arXiv subjects

Evgeny Shinder

Publications and source records attributed to Evgeny Shinder.

30 records · Page 2Linked to original sources

arXiv2019

K-theory and the singularity category of quotient singularities

In this paper we study Schlichting's K-theory groups of the Buchweitz-Orlov singularity category $\mathcal{D}^{sg}(X)$ of a quasi-projective algebraic scheme $X/k$ with applications to Algebraic K-theory. We prove that for isolated quotient singularities $\mathrm{K}_0(\mathcal{D}^{sg}(X))$ is finite torsion, and that $\mathrm{K}_1(\mathcal{D}^{sg}(X)) = 0$. One of the main applications is that algebraic varieties with isolated quotient singularities satisfy rational Poincare duality on the level of the Grothendieck group; this allows computing the Grothendieck group of such varieties in terms of their resolution of singularities. Other applications concern the Grothendieck group of perfect complexes supported at a singular point and topological filtration on the Grothendieck groups.

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arXiv2019

The motivic nearby fiber and degeneration of stable rationality

We prove that stable rationality specializes in regular families whose fibers are integral and have at most ordinary double points as singularities. Our proof is based on motivic specialization techniques and the criterion of Larsen and Lunts for stable rationality in the Grothendieck ring of varieties.

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arXiv2018

On the geometry of the LLSvS eightfold

In this note we make a few remarks about the geometry of the holomorphic symplectic manifold Z constructed by C.Lehn, M.Lehn, C.Sorger and D. van Straten as a two-step contraction of the variety of twisted cubic curves on a cubic fourfold Y in P^5. We show that Z is birational to a component of a moduli space of stable sheaves in the Calabi-Yau subcategory of the derived category of Y. Using this description we deduce that the twisted cubics contained in a hyperplane section Y_H of Y give rise to a Lagrangian subvariety Z_H in Z. For a generic choice of the hyperplane, Z_H is birational to the theta-divisor in the intermediate Jacobian of Y_H.

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arXiv2017

Grothendieck ring of varieties, D- and L-equivalence, and families of quadrics

We discuss a conjecture saying that derived equivalence of simply connected smooth projective varieties implies that the difference of their classes in the Grothendieck ring of varieties is annihilated by a power of the affine line class. We support the conjecture with a number of known examples, and one new example. We consider a smooth complete intersection $X$ of three quadrics in ${\mathbf P}^5$ and the corresponding double cover $Y \to {\mathbf P}^2$ branched over a sextic curve. We show that as soon as the natural Brauer class on $Y$ vanishes, so that $X$ and $Y$ are derived equivalent, the difference $[X] - [Y]$ is annihilated by the affine line class.

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arXiv2017

Group actions on categories and Elagin's Theorem Revisited

After recalling basic definitions and constructions for a finite group $G$ action on a $k$-linear category we give a concise proof of the following theorem of Elagin: if $\mathcal{C} = \langle \mathcal{A}, \mathcal{B} \rangle$ is a semiorthogonal decomposition of a triangulated category which is preserved by the action of $G$, and $\mathcal{C}^G$ is triangulated, then there is a semiorthogonal decomposition $\mathcal{C}^G = \langle \mathcal{A}^G, \mathcal{B}^G \rangle$. We also prove that any $G$-action on $\mathcal{C}$ is weakly equivalent to a strict $G$-action which is the analog of the Coherence Theorem for monoidal categories.

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arXiv2016

The Bogomolov-Prokhorov invariant of surfaces as equivariant cohomology

For a complex smooth projective surface $M$ with an action of a finite cyclic group $G$ we give a uniform proof of the isomorphism between the invariant $H^1(G, H^2(M, {\mathbb Z}))$ and the first cohomology of the divisors fixed by the action, using $G$-equivariant cohomology. This generalizes the main result of Bogomolov and Prokhorov "On stable conjugacy of finite subgroups of the plane Cremona group, I".

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arXiv2015

On a zeta-function of a dg-category

We define a zeta-function of a pre-triangulated dg-category and investigate its relationship with the motivic zeta-function in the geometric case.

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arXiv2014

The Fano variety of lines and rationality problem for a cubic hypersurface

We find a relation between a cubic hypersurface $Y$ and its Fano variety of lines $F(Y)$ in the Grothendieck ring of varieties. We prove that if the class of an affine line is not a zero-divisor in the Grothendieck ring of varieties, then Fano variety of lines on a smooth rational cubic fourfold is birational to a Hilbert scheme of two points on a K3 surface; in particular, general cubic fourfold is irrational.

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arXiv2014

Linear Mahler Measures and Double L-values of Modular Forms

We consider the Mahler measure of the polynomial 1+x_1+x_2+x_3+x_4, which is the first case not yet evaluated explicitly. A conjecture due to F. Rodriguez-Villegas represents this Mahler measure as a special value at the point 4 of the L-function of a modular modular form of weight 3. We prove that this Mahler measure is equal to a linear combination of double L-values of certain meromorphic modular forms of weight 4.

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arXiv2013

Minifolds and Phantoms

A minifold is a smooth projective $n$-dimensional variety such that its bounded derived category of coherent sheaves $\D^b(X)$ admits a semi-orthogonal decomposition into an exceptional collection of $n+1$ exceptional objects. In this paper we classify minifolds of dimension $n \leq 4$. We conjecture that the derived category of fake projective spaces have a similar semi-orthogonal decomposition into a collection of $n+1$ exceptional objects and a category with vanishing Hochschild homology. We prove this for fake projective planes with non-abelian automorphism group. We construct new examples of phantom categories with both Hochschild homology and Grothendieck group vanishing.

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arXiv2013

Exceptional collections of line bundles on the Beauville surface

We construct quasi-phantom admissible subcategories in the derived category of coherent sheaves on the Beauville surface $S$. These quasi-phantoms subcategories appear as right orthogonals to subcategories generated by exceptional collections of maximal possible length 4 on $S$. We prove that there are exactly 6 exceptional collections consisting of line bundles (up to a twist) and these collections are spires of two helices.

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arXiv2012

On motives of algebraic groups associated to a divsion algebra

We consider algebraic groups GL_1(A), SL_1(A), where A is a division algebra of prime degree over a field F, and associated motives in the category of motivic complexes DM(F). Following an idea of Suslin, we relate motives of these groups to the motive of Voevodsky's simplicial scheme X, associated to the Severi-Brauer variety of A.

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