Open Access
Access granted
Subscription Access
Vol 212, No 3 (2021)
Editorial
Matematicheskii Sbornik. 2021;212(3):5-5
5-5
Versal families of elliptic curves with rational 3-torsion
Abstract
For an arbitrary field of characteristic different from 2 and 3, we construct versal families of elliptic curves whose 3-torsion is either rational or isomorphic to $\mathbb Z/3\mathbb Z\oplus \mu_3$ as a Galois module. Bibliography: 10 titles.
Matematicheskii Sbornik. 2021;212(3):6-19
6-19
Singularities on toric fibrations
Abstract
In this paper we investigate singularities on toric fibrations. In this context we study a conjecture of Shokurov (a special case of which is due to M\textsuperscript{c}Kernan) which roughly says that if $(X,B)\to Z$ is an $\varepsilon$-lc Fano-type log Calabi-Yau fibration, then the singularities of the log base $(Z,B_Z+M_Z)$ are bounded in terms of $\varepsilon$ and $\dim X$ where $B_Z$ and $M_Z$ are the discriminant and moduli divisors of the canonical bundle formula. A corollary of our main result says that if $X\to Z$ is a toric Fano fibration with $X$ being $\varepsilon$-lc, then the multiplicities of the fibres over codimension one points are bounded depending only on $\varepsilon$ and $\dim X$. Bibliography: 20 titles.
Matematicheskii Sbornik. 2021;212(3):20-38
20-38
Hermitian-Yang-Mills approach to the conjecture of Griffiths on the positivity of ample vector bundles
Abstract
Given a vector bundle of arbitrary rank with ample determinant line bundle on a projective manifold, we propose a new elliptic system of differential equations of Hermitian-Yang-Mills type for the curvature tensor. The system is designed so that solutions provide Hermitian metrics with positive curvature in the sense of Griffiths — and even in the dual Nakano sense. As a consequence, if an existence result could be obtained for every ample vector bundle, the Griffiths conjecture on the equivalence between ampleness and positivity of vector bundles would be settled. Bibliography: 15 titles.
Matematicheskii Sbornik. 2021;212(3):39-53
39-53
54-67
68-87
General elephants for threefold extremal contractions with one-dimensional fibres: exceptional case
Abstract
Let $(X, C)$ be a germ of a threefold $X$ with terminal singularities along a connected reduced complete curve $C$ with a contraction $f \colon (X, C) \to (Z, o)$ such that $C = f^{-1} (o)_{\mathrm{red}}$ and $-K_X$ is $f$-ample. Assume that each irreducible component of $C$ contains at most one point of index ${>2}$. We prove that a general member $D\in |{-}K_X|$ is a normal surface with Du Val singularities. Bibliography: 16 titles.
Matematicheskii Sbornik. 2021;212(3):88-111
88-111
On automorphisms of quasi-smooth weighted complete intersections
Abstract
We show that every reductive subgroup of the automorphism group of a quasi-smooth well-formed weighted complete intersection of dimension at least $3$ is a restriction of a subgroup in the automorphism group in the ambient weighted projective space. Also, we provide examples demonstrating that the automorphism group of a quasi-smooth well-formed Fano weighted complete intersection may be infinite and even non-reductive. Bibliography: 25 titles.
Matematicheskii Sbornik. 2021;212(3):112-127
112-127
Mironov Lagrangian cycles in algebraic varieties
Abstract
We generalize a construction due to Mironov. Some time ago he presented new examples of minimal and Hamiltonian minimal Lagrangian submanifolds in $\mathbb{C}^n$ and $\mathbb{C} \mathbb{P}^n$. His construction is based on the considerations of a noncomplete toric action of $T^k$, where $k < n$, on subspaces that are invariant with respect to the action of a natural antiholomorphic involution. This situation takes place for a rather broad class of algebraic varieties: complex quadrics, Grassmannians, flag varieties and so on, which makes it possible to construct many new examples of Lagrangian submanifolds in these algebraic varieties. Bibliography: 4 titles.
Matematicheskii Sbornik. 2021;212(3):128-138
128-138
139-156
157-174
On a conjecture of Teissier: the case of log canonical thresholds
Abstract
For a smooth germ of an algebraic variety $(X,0)$ and a hypersurface $(f=0)$ in $X$, with an isolated singularity at $0$, Teissier conjectured a lower bound for the Arnold exponent of $f$ in terms of the Arnold exponent of a hyperplane section $f|_H$ and the invariant $\theta_0(f)$ of the hypersurface. By building on an approach due to Loeser, we prove the conjecture in the case of log canonical thresholds. Bibliography: 21 titles.
Matematicheskii Sbornik. 2021;212(3):175-192
175-192