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卷 87, 编号 3 (2023)
Articles
Igor' Rostislavovich Shafarevich
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2023;87(3):3-4
3-4
Ramification filtration and differential forms
摘要
Let $L$ be a complete discrete valuation field of prime characteristic $p$ with finite residue field. Denote by $\Gamma_{L}^{(v)}$ the ramification subgroups of $\Gamma_{L}=\operatorname{Gal}(L^{\mathrm{sep}}/L)$. We consider the category $\operatorname{M\Gamma}_{L}^{\mathrm{Lie}}$ of finite $\mathbb{Z}_p[\Gamma_{L}]$-modules $H$, satisfying some additional (Lie)-condition on the image of $\Gamma_L$ in $\operatorname{Aut}_{\mathbb{Z}_p}H$. In the paper it is proved that all information about the images of the groups $\Gamma_L^{(v)}$ in $\operatorname{Aut}_{\mathbb{Z}_p}H$ can be explicitly extracted from some differential forms $\widetilde{\Omega} [N]$ on the Fontaine etale $\phi $-module $M(H)$ associated with $H$. The forms $\widetilde{\Omega}[N]$ are completely determined by a canonical connection $\nabla $ on $M(H)$. In the case of fields $L$ of mixed characteristic, which contain a primitive $p$th root of unity, we show that a similar problem for $\mathbb{F}_p[\Gamma_L]$-modules also admits a solution. In this case we use the field-of-norms functor to construct the corresponding $\phi $-module together with the action of the Galois group of a cyclic extension $L_1$ of $L$ of degree $p$. Then our solution involves the characteristic $p$ part (provided by the field-of-norms functor) and the condition for a “good” lift of a generator of $\operatorname{Gal}(L_1/L)$. Apart from the above differential forms the statement of this condition uses the power series coming from the $p$-adic period of the formal group $\mathbb{G}_m$.Bibliography: 21 titles.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2023;87(3):5-22
5-22
Coherent Sheaves, Chern Classes, and Superconnections on compact complex-analytic manifolds
摘要
We construct a twist-closed enhancement of the category $\mathcal D^b_{coh}(X)$, the boundedderived category of complexes of $\mathcal O_X$-modules with coherent cohomology, by meansof the DG-category of $\bar\partial$-superconnections. Then we apply the techniques of $\bar\partial$-superconnections to dene Chern classes and Bott–Chern classes of objects in the category, in particular, of coherent sheaves.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2023;87(3):23-55
23-55
Isogeny classes and endomorphism algebras of abelian varieties over finite fields
摘要
We construct non-isogenous simple ordinary abelian varieties over an algebraic closure of a finite field with isomorphic endomorphism algebras.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2023;87(3):56-74
56-74
On higher-dimensional del Pezzo varieties
摘要
We study del Pezzo varieties, higher-dimensional analogues of del Pezzo surfaces. In particular, we introduce ADE classification of del Pezzo varieties, show that in type $\mathrm A$ the dimension of non-conical del Pezzo varieties is bounded by $12 - d - r$, where $d$ is the degree and $r$ is the rank of the class group, and classify maximal del Pezzo varieties.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2023;87(3):75-148
75-148
On the local fundamental group of the complement to a curve in a normal surface
摘要
In the article, we give a presentation of the fundamental group of the complement to a curve $C$ in its "tubular" neighborhood in a normal surface $S$. The presentation is given in terms of the double weighted dual graph of a resolution of singularities of $C$ (and $S$)and it is a generalization of the presentation of the fundamental group of the complement to a normal singularity in its neighborhood given by Mumford in the case when the dual graph of resolution is a tree and all exceptional curves of the resolution are rational curves.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2023;87(3):149-174
149-174
Rational points of algebraic varieties: a homotopical approach
摘要
This article, dedicated to the 100-th anniversary of I. R. Shafarevich, is a survey of techniques of homotopical algebra, applied to the problem of distribution of rational points on algebraic varieties.We due to I. R. Shafarevich, jointly with J. Tate, one of the breakthrough discoveries in this domain: construction of the so-called Shafarevich–Tate groups and the related obstructions to the existence of rational points. Later it evolved into the theory of Brauer–Manin obstructions.Here we focus on some facets of the later developments in Diophantine geometry: the study of the distribution of rational points on them.More precisely, we show how the definition of accumulating subvarieties, based upon counting the number of points whose height is bounded by varying $H$, can be encoded by a special class of categories in such a way that the arithmetical invariants of varieties are translated into homotopical invariants of objects and morphisms of these categories.The central role in this study is played by the structure of an assembler (I. Zakharevich) in general, and a very particular case of it, anassembler on the family of unions of half-open intervals $(a,b]$ with rational ends.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2023;87(3):175-183
175-183
Special Bohr–Sommerfeld geometry: variations
摘要
Present paper continues the investigations in special Bohr–Sommerfeld lagrangian geometry of compact symplectic manifolds. Using natural deformation parameters we avoid the difficulties arose in the definition of the moduli space of special Bohr–Sommerfeld cycles for compact simply connected algebraic varieties. As an application one presents remarks which show how our construction can be exploited in the study of the Weinstein structures and a conjecture of Eliashberg.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2023;87(3):184-205
184-205
Log adjunction: moduli part
摘要
Upper moduli part of adjunction is introduced and its basic property are discussed. The moduli part is b-Cartier in the case of rational multiplicities and is b-nef in the maximal case.Bibliography: 17 titles.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2023;87(3):206-230
206-230