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N. G. Kruzhilin

Publications and source records attributed to N. G. Kruzhilin.

7 recordsLinked to original sources
arXiv2020

Plane algebraic curves in fancy balls

Boileau and Rudolph called a link $L$ in the $3$-sphere a $\bf C$-boundary if it can be realized as the intersection of an algebraic curve $A$ in $\bf C^2$ with the boundary of a smooth embedded $4$-ball $B$. They showed that some links are not $\bf C$-boundaries. We say that $L$ is a strong $\bf C$-boundary if $A\setminus B$ is connected. In particular, all quasipositive links are strong $\bf C$-boundaries. In this paper we give examples of non-quasipositive strong $\bf C$-boundaries and non-strong $\bf C$-boundaries. We give a complete classification of (strong) $\bf C$-boundaries with at most 5 crossings.

math.GT↗
arXiv2014

Associated forms of binary quartics and ternary cubics

Let ${\mathcal Q}_n^d$ be the vector space of forms of degree $d\ge 3$ on ${\mathbb C}^n$, with $n\ge 2$. The object of our study is the map $Φ$, introduced in papers [EI], [AI1], that assigns every nondegenerate form in ${\mathcal Q}_n^d$ the so-called associated form, which is an element of ${\mathcal Q}_n^{n(d-2)*}$. We focus on two cases: those of binary quartics ($n=2$, $d=4$) and ternary cubics ($n=3$, $d=3$). In these situations the map $Φ$ induces a rational equivariant involution on the projectivized space ${\mathbb P}({\mathcal Q}_n^d)$, which is in fact the only nontrivial rational equivariant involution on ${\mathbb P}({\mathcal Q}_n^d)$. In particular, there exists an equivariant involution on the space of elliptic curves with nonvanishing $j$-invariant. In the present paper, we give a simple interpretation of this involution in terms of projective duality. Furthermore, we express it via classical contravariants.

math.AG↗
arXiv2012

Explicit reconstruction of homogeneous isolated hypersurface singularities from their Milnor algebras

By the Mather-Yau theorem, a complex hypersurface germ $V$ with isolated singularity is completely determined by its moduli algebra $A(V)$. The proof of the theorem does not provide an explicit procedure for recovering $V$ from $A(V)$, and finding such a procedure is a long-standing open problem. In this paper, we present an explicit way for reconstructing $V$ from $A(V)$ up to biholomorphic equivalence under the assumption that the singularity of $V$ is homogeneous, in which case $A(V)$ coincides with the Milnor algebra of $V$.

math.CV↗
arXiv2007

Proper actions of Lie groups of dimension $n^2+1$ on $n$-dimensional complex manifolds

In this paper we continue to study actions of high-dimensional Lie groups on complex manifolds. We give a complete explicit description of all pairs $(M,G)$, where $M$ is a connected complex manifold $M$ of dimension $n\ge 2$, and $G$ is a connected Lie group of dimension $n^2+1$ acting effectively and properly on $M$ by holomorphic transformations. This result complements a classification obtained earlier by the first author for $n^2+2\le\hbox{dim} G<n^2+2n$ and a classical result due to W. Kaup for the maximal group dimension $n^2+2n$.

math.CV↗
arXiv2000

Effective actions of the unitary group on complex manifolds

We classify all connected $n$-dimensional complex manifolds admitting an effective action of the unitary group $U_n$ by biholomorphic transformations. One consequence of this classification is a characterization of ${\bf C}^n$ by its automorphism group.

math.CV↗