Proceedings of the Steklov Institute of Mathematics

A brief biography of Igor Rostislavovich Shafarevich and an overview of all his main scientific publications are presented.

We revisit the non-commutative Hodge-to-de Rham degeneration theorem of the first author and present its proof in a somewhat streamlined and improved form that explicitly uses spectral algebraic geometry. We also try to explain why topology is essential to the proof.

Questions related to deformations of germs of finite morphisms of smooth surfaces are discussed. Four-sheeted finite cover germs F: (U, o′) → (V, o), where (U, o′) and (V, o) are two germs of smooth complex analytic surfaces, are classified up to smooth deformations. The singularity types of branch curves and the local monodromy groups of these germs are also investigated.

A fractal group is a group with unbounded iterated identity. In this paper a finitely generated fractal solvable group is constructed.

We reconstruct hermitian K-theory via algebraic symplectic cobordism. In the motivic stable homotopy category SH(S), there is a unique morphism ϕ: MSp → BO of commutative ring T-spectra which sends the Thom class th^MSp to the Thom class th^BO. Using ϕ we construct an isomorphism of bigraded ring cohomology theories on the category \({\mathop{\rm Sm}\nolimits} {\mathcal O}p/S,\bar \varphi :{{\mathop{\rm MSp}\nolimits} ^{*,*}}(X,U){ \otimes _{{\rm{MS}}{{\rm{p}}^{4*,0*}}({\rm{pt}})}}{\rm{B}}{{\rm{O}}^{4*,2*}}({\rm{pt}}) \cong {\rm{B}}{{\rm{O}}^{*,*}}(X,U)\). The result is an algebraic version of the theorem of Conner and Floyd reconstructing real K-theory using symplectic cobordism. Rewriting the bigrading as MSp^p,q = MSp_1q−p^[q], we have an isomorphism \(\bar \varphi :{{\mathop{\rm MSp}\nolimits} _*}^{[*]}(X,U){ \otimes _{{\rm{MSp}}_0^{[2*]}({\rm{pt}})}}{\rm{KO}}_0^{[2*]}({\rm{pt}}) \cong {\rm{K}}{{\rm{O}}_*}^{[*]}(X,U)\), where the KO_i^[n](X,U) are Schlichting’s hermitian K-theory groups.

We prove that for any prime p there exists an algebraic action of the two-dimensional Witt group W₂ (p) on an algebraic variety X such that the closure in X of the W₂(p)-orbit of some point x ∈ X contains infinitely many W₂(p)-orbits. This is related to the problem of extending, from the case of characteristic zero to the case of characteristic p, the classification of connected affine algebraic groups G such that every algebraic G-variety with a dense open G-orbit contains only finitely many G-orbits.

We classify some special classes of nonrational Fano threefolds with terminal singularities. In particular, all such hyperelliptic and trigonal varieties are found.

Over an algebraically closed field of characteristic p, there are three group schemes of order p, namely the ordinary cyclic group ℤ/p, the multiplicative group \(\mu_{p}\subset\mathbb{G}_{m}\), and the additive group α_p ⊂ \(\mathbb{G}_{a}\). The Tate-Oort group scheme \(\mathbb{TO}_p\) puts these into one happy family, together with the cyclic group of order p in characteristic zero. This paper studies a simplified form of \(\mathbb{TO}_p\), focusing on its representation theory and basic applications in geometry. A final section describes more substantial applications to varieties having p-torsion in Pic^τ, notably the 5-torsion Godeaux surfaces and Calabi-Yau threefolds obtained from \(\mathbb{TO}_5\)-invariant quintics.

For the rational functions on algebraic curves, the number of critical values typical for the Belyi functions is three. This number is replaced by four. The emerging objects, unlike the rigid Belyi pairs, can be deformed; they constitute one-parameter families. The main properties of the corresponding curves in the moduli spaces are established, and the relation of these curves to the Belyi pairs is considered. The case of the clean Belyi pairs of genus 2 of minimal degree 8 is analyzed.