arXiv subjects

Nikolay Konovalov

Publications and source records attributed to Nikolay Konovalov.

6 recordsLinked to original sources

arXiv2026

On the universality of multiexcisive functors

We provide a multiplicative classification of polynomial endofunctors on spectra in terms of their Mackey functors of cross--effects. More precisely, we prove that various categories of multivariable excisive functors from spectra to spectra are symmetric monoidally equivalent to the corresponding variants of spectral Mackey functors. The symmetric monoidal structures appearing here are the Day convolutions on both sides, and the Mackey functors we consider involve variations on the category of finite sets and surjections. The method is first to introduce certain multivariable functors we call subdiagonal functors. By considering them all at once using parametrised category theory, we prove inductively that they all admit Mackey functor descriptions as symmetric monoidal categories, endowing them with a universal property along the way. In particular, specialising this to univariate functors gives a new proof and strengthening of Glasman's result about d-excisive endofunctors on spectra. As application of our perspective, we prove a ``Segal conjecture'' in the context of Goodwillie calculus when d is a prime number.

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arXiv2025

Algebraic Goodwillie spectral sequence

Let $\mathit{s}\mathcal{L}$ be the $\infty$-category of simplicial restricted Lie algebras over $\mathbf{F} = \overline{\mathbf{F}}_p$, the algebraic closure of a finite field $\mathbf{F}_p$. By the work of A. K. Bousfield et al. on the unstable Adams spectral sequence, the category $\mathit{s}\mathcal{L}$ can be viewed as an algebraic approximation of the $\infty$-category of pointed $p$-complete spaces. We study the functor calculus in the category $\mathit{s}\mathcal{L}$. More specifically, we consider the Taylor tower for the functor $L^r\colon \mathcal{M}\mathrm{od}^{\geq 0}_{\mathbf{F}} \to \mathit{s}\mathcal{L}$ of a free simplicial restricted Lie algebra together with the associated Goodwillie spectral sequence. We show that this spectral sequence evaluated at $Σ^l \mathbf{F}$, $l\geq 0$ degenerates on the third page after a suitable re-indexing, which proves an algebraic version of the Whitehead conjecture. In our proof we compute explicitly the differentials of the Goodwillie spectral sequence in terms of the $Λ$-algebra of A. K. Bousfield et al. and the Dyer-Lashof-Lie power operations, which naturally act on the homology groups of a spectral Lie algebra. As an essential ingredient of our calculations, we establish a general Leibniz rule in functor calculus associated to the composition of mapping spaces, which conceptualizes certain formulas of W. H. Lin. Also, as a byproduct, we identify previously unknown Adem relations for the Dyer-Lashof-Lie operations in the odd-primary case.

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arXiv2024

Koszul duality for simplicial restricted Lie algebras

Let $\mathsf{s}_0\mathsf{Lie}^r$ be the category of $0$-reduced simplicial restricted Lie algebras over a fixed perfect field of positive characteristic $p$. We prove that there is a full subcategory $\mathrm{Ho}(\mathsf{s}_0\mathsf{Lie}^r_ξ)$ of the homotopy category $\mathrm{Ho}(\mathsf{s}_0\mathsf{Lie}^r)$ and an equivalence $\mathrm{Ho}(\mathsf{s}_0\mathsf{Lie}^r_ξ)\simeq\mathrm{Ho}(\mathsf{s}_1\mathsf{CoAlg}^{tr})$. Here $\mathsf{s}_1\mathsf{CoAlg}^{tr}$ is the category of $1$-reduced simplicial truncated coalgebras; informally, a coaugmented cocommutative coalgebra $C$ is truncated if $x^p=0$ for any $x$ from the augmentation ideal of the dual algebra $C^*$. Moreover, we provide a sufficient and necessary condition in terms of the homotopy groups $π_*(L_\bullet)$ for $L_\bullet \in \mathrm{Ho}(\mathsf{s}_0\mathsf{Lie}^r)$ to lie in the full subcategory $\mathrm{Ho}(\mathsf{s}_0\mathsf{Lie}^r_ξ)$. As an application of the equivalence above, we construct and examine an analog of the unstable Adams spectral sequence of A. K. Bousfield and D. Kan in the category $\mathsf{s}\mathsf{Lie}^r$. We use this spectral sequence to recompute the homotopy groups of a free simplicial restricted Lie algebra.

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arXiv2024

On the automorphism groups of smooth Fano threefolds

Let $\mathcal{X}$ be a smooth Fano threefold over the complex numbers of Picard rank $1$ with finite automorphism group. We give numerical restrictions on the order of the automorphism group $\mathrm{Aut}(\mathcal{X})$ provided the genus $g(\mathcal{X})\leq 10$ and $\mathcal{X}$ is not an ordinary smooth Gushel-Mukai threefold. More precisely, we show that the order $|\mathrm{Aut}(\mathcal{X})|$ divides a certain explicit number depending on the genus of $\mathcal{X}$. We use a classification of Fano threefolds in terms of complete intersections in homogeneous varieties and the previous paper of A. Gorinov and the author regarding the topology of spaces of regular sections.

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arXiv2022

A Division Theorem for Nodal Projective Hypersurfaces

Let $V_{n,d}$ be the variety of equations for hypersurfaces of degree $d$ in $\mathbb{P}^n(\mathbb{C})$ with singularities not worse than simple nodes. We prove that the orbit map $G'=SL_{n+1}(\mathbb{C}) \to V_{n,d}$, $g\mapsto g\cdot s_0$, $s_0\in V_{n,d}$ is surjective on the rational cohomology if $n>1$, $d\geq 3$, and $(n,d)\neq (2,3)$. As a result, the Leray-Serre spectral sequence of the map from $V_{n,d}$ to the homotopy quotient $(V_{n,d})_{hG'}$ degenerates at $E_2$, and so does the Leray spectral sequence of the quotient map $V_{n,d}\to V_{n,d}/G'$ provided the geometric quotient $V_{n,d}/G'$ exists. We show that the latter is the case when $d>n+1$.

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arXiv2021

Topology of spaces of regular sections and applications to automorphism groups

Let $G$ be a complex connected reductive algebraic group that acts on a smooth complex algebraic variety $X$, and let $E$ be a $G$-equivariant algebraic vector bundle over $X$. A section of $E$ is regular if it is transversal to the zero section. Let $U\subsetΓ(X,E)$ be the subset of regular sections. We give a sufficient condition in terms of topological invariants of $E$ and $X$ that implies that every orbit map $O\colon G\to U$ induces a surjection in rational cohomology. Under natural assumptions on $X$ and $E$ this condition is also necessary. If the condition is satisfied, then (1) the geometric quotient $U/G$ exists; (2) there is an isomorphism $H^*(U,\mathrm{Q})\cong H^*(G,\mathrm{Q})\otimes H^*(U/G,\mathrm{Q})$ of cohomology rings; (3) the order of the stabiliser $G_s,s\in U$ divides a certain expression that can be explicitly calculated e.g. if $X$ is a compact homogeneous space. In some cases (e.g. if $E$ is a line bundle) we also prove similar statements for the space of the zero loci of $s\in U$. We apply these results to several explicit examples which include hypersurfaces in projective spaces, non-degenerate quadrics and complete flag varieties of the simple Lie groups of rank 2, and also certain Fano varieties of dimension 3 and 4.

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