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Vladimir Guletskii

Publications and source records attributed to Vladimir Guletskii.

17 recordsLinked to original sources
arXiv2026

Motivic obstruction to rationality of a general cubic hypersurface in $\mathbb P^5$

We introduce integrally essentially indecomposable motives and prove that, if the integral motive of a smooth projective surface over a field of sufficiently big transcendence degree is integrally essentially indecomposable, then a very general cubic fourfold in $\mathbb P^5$ is not rational. We also prove a lifting theorem saying that, given a smooth projective family of surfaces over a Henselian DVR, if the motive of the special fibre is integrally essentially indecomposable, then so is the motive of the generic fibre. This suggests a possible reduction of the cubic fourfold conjecture to certain arithmetic phenomena in positive characteristic.

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arXiv2026

The tangent space to the space of 0-cycles

Let $S$ be a Noetherian scheme, and let $X$ be a scheme over $S$. Under mild assumptions, one can construct the connected infinite symmetric power ${\rm Sym}^{\infty }(X/S)$, whose group completion ${\rm Sym}^{\infty }(X/S)^+$ is an abelian group object in the category of set valued sheaves on the Nisnevich site over $S$. Viewing this completion as the space of relative $0$-cycles on $X/S$, we construct the sheaf of Kähler differentials $Ω^1_{{\rm Sym}^{\infty }(X/S)^+}$, and the tangent sheaf $T_{{\rm Sym}^{\infty }(X/S)^+}$. We prove that the category of étale neighbourhoods at a point $P$ on the space of $0$-cycles is cofiltered. Applying the stalk functor, we also obtain the stalk of the tangent sheaf at $P$, whose tensor product with the residue field is the needed tangent space to the space of $0$-cycles at $P$.

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arXiv2026

Arithmetic of Gysin kernels

Let $k$ be a field, and let $X$ be a smooth projective surface over $k$. Fix a Lefschetz pencil on $X$, and let $C$ be its fibre at the generic point of $\mathbb P^1$. The closed immersion of $C$ in to $X_{k(t)}$ induces the Gysin homomorphism from the Jacobian $A$ of the curve $C$ to the Chow group $A_0(X_{k(t)})$ of $0$-cycles of degree $0$ on $X_{k(t)}$. Embedding $k(t)$ in to an uncountable universal domain $\Bbbk $, we obtain the corresponding homomorphism from $A(\Bbbk )$ to $A_0(X_{\Bbbk })$, whose kernel is either countable or the union of translates of a certain abelian subvariety inside $A_{\Bbbk }$, due to the Deligne-Katz irreducibility of monodromy action on vanishing cycles. We prove three dichotomy theorems on the structure of the kernel of Gysin homomorphism on $0$-cycles, in terms of étale monodromy action, and working over a field of arbitrary characteristic.

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arXiv2025

The Bloch conjecture

We prove the conjecture stated by Spencer Bloch in 1975 and saying that the Albanese kernel of a smooth projective surface is 0, provided its second cohomology group is algebraic.

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arXiv2018

Étale monodromy and rational equivalence for $1$-cycles on cubic hypersurfaces in $\mathbb P^5$

Let $k$ be an uncountable algebraically closed field of characteristic $0$, and let $X$ be a smooth projective connected variety of dimension $2p$, appropriately embedded into $\mathbb P^m$ over $k$. Let $Y$ be a hyperplane section of $X$, and let $A^p(Y)$ and $A^{p+1}(X)$ be the groups of algebraically trivial algebraic cycles of codimension $p$ and $p+1$ modulo rational equivalence on $Y$ and $X$ respectively. Assume that, whenever $Y$ is smooth, the group $A^p(Y)$ is regularly parametrized by an abelian variety $A$ and coincides with the subgroup of degree $0$ classes in the Chow group $CH^p(Y)$. In the paper we prove that the kernel of the push-forward homomorphism from $A^p(Y)$ to $A^{p+1}(X)$ is the union of a countable collection of shifts of a certain abelian subvariety $A_0$ inside $A$. For a very general section $Y$ either $A_0=0$ or $A_0$ coincides with an abelian subvariety $A_1$ in $A$ whose tangent space is the group of vanishing cycles $H^{2p-1}(Y)_{\rm van}$. Then we apply these general results to sections of a smooth cubic fourfold in $\mathbb P^5$.

math.AG↗
arXiv2017

Bloch's conjecture for surfaces with involutions and of geometric genus zero

Let $S$ be a smooth projective surface with $p_g=0$, let $ι$ be a regular involution acting on $S$, and let $W$ be the resolution of singularities of the quotient surface $S/ι$. In the paper we prove that Bloch's conjecture holds for the surface $S$ if and only if it holds for the surface $W$. This yields Bloch's conjecture for all surfaces $S$ whenever the same conjecture is true for the desingularized quotient $W$. In particular, Bloch's conjecture holds true for all numerical Godeaux surfaces with involutions, a "half" of Campedelli surfaces with involutions, the surface of Craighero and Gattazzo, some Catanese surfaces and other examples. Applying the same method to $K3$-surfaces, we prove that if a $K3$-surface $S$ admits a regular involution whose quotient is of Enriques type, then the motive $M(S)$ is finite-dimensional.

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arXiv2015

$\mathbb A^1$-connectivity on Chow monoids v.s. rational equivalence of algebraic cycles

Let $k$ be a field of characteristic zero, and let $X$ be a projective variety embedded into a projective space over $k$. For two natural numbers $r$ and $d$ let $C_{r,d}(X)$ be the Chow scheme parametrizing effective cycles of dimension $r$ and degree $d$ on the variety $X$. An effective $r$-cycle of minimal degree on $X$ gives rise to a chain of embeddings of $C_{r,d}(X)$ into $C_{r,d+1}(X)$, whose colimit is the connective Chow monoid $C_r^{\infty }(X)$ of $r$-cycles on $X$. Let $BC_r^{\infty }(X)$ be the motivic classifying space of this monoid. In the paper we establish an isomorphism between the Chow group $CH_r(X)_0$ of degree $0$ dimension $r$ algebraic cycles modulo rational equivalence on $X$, and the group of sections of the sheaf of $\mathbb A^1$-path connected components of the loop space of $BC_r^{\infty }(X)$ at $Spec(k)$. Equivalently, $CH_r(X)_0$ is isomorphic to the group of sections of the $S^1\wedge \mathbb A^1$-fundamental group $Π_1^{S^1\wedge \mathbb A^1}(BC_r^{\infty }(X))$ at $Spec(k)$.

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arXiv2015

Positive model structures for abstract symmetric spectra

We give a general method of constructing positive stable model structures for symmetric spectra over an abstract simplicial symmetric monoidal model category. The method is based on systematic localization, in Hirschhorn's sense, of a ceratin positive projective model structure on spectra, where positivity basically means the truncation of the zero slice. The localization above is by the set of stabilizing morphisms, or their truncated version.

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arXiv2014

Symmetric powers in abstract homotopy categories

We study symmetric powers in the homotopy categories of abstract closed symmetric monoidal model categories, in both unstable and stable settings. As an outcome, we prove that symmetric powers preserve the Nisnevich and etale homotopy type in the unstable and stable motivic homotopy theories of schemes over a base. More precisely, if f is a weak equivalence of motivic spaces, or a weak equivalence between positively cofibrant motivic spectra, with respect to the Nisnevich or etale topology on schemes, then all symmetric powers Sym^n(f) are weak equivalences too. This gives left derived symmetric powers which aggregate into a categorical lambda-structures on the corresponding motivic homotopy categories of schemes over a base.

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arXiv2010

Transcendence degree of zero-cycles and the structure of Chow motives

We show how the notion of the transcendence degree of a zero-cycle on a smooth projective variety X is related to the structure of the motive M(X). This can be of particular interest in the context of Bloch's conjecture, especially for Godeaux surfaces, when the surface is given as a finite quotient of a suitable quintic in P^3.

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arXiv2007

On the continuous part of codimension two algebraic cycles on threefolds over a field

Let $X$ be a non-singular projective threefold over an algebraically closed field of any characteristic, and let $A^2(X)$ be the group of algebraically trivial codimension 2 algebraic cycles on $X$ modulo rational equivalence with coefficients in $\mathbb Q$. Assume $X$ is birationally equivalent to another threefold $X'$ admitting a fibration over an integral curve $C$ whose generic fiber $X'_{\bar η}$, where $\bar η=Spec(\bar {k(C)})$, satisfies the following three conditions: (i) the motive $M(X'_{\bar η})$ is finite-dimensional, (ii) $H^1_{et}(X_{\bar η},\mathbb Q_l)=0$ and (iii) $H^2_{et}(X_{\bar η},\mathbb Q_l(1))$ is spanned by divisors on $X_{\bar η}$. We prove that, provided these three assumptions, the group $A^2(X)$ is representable in the weak sense: there exists a curve $Y$ and a correspondence $z$ on $Y\times X$, such that $z$ induces an epimorphism $A^1(Y)\to A^2(X)$, where $A^1(Y)$ is isomorphic to $Pic^0(Y)$ tensored with $\mathbb Q$. In particular, the result holds for threefolds birational to three-dimensional Del Pezzo fibrations over a curve.

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arXiv2006

Zeta functions in triangulated categories

We prove 2-out-of-3 property for rationality of motivic zeta function in distinguished triangles in Voevodsky's category DM. As an application, we show rationality of motivic zeta functions for all varieties whose motives are in the thick triangulated monoidal subcategory generated by motives of quasi-projective curves in DM. Joint with a result of P.O'Sullivan it also gives an example of a variety whose motive is not finite-dimensional while the motivic zeta function is rational.

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arXiv2005

Motives of smooth families and cycles on threefolds

Let X --> S be a smooth projective family of surfaces over a smooth curve S such that the generic fiber is a surface with Weil H^2 spanned by divisors and trivial H^1. We prove that if the relative motive of X/S is finite-dimensional the Chow group CH^2(X) with coefficients in Q is generated by a multisection and vertical cycles, i.e. one-dimensional cycles lying in closed fibers of the map X --> S. If S is the projective line P^1 then CH^2(X) is a direct sum of n+1 copies of Q where n<=b_2 and b_2 is the second Betti number of the generic fiber. Vertical generators in CH^2(X) can be concretely expressed in terms of spreads of algebraic generators of the above H^2. We also show where such families are naturally arising from by spreading out surfaces over C.

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arXiv2003

Finite dimensional objects in distinguished triangles

We prove an additivity for evenly (oddly) finite dimensional objects in distinguished triangles in a triangulated monoidal category structured by an underlying model monoidal category. In particular, the result holds in the Q-localized motivic stable homotopy category of spectra and in Q-localized Voevodsky's category of motives over a field, char=0. As an application, we show that the motives of schemes of dimension one (separated and of finite type over a field, char=0) are finite dimensional.

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arXiv2003

Finite dimensional motives and the Conjectures of Beilinson and Murre

We relate the notion of finite dimensionality of the Chow motive M(X) of a smooth projective variety X (as defined by S. Kimura) with the Conjectures of Beilinson, Bloch and Murre on the existence of a filtration on the Chow ring CH(X). We show (Theorem 14) that finite dimensionality of M(X) implies uniqueness, up to isomorphism, of Murre's decomposition of M(X). Conversely (Theorem 17), Murre's Conjecture for a suitable m-fold product of X by itself implies finite dimensionality of M(X). We also show (Theorem 27) that, for a surface X with trivial geometrical genus, the motive M(X) is finite dimensional if and only if the Chow group of 0-cycles of X is finite dimensional in the sense of Mumford, i.e. iff the Bloch Conjecture holds for X.

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