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Of all publications in the section: 48
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Article
Polishchuk A. Selecta Mathematica, New Series. 2018. Vol. 24. No. 1. P. 563-589.
Article
Feigin B. L., Makhlin I. Selecta Mathematica, New Series. 2016. Vol. 22. No. 3. P. 1703-1747.

We present a new combinatorial formula for Hall–Littlewood functions associated with the affine root system of type (Formula presented.), i.e., corresponding to the affine Lie algebra (Formula presented.). Our formula has the form of a sum over the elements of a basis constructed by Feigin, Jimbo, Loktev, Miwa and Mukhin in the corresponding irreducible representation. Our formula can be viewed as a weighted sum of exponentials of integer points in a certain infinite-dimensional convex polyhedron. We derive a weighted version of Brion’s theorem and then apply it to our polyhedron to prove the formula. © 2016 Springer International Publishing

Article
Verbitsky M. Selecta Mathematica, New Series. 2012. Vol. 18. No. 3. P. 539-555.

A knot space in a manifold M is a space of oriented immersions S1M up to Diff(S 1). J.-L. Brylinski has shown that a knot space of a Riemannian threefold is formally Kähler. We prove that a space of knots in a holonomy G 2 manifold is formally Kähler.

Article
Gorsky Evgeny. Selecta Mathematica, New Series. 2013. Vol. 19. No. 1. P. 125-140.

A theorem of Y. Berest, P. Etingof and V. Ginzburg states that finite-dimensional irreducible representations of a type A rational Cherednik algebra are classified by one rational number m/n. Every such representation is a representation of the symmetric group Sn . We compare certain multiplicity spaces in its decomposition into irreducible representations of Sn with the spaces of differential forms on a zero-dimensional moduli space associated with the plane curve singularity x^my^n .

Article
Cheltsov I., Yanir A. R., Zhang K. Selecta Mathematica, New Series. 2019. Vol. 25: 34. P. 1-36.

The purpose of this article is to develop techniques for estimating basis log canonical thresholds on logarithmic surfaces. To that end, we develop new local intersection estimates that imply log canonicity. Our main motivation and application is to show the existence of Kähler–Einstein edge metrics on all but finitely many families of asymptotically log del Pezzo surfaces, partially confirming a conjecture of two of us. In an appendix we show that the basis log canonical threshold of Fujita–Odaka coincides with the greatest lower Ricci bound invariant of Tian.

Article
Positselski L. Selecta Mathematica, New Series. 2016.

For a left coherent ring A with every left ideal having a countable set of generators, we show that the coderived category of left A-modules is compactly generated by the bounded derived category of finitely presented left A-modules (reproducing a particular case of a recent result of Št’ovíček with our methods). Furthermore, we present the definition of a dualizing complex of fp-injective modules over a pair of noncommutative coherent rings A and B, and construct an equivalence between the coderived category of A-modules and the contraderived category of B-modules. Finally, we define the notion of a relative dualizing complex of bimodules for a pair of noncommutative ring homomorphisms (Formula presented.) and (Formula presented.), and obtain an equivalence between the R / A-semicoderived category of R-modules and the S / B-semicontraderived category of S-modules. For a homomorphism of commutative rings (Formula presented.), we also construct a tensor structure on the R / A-semicoderived category of R-modules. A vision of semi-infinite algebraic geometry is discussed in the introduction.

Article
Feigin B. L., Kedem R., Loktev S. et al. Selecta Mathematica, New Series. 2002. Vol. 8. No. 3. P. 419-474.
Article
Danilov V., Karzanov A., G.A.Koshevoy. Selecta Mathematica, New Series. 2017. Vol. 23. No. 2. P. 1175-1203.

In 1998, Leclerc and Zelevinsky introduced the notion of weakly separated collections of subsets of the ordered n-element set [n] (using this notion to give a combinatorial characterization for quasi-commuting minors of a quantum matrix). They conjectured the purity of certain natural domains D⊆2[n]D⊆2[n] (in particular, of the hypercube 2[n]2[n] itself, and the hyper-simplex {X⊆[n]:|X|=m}{X⊆[n]:|X|=m} for m fixed), where DD is called pure if all maximal weakly separated collections in DD have the same cardinality. These conjectures have been answered affirmatively. In this paper, generalizing those earlier results, we reveal wider classes of pure domains in 2[n]2[n]. This is obtained as a consequence of our study of a novel geometric–combinatorial model for weakly separated set-systems, so-called combined (polygonaltilingson a zonogon, which yields a new insight in the area.

Article
Alexander Kuznetsov, Perry A. Selecta Mathematica, New Series. 2017. Vol. 23. No. 1. P. 389-423.

Given a variety Y with a rectangular Lefschetz decomposition of its derived category, we consider a degree n cyclic cover X→Y ramified over a divisor Z⊂Y. We construct semiorthogonal decompositions of Db(X) and Db(Z) with distinguished components AX and AZ and prove the equivariant category of AX (with respect to an action of the nth roots of unity) admits a semiorthogonal decomposition into n−1n−1 copies of AZ. As examples, we consider quartic double solids, Gushel–Mukai varieties, and cyclic cubic hypersurfaces.

Article
Khoroshkin S. M., Nazarov M. Selecta Mathematica, New Series. 2009. Vol. 14. No. 2. P. 321.
Article
Cheltsov I., Shramov K. Selecta Mathematica, New Series. 2019. Vol. 25. No. 5. P. 71.

We classify finite groups $G$ in $\mathrm{PGL}_{4}(\mathbb{C})$ such that $\mathbb{P}^3$ is $G$-birationally rigid.

Article
Feigin B. L., Finkelberg M. V., Rybnikov L. G. et al. Selecta Mathematica, New Series. 2011. Vol. 17. No. 2. P. 337-361.

Laumon moduli spaces are certain smooth closures of the moduli spaces of maps from the projective line to the flag variety of GLn. We calculate the equivariant cohomology rings of the Laumon moduli spaces in terms of Gelfand-Tsetlin subalgebra of U(gln), and formulate a conjectural answer for the small quantum cohomology rings in terms of certain commutative shift of argument subalgebras of U(gln).

Article
Feigin E., Makedonskyi I. Selecta Mathematica, New Series. 2017. Vol. 23. No. 4. P. 2863-2897.

Classical local Weyl modules for a simple Lie algebra are labeled by dominant weights. We generalize the definition to the case of arbitrary weights and study the properties of the generalized modules. We prove that the representation theory of the generalized Weyl modules can be described in terms of the alcove paths and the quantum Bruhat graph. We make use of the Orr–Shimozono formula in order to prove that the $t=\infty$ specializations of the nonsymmetric Macdonald polynomials are equal to the characters of certain generalized Weyl modules.

Article
Kuznetsov A. G., Shinder E. Selecta Mathematica, New Series. 2018. Vol. 24. No. 4. P. 3475-3500.

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 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.

Article
Feigin E. Selecta Mathematica, New Series. 2012. Vol. 18. No. 3. P. 513-537.

Let Fλ be a generalized flag variety of a simple Lie group G embedded into the projectivization of an irreducible G-module Vλ. We define a flat degeneration Fλa, which is a GaM variety. Moreover, there exists a larger group Ga acting on Fλa, which is a degeneration of the group G. The group Ga contains GaM as a normal subgroup. If G is of type A, then the degenerate flag varieties can be embedded into the product of Grassmannians and thus to the product of projective spaces. The defining ideal of Fλa is generated by the set  of degenerate Plüker relations. We prove that the coordinate ring of Fλa is isomorphic to a direct sum of dual PBW-graded g-modules. We also prove that there exist bases in multi-homogeneous components of the coordinate rings, parametrized by the semistandard PBW-tableux, which are analogues of semistandard tableaux.

Article
Neguț A., Gorsky E. Selecta Mathematica, New Series. 2017. Vol. 23. No. 3. P. 1909-1930.

We study the Maulik-Okounkov K-theoretic stable basis for the Hilbert scheme of points on the plane, and its dependence of the slope parameter.

Article
Mirkovic I., Finkelberg M. V., Braverman A. et al. Selecta Mathematica, New Series. 2002. Vol. 8. No. 3. P. 381-418.
Article
Finkelberg M. V., Ginzburg V., Ionov A. et al. Selecta Mathematica, New Series. 2016. Vol. 22. No. 4. P. 2491-2534.

We study the natural Gieseker and Uhlenbeck compactifications of the rational Calogero–Moser phase space. The Gieseker compactification is smooth and provides a small resolution of the Uhlenbeck compactification. We use the resolution to compute the stalks of the IC-sheaf of the Uhlenbeck compactification.

Article
Bufetov A., Петров Л. Selecta Mathematica, New Series. 2015. Vol. 21. No. 4. P. 1271-1338.
Asymptotic representation theory of general linear groups GL(n,q) over a finite field leads to studying probability measures \rho on the group U of all infinite uni-uppertriangular matrices over F_q, with the condition that \rho is invariant under conjugations by arbitrary infinite matrices. Such probability measures form an infinite-dimensional simplex, and the description of its extreme points was conjectured by Kerov in connection with nonnegative specializations of Hall-Littlewood symmetric functions. Vershik and Kerov also conjectured the Law of Large Numbers for random Young diagrams distributed according to these measures. Our main result is the proof of this Law of Large Numbers. We achieve it by analyzing a new randomized Robinson-Schensted-Knuth (RSK) insertion algorithm which samples random Young diagrams \lambda(n) coming from ergodic measures.