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Of all publications in the section: 324
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Working paper
Bogomolov F. A., Kurnosov N., Kuznetsova A. et al. math. arxive. Cornell University, 2020
We consider the only one known class of non-Kähler irreducible holomorphic symplectic manifolds, described in the works of D. Guan and the first author. Any such manifold Q of dimension 2n−2 is obtained as a finite degree n2 cover of some non-Kähler manifold WF which we call the base of Q. We show that the algebraic reduction of Q and its base is the projective space of dimension n−1. Besides, we give a partial classification of submanifolds in Q, describe the degeneracy locus of its algebraic reduction, and prove that the automorphism group of Q satisfies the Jordan property.
Working paper
Eliyashev Y. math. arxive. Cornell University, 2016. No. 1608.06077.
Recently Krichever proposed a generalization of the amoeba and the Ronkin function of a plane algebraic curve. In our paper higher-dimensional version of this generalization is studied. We translate to the generalized case different geometric results that are knows in the standard amoebas case.
Working paper
Kuznetsov A., Shinder E. math. arxive. Cornell University, 2016
We discuss a conjecture saying that derived equivalence of 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 P5 and the corresponding double cover Y→P2 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.
Working paper
Kuznetsov A., Debarre O. math. arxive. Cornell University, 2020
We describe intermediate Jacobians of Gushel-Mukai varieties X of dimensions 3 or 5: if A is the Lagrangian space associated with X, we prove that the intermediate Jacobian of X is isomorphic to the Albanese variety of the canonical double covering of any of the two dual Eisenbud-Popescu-Walter surfaces associated with A. As an application, we describe the period maps for Gushel-Mukai threefolds and fivefolds.
Working paper
Debarre O., Kuznetsov A. math. arxive. Cornell University, 2016
Beauville and Donagi proved in 1985 that the primitive middle cohomology of a smooth complex cubic fourfold and the primitive second cohomology of its variety of lines, a smooth hyperk\"ahler fourfold, are isomorphic as polarized integral Hodge structures. We prove analogous statements for smooth complex Gushel-Mukai varieties of dimension 4 (resp. 6), i.e., smooth dimensionally transverse intersections of the cone over the Grassmannian Gr(2,5), a quadric, and two hyperplanes (resp. of the cone over Gr(2,5) and a quadric). The associated hyperk\"ahler fourfold is in both cases a smooth double cover of a hypersurface in P5 called an EPW sextic.
Working paper
Kuznetsov A., Debarre O. math. arxive. Cornell University, 2018
We describe the moduli stack of Gushel-Mukai varieties as a global quotient stack and its coarse moduli space as the corresponding GIT quotient. The construction is based on a comprehensive study of the relation between this stack and the stack of Lagrangian data; roughly speaking, we show that the former is a generalized root stack of the latter. As an application, we define the period map for Gushel-Mukai varieties and construct some complete nonisotrivial families of smooth Gushel-Mukai varieties. In an appendix, we describe a generalization of the root stack construction used in our approach to the moduli space.
Working paper
Smilga I. math. arxive. Cornell University, 2012. No. 1205.4442.
In this paper, we give a few results on the local behavior of harmonic functions on the Sierpinski triangle - more precisely, of their restriction to a side of the triangle. First we present a general formula that gives the Hölder exponent of such a function in a given point. From this formula, we deduce an explicit algorithm to calculate this exponent in any rational point, and the fact that the derivative of such a function is always equal to 0, infinity or undefined.
Working paper
Semyon Abramyan, Panov T. E. math. arxive. Cornell University, 2019. No. 1901.07918.
We study the question of realisability of iterated higher Whitehead products with a given form of nested brackets by simplicial complexes, using the notion of the moment-complex $\mathcal{Z_K}$. Namely, we say that a simplicial complex K realises an iterated higher Whitehead product w if w is a nontrivial element of the homotopy group $\pi_*(\mathcal{Z_K})$. The combinatorial approach to the question above uses the operation of substitution of simplicial complexes: for any iterated higher Whitehead product w we describe a simplicial complex $\partial\Delta_w$ that realises $w$. Furthermore, for a particular form of brackets inside w, we prove that $\partial\Delta_w$ is the smallest complex that realises $w$. We also give a combinatorial criterion for the nontriviality of the product w. In the proof of nontriviality we use the Hurewicz image of w in the cellular chains of $\mathcal{Z_K}$ and the description of the cohomology product of $\mathcal{Z_K}$. The second approach is algebraic: we use the coalgebraic versions of the Koszul complex and the Taylor resolution of the face coalgebra of $\mathcal{K}$ to describe the canonical cycles corresponding to iterated higher Whitehead products $w$. This gives another criterion for realisability of $w$.
Working paper
Gorsky E., Hogancamp M. math. arxive. Cornell University, 2017
We define a deformation of the triply graded Khovanov-Rozansky homology of a link L depending on a choice of parameters for each component of L.  We conjecture that this invariant restores the missing symmetry of the triply graded Khovanov-Rozansky homology, and in addition satisfies a number of predictions coming from a conjectural connection with Hilbert schemes of points in the plane. We compute this invariant for all positive powers of the full twist and match it to the family of ideals appearing in Haiman's description of the isospectral Hilbert scheme. symmetry of the triply graded Khovanov-Rozansky homology, and in addition satisfies a number of predictions coming from a conjectural connection with Hilbert schemes of points in the plane. We compute this invariant for all positive powers of the full twist and match it to the family of ideals appearing in Haiman's description of the isospectral Hilbert scheme.
Working paper
Kuznetsov A., Prokhorov Y., Shramov K. math. arxive. Cornell University, 2016
We discuss various results on Hilbert schemes of lines and conics and automorphism groups of smooth Fano threefolds with Picard rank 1. Besides a general review of facts well known to experts, the paper contains some new results, for instance, we give a description of the Hilbert scheme of conics on any smooth Fano threefold of index 1 and genus 10. We also show that the action of the automorphism group of a Fano threefold X of index 2 (respectively, 1) on an irreducible component of its Hilbert scheme of lines (respectively, conics) is faithful if the anticanonical class of X is very ample with a possible exception of several explicit cases. We use these faithfulness results to prove finiteness of the automorphism groups of most Fano threefolds and classify explicitly all Fano threefolds with infinite automorphism group. We also discuss a derived category point of view on the Hilbert schemes of lines and conics, and use this approach to identify some of them.
Working paper
Kaledin D. math. arxive. Cornell University, 2016
In arxiv:1602.04254, we have defined polynomial Witt vectors functor from vector spaces over a perfect field k of positive characteristic p to abelian groups. In this paper, we use polynomial Witt vectors to construct a functorial Hochschild-Witt complex WCH∗(A) for any associative unital k-algebra A, with homology groups WHH∗(A). We prove that the group WHH0(A) coincides with the group of non-commutative Witt vectors defined by Hesselholt, while if A is commutative, finitely generated, and smooth, the groups WHHi(A) are naturally identified with the terms WΩiA of the de Rham-Witt complex of the spectrum of A.
Working paper
Verbitsky M., Kamenova L. math. arxive. Cornell University, 2019
Let M be a holomorphic symplectic Kähler manifold equipped with a Lagrangian fibration π with compact fibers. The base of this manifold is equipped with a special Kähler structure, that is, a Kähler structure (I,g,ω) and a symplectic flat connection ∇ such that the metric g is locally the Hessian of a function. We prove that any Lagrangian subvariety Z⊂M which intersects smooth fibers of π and smoothly projects toπ(Z) is a toric fibration over its image π(Z) in B, and this image is also special Kähler. This answers a question of N. Hitchin related to Kapustin-Witten BBB/BAA duality.
Working paper
Polishchuk A., Lekili Y. math. arxive. Cornell University, 2018
Using Auroux's description of Fukaya categories of symmetric products of punctured surfaces, we compute the partially wrapped Fukaya category of the complement of k+1 generic hyperplanes in ℂℙ^n, for k≥n, with respect to certain stops in terms of the endomorphism algebra of a generating set of objects. The stops are chosen so that the resulting algebra is formal. In the case of the complement of (n+2)-generic hyperplanes in ℂP^n (n-dimensional pair-of-pants), we show that our partial wrapped Fukaya category is equivalent to a certain categorical resolution of the derived category of the singular affine variety x_1x_2..x_{n+1}=0. By localizing, we deduce that the (fully) wrapped Fukaya category of n-dimensional pants is equivalent to the derived category of x_1x_2...x_{n+1}=0.
Working paper
Kuznetsov A., Perry A. math. arxive. Cornell University, 2019
We show that over an algebraically closed field of characteristic not equal to 2, homological projective duality for smooth quadric hypersurfaces and for double covers of projective spaces branched over smooth quadric hypersurfaces is a combination of two operations: one interchanges a quadric hypersurface with its classical projective dual and the other interchanges a quadric hypersurface with the double cover branched along it.
Working paper
Bershtein M., Tsymbaliuk A. math. arxive. Cornell University, 2015. No. 1512.09109.
This paper concerns the relation between the quantum toroidal algebras and the affine Yangians of $\mathfrak{sl}_n$, denoted by  $\mathcal{U}^{(n)}_{q_1,q_2,q_3}$ and $\mathcal{Y}^{(n)}_{h_1,h_2,h_3}$, respectively.  Our motivation arises from the milestone work Gautam and Toledano Laredo, where a similar relation between the quantum loop algebra U_q(L\\mathfrak{g})$and the Yangian$Y_h(\mathfrak{g})$has been established by constructing an isomorphism of$\mathbb{C}[[\hbar]]$-algebras$\Phi:\widehat{U}_{\exp(\hbar)}(L\mathfrak{g})\to \widehat{Y}_\hbar(\mathfrak{g})$(with$\ \widehat{}\ $standing for the appropriate completions). These two completions model the behavior of the algebras in the formal neighborhood of$h=0$. The same construction can be applied to the toroidal setting with$q_i=\exp(\hbar_i)$for$i=1,2,3$. In the current paper, we are interested in the more general relation:$\mathrm{q}_1=\omega_{mn}e^{h_1/m}, \mathrm{q}_2=e^{h_2/m}, \mathrm{q}_3=\omega_{mn}^{-1}e^{h_3/m}$, where$m,n\in \mathbb{N}$and$\omega_{mn}$is an$mn$th root of$1$. For any such choice of$m,n,\omega_{mn}$and the corresponding values$\mathrm{q}_1,\mathrm{q}_2,\mathrm{q}_3$, we construct a homomorphism$\Phi^{\omega_{mn}}_{m,n}$from the completion of the formal version of$\mathcal{U}^{(m)}_{\mathrm{q}_1,\mathrm{q}_2,\mathrm{q}_3}$to the completion of the formal version of$\mathcal{Y}^{(mn)}_{h_1/mn,h_2/mn,h_3/mn}$. We also construct homomorphisms$\Psi^{\omega',\omega}_{m,n}$between the completions of the formal versions of$\mathcal{U}^{(m)}_{\mathrm{q}_1,\mathrm{q}_2,\mathrm{q}_3}$with different parameters$m$and$\omega_{mn}\$.
Working paper
Ornea L., Verbitsky M. math. arxive. Cornell University, 2016
An LCK manifold with potential is a compact quotient M of a Kahler manifold X equipped with a positive plurisubharmonic function f, such that the monodromy group acts on X by holomorphic homotheties and maps f to a function proportional to f. It is known that M admits an LCK potential if and only if it can be holomorphically embedded to a Hopf manifold. We prove that any non-Vaisman LCK manifold with potential contains a Hopf surface H. Moreover, H can be chosen non-diagonal, hence, also not admitting a Vaisman structure.
Working paper
Kiselev S., Cherkashin D. math. arxive. Cornell University, 2019
We consider a family of distance graphs in R n and find its independent numbers in some cases. Define graph J±(n, k, t) in the following way: the vertex set consists of all vectors from {−1, 0, 1} n with k nonzero coordinates; edges connect the pairs of vertices with scalar product t. We find the independence number of J±(n, k, t) for n > n0(k, t) in the cases t = 0 and t = −1; these cases for k = 3 are solved completely. Also the independence number is found for negative odd t and n > n0(k, t).
Working paper
Pogrebkov A. math. arxive. Cornell University, 2019. No. 1904.09469.