## Publications

Publications (59)

Let $Y/S$ be a morphism of crystalline prisms, i.e., a $ p$-torsion free $p$-adic formal schemes endowed with a Frobenius lift, and let $\overline Y/\overline S$ denote its reduction modulo $p$. We show that the category of crystals on the prismatic site of $\overline Y/S$ is equivalent to the category of $\mathcal{O}_Y$-modules with integrable and...

Cambridge Core - Geometry and Topology - Lectures on Logarithmic Algebraic Geometry - by Arthur Ogus

A now classical construction due to Kato and Nakayama attaches a topological space (the "Betti realization") to a log scheme over $\mathbf{C}$. We show that in the case of a log smooth degeneration over the standard log disc, this construction allows one to recover the topology of the germ of the family from the log special fiber alone. We go on to...

Let FF be a differential field whose field of constants is algebraically closed and let A be a matrix with coefficients in FF which commutes with its derivative DA. We show that all the eigenvalues of A lie in FF, answering open problem 22 of [1]. We also give a simple proof of a theorem of Schur characterizing matrices A with the property that the...

Let A be a local ring of dimension d. If A is a quotient of a regular local ring of dimension n = d+r, then we say that A has embedding codimension ≤ r. This paper investigates some special properties of local rings of small embedding codimension. The main idea is to exploit a result in [17], which says that local rings of small embedding codimensi...

v2: We improved a little bit according to the referee's wishes. v1: On $X$ projective smooth over a field $k$, Pink and Roessler conjecture that the dimension of the Hodge cohomology of an invertible $n$-torsion sheaf $L$ is the same as the one of its $a$-th power $L^a$ if $a$ is prime to $n$, under the assumptions that $X$ lifts to $W_2(k)$ and $d...

Given a scheme in characteristic p together with a lifting modulo p
2, we construct a functor from a category of suitably nilpotent modules with connection to the category of Higgs modules. We
use this functor to generalize the decomposition theorem of Deligne-Illusie to the case of de Rham cohomology with coefficients.

Let X/S be a smooth morphism of schemes in characteristic p and let (E,) be a sheaf of O X -modules with integrable connection on X. We give a formula for the cohomology sheaves of the De Rham complex of (E,) in terms of a Higgs complex constructed from the p-curvature of (E,). This formula generalizes the classical Cartier isomorphism, with which...

We construct a classification of coherent sheaves with an integrable log connection, or, more precisely, sheaves with an inte-grable connection on a smooth log analytic space X over C. We do this in three contexts: sheaves and connections which are equivariant with respect to a torus action, germs of holomorphic connections, and finally global log...

relates the Hodge numbers of X k to the action of Frobenius on the crys-talline cohomology H,cris(X W) of X over the Witt ring W of k, which can be viewed as a linear map 8: F*WH,cris(X W) H,cris(X W). If H,cris(X W) is torsion free, then since W is a discrete valuation ring, the source and target of 8 admit (unrelated) bases with respect to which...

Moduli spaces of varieties in characteristic p are stratified by Newton polygons. For example, the moduli space of polarized K3 surfaces of degreedadmits a stratification by the height h. Using the classification of supersingular K3 crystals, we determine the dimension and singular locus of each stratum. For example, ifdis prime topthe singular loc...

This paper extends the proof [ 16 ] of the Tate conjecture for ordinary K3 surfaces over a finite field to the more general case of all K3's of finite height. As in [ 16 ], our method is to find a lifting of the K3 to characteristic zero with sufficiently many Hodge cycles. In the ordinary case, the so-called "canonical lifting" of Deligne and Illu...

I like to argue that crystalline cohomology will play a role in characteristic p analogous to the role of Ilodge theory in characteristic zero. One aspect of this analogy is that the F-crystal structure on crystalline cohomology should reflect deep geometric properties of varieties. This should be especially true of varieties for which the “p-adic...

This paper is a collection of musings about several questions related to crystalline cohomology that have plagued me for the
past few years. It contains many more conjectures than proofs, and my justification for publishing is the hope that others
will find the problems as intriguing as I did but perhaps have more success in solving them.

Let X be a smooth projective variety over ℂ. Hodge conjectured that certain cohomology classes on X are algebraic. The work of Deligne that is described in the first article of this volume shows that, when X is an abelian variety, the classes considered by Hodge have many of the properties of algebraic classes.

The ring of finite adèles,
⊗ φ, of φ is denoted by IAf, and IA denotes the full ring of adèles IR. × IAf. For E a number field, IA
Ef and IAE denote E ⊗φ IAf and E ⊗ IA. The group of idèles of E is IA
xE, and the idèle class group is CE =IA
Ef/Ex.

The ring of finite adèles,
⊗ φ, of φ is denoted by IAf, and IA denotes the full ring of adèles IR. × IAf. For E a number field, IA
Ef and IAE denote E ⊗φ IAf and E ⊗ IA. The group of idèles of E is IA
xE, and the idèle class group is CE =IA
Ef/Ex.

Let X be a smooth projective variety over ℂ. Hodge conjectured that certain cohomology classes on X are algebraic. The work of Deligne that is described in the first article of this volume shows that, when X is an abelian variety, the classes considered by Hodge have many of the properties of algebraic classes.

We give a modern and fairly easy proof of (a slight improvement of) an important theorem of Zariski. The result gives conditions under which certain multigraded rings and modules associated with n linear systems are finitely generated, in a very strong sense.

We give a modern and fairly easy proof of (a slight improvement of) an important theorem of Zariski. The result gives conditions under which certain multigraded rings and modules associated with n linear systems are finitely generated, in a very strong sense.

We prove an analog of Nakayama's Lemma, in which the finitely generated module is replaced by a half-exact functor from modules to modules. As applications, we obtain simple proofs of Grothendieck's "property of exchange" for a sheaf of modules under base change, and of the "local criterion for flatness."

The purpose of this note is to investigate some of the foundational questions concerning convergent cohomology as introduced in [?] and [?], using the language and techniques of Grothendieck topologies. In particular, if X is a scheme of finite type over a perfect field k of characteristic p and with Witt ring W, we define the “convergent topos (X/...