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Of all publications in the section: 482
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Working paper
Deviatov R. arxiv.org. math. Cornell University, 2010. No. arXiv:1007.1353v1.

Let G be a semisimple algebraic group whose decomposition into the product of simple components does not contain simple groups of type A, and P⊆G be a parabolic subgroup. Extending the results of Popov [7], we enumerate all triples (G, P, n) such that (a) there exists an open G-orbit on the multiple flag variety G/P × G/P × . . . × G/P (n factors), (b) the number of G-orbits on the multiple flag variety is finite.

Added: Jun 28, 2012
Working paper
Kiritchenko V. arxiv.org. math. Cornell University, 2014
We describe an elementary convex geometric algorithm for realizing Schubert cycles in complete flag varieties by unions of faces of polytopes. For GL_n and Gelfand--Zetlin polytopes, combinatorics of this algorithm coincides with that of  the  mitosis on pipe dreams introduced by Knutson and Miller. For Sp_4 and a Newton--Okounkov polytope of the symplectic flag variety, the algorithm yields a new combinatorial rule that extends to Sp_{2n}.
Added: Sep 17, 2014
Working paper
Gavrilovich M., Pimenov K. arxiv.org. math. Cornell University, 2020
We interpret a construction of geometric realisation by [Besser], [Grayson], and [Drinfeld] of a simplicial set as constructing a space of maps from the interval to a simplicial set, in a certain formal sense, reminiscent of the Skorokhod space of semi-continuous functions; in particular, we show the geometric realisation functor factors through an endofunctor of a certain category. Our interpretation clarifies the explanation of [Drinfeld] "why geometric realization commutes with Cartesian products and why the geometric realization of a simplicial set [...] is equipped with an action of the group of orientation preserving homeomorphisms of the segment [0,1]".
Added: Oct 29, 2020
Working paper
Berger G., Beklemishev L. D., Tompits H. arxiv.org. math. Cornell University, 2016. No. 1601.02857.
We consider a many-sorted variant of Japaridze's polymodal provability logic GLP. In this variant, propositional variables are assigned sorts n< omega, where variables of sort n are arithmetically interpreted as Pi_{n+1}-sentences of the arithmetical hierarchy. We prove that the many-sorted variant is arithmetically complete with respect to this interpretation.
Added: Mar 13, 2016
Working paper
Galkin S., Belmans P., Mukhopadhyay S. arxiv.org. math. Cornell University, 2020. No. 2009.05568.
We introduce graph potentials, which are Laurent polynomials associated to (colored) trivalent graphs. These graphs encode degenerations of curves to rational curves, and graph potentials encode degenerations of the moduli space of rank 2 bundles with fixed determinant. We show that the birational type of the graph potential only depends on the homotopy type of the colored graph, and thus define a topological quantum field theory. By analyzing toric degenerations of the moduli spaces we explain how graph potentials are related to these moduli spaces in the setting of mirror symmetry for Fano varieties.  On the level of enumerative mirror symmetry this shows how invariants of graph potentials are related to Gromov-Witten invariants of the moduli space. In the context of homological mirror symmetry we formulate a conjecture regarding the shape of semiorthogonal decompositions for the derived category. Studying the properties of graph potentials we provide evidence for this conjecture. Finally, by studying the Grothendieck rings of varieties and categories we will give further geometric evidence.
Added: Apr 15, 2021
Working paper
Kleptsyn V., Alvarez S., Malicet D. et al. arxiv.org. math. Cornell University, 2015
Added: Jun 22, 2016
Working paper
Kuznetsov A., Debarre O. arxiv.org. math. Cornell University, 2015
This paper performs a systematic study of Gushel–Mukai varieties—Fano manifolds with Picard number 1, coindex 3, and degree 10 (higher-dimensional analogues of prime Fano threefolds of genus 6). We introduce a new approach to the classification of these varieties which includes mildly singular varieties, gives a criterion for an isomorphism of such varieties, and describes their automorphisms groups. We carefully develop the relation between Gushel–Mukai varieties and Eisenbud–Popescu– Walter sextics introduced earlier by Iliev–Manivel and O’Grady. We describe explicitly all Gushel–Mukai varieties whose associated EPW sextics are isomorphic or dual (we call them period partners or dual varieties respectively). Finally, we show that in dimension 3 and higher, period partners/dual varieties are always birationally isomorphic.
Added: Nov 15, 2015
Working paper
Takebe T., Takasaki K. arxiv.org. math. Cornell University, 2009. No. 0912.4867.
Added: Nov 2, 2012
Working paper
Levin A., Olshanetsky M., Smirnov A. et al. arxiv.org. math. Cornell University, 2012
Added: Feb 6, 2013
Working paper
Kuznetsov A. arxiv.org. math. Cornell University, 2012. No. 1211.4693.
We define the normal Hochschild cohomology of an admissible subcategory of the derived category of coherent sheaves on a smooth projective variety $X$ --- a graded vector space which controls the restriction morphism from the Hochschild cohomology of $X$ to the Hochschild cohomology of the orthogonal complement of this admissible subcategory. When the subcategory is generated by an exceptional collection, we define its new invariant (the height) and show that the orthogonal to an exceptional collection of height $h$ in the derived category of a smooth projective variety $X$ has the same Hochschild cohomology as $X$ in degrees up to $h - 2$. We use this to describe the second Hochschild cohomology of quasiphantom categories in the derived categories of some surfaces of general type. We also give necessary and sufficient conditions of fullness of an exceptional collection in terms of its height and of its normal Hochschild cohomology.
Added: Oct 4, 2013
Working paper
Kolesnikov A. arxiv.org. math. Cornell University, 2012. No. 1201.2342.
We study the optimal transportation mapping $\nabla \Phi : \mathbb{R}^d \mapsto \mathbb{R}^d$ pushing forward a probability measure $\mu = e^{-V} \ dx$ onto another probability measure $\nu = e^{-W} \ dx$. Following a classical approach of E. Calabi we introduce the Riemannian metric $g = D^2 \Phi$ on $\mathbb{R}^d$ and study spectral properties of the metric-measure space $M=(\mathbb{R}^d, g, \mu)$. We prove, in particular, that $M$ admits a non-negative Bakry-{\'E}mery tensor provided both $V$ and $W$ are convex. If the target measure $\nu$ is the Lebesgue measure on a convex set $\Omega$ and $\mu$ is log-concave we prove that $M$ is a $CD(K,N)$ space. Applications of these results include some global dimension-free a priori estimates of $\| D^2 \Phi\|$. With the help of comparison techniques on Riemannian manifolds and probabilistic concentration arguments we proof some diameter estimates for $M$.
Added: Mar 28, 2013
Working paper
Yurii M. Burman. arxiv.org. math. Cornell University, 2015. No. 1508.02245.
We prove a generalization of the (nonsymmetric) matrix-tree theorem containing no trees and essentially no matrices. Instead of trees we consider acyclic directed graphs with a prescribed set of sinks, and instead of determinant, a polynomial invariant of the matrix determined by directed graph such that any two vertices of the same connected component are mutually reacheable (according to arrows). 
Added: Oct 9, 2015
Working paper
Burman Y. M., Ploskonosov A., Trofimova A. arxiv.org. math. Cornell University, 2011. No. 1109.6625.

We calculate determinants of weighted sums of reflections and of (nested) commutators of reflections. The results obtained generalize the matrix-tree theorem by Kirchhoff and the Pfaffian-hypertree theorem by Massbaum and Vaintrob.

Added: Nov 7, 2012
Working paper
Netay I. V. arxiv.org. math. Cornell University, 2016
    It is well known that the two-parametric Todd genus and elliptic functions of level~$d$ define $n$-multiplicative Hirzebruch genera, if~$d$ divides~$n+1$.     Both these cases are particular cases of Krichever genera defined by the Baker--Akhiezer functions.     In this work the inverse problem is solved.     Namely, it is proved that only these families of functions define $n$-multiplicative Hirzebruch genera among all the Krichever genera for all~$n$.  
Added: Oct 19, 2016
Working paper
Shramov K., Przyjalkowski V. arxiv.org. math. Cornell University, 2018
We give lower bounds for Hodge numbers of smooth well formed Fano weighted complete intersections. In particular, we compute their Hodge complexity, that is, the maximal distance between non-trivial Hodge numbers. This allows us to classify varieties of such type whose Hodge numbers are like that of a projective space, of a curve, or of a Calabi-Yau variety of low dimension.
Added: Oct 21, 2018
Working paper
Verbitsky M. arxiv.org. math. Cornell University, 2012
Added: Feb 6, 2013
Working paper
Pirkovskii A. Y. arxiv.org. math. Cornell University, 2013. No. 1304.1991.
We introduce and study holomorphically finitely generated (HFG) Fr\'echet algebras, which are analytic counterparts of affine (i.e., finitely generated) complex algebras. Using a theorem of O. Forster, we prove that the category of commutative HFG algebras is anti-equivalent to the category of Stein spaces of finite embedding dimension. We also show that the class of HFG algebras is stable under some natural constructions. This enables us to give a series of concrete examples of HFG algebras, including Arens-Michael envelopes of affine algebras (such as the algebras of holomorphic functions on the quantum affine space and on the quantum torus), the algebras of holomorphic functions on the free polydisk, on the quantum polydisk, and on the quantum polyannulus.
Added: May 2, 2013
Working paper
Andrey Soldatenkov, Verbitsky M. arxiv.org. math. Cornell University, 2013
A hypercomplex manifold is a manifold equipped with a triple of complex structures satisfying the quaternionic relations. A holomorphic Lagrangian variety on a hypercomplex manifold with trivial canonical bundle is a holomorphic subvariety which is calibrated by a form associated with the holomorphic volume form; this notion is a generalization of the usual holomorphic Lagrangian subvarieties known in hyperkaehler geometry. An HKT (hyperkaehler with torsion) metric on a hypercomplex manifold is a metric determined by a local potential, in a similar way to the Kaehler metric. We prove that a base of a holomorphic Lagrangian fibration is always Kaehler, if its total space is HKT. This is used to construct new examples of hypercomplex manifolds which do not admit an HKT structure.
Added: Dec 22, 2013
Working paper
Lebedev V., Olevskii A. arxiv.org. math. Cornell University, 2019. No. arXiv:1803.02177v2.
We consider the algebras M_p of Fourier multipliers and show that for every bounded continuous function f on R^d there exists a self-homeomorphism h of R^d such that the superposition f oh is in M_p(R^d) for all p, 1 < p < \infty. Moreover, under certain assumptions on a family K of continuous functions, one h will sffice for all f in K. A similar result holds for functions on the torus T^d. This may be contrasted with the known solution of Luzin's problem related to the Wiener algebra.  
Added: May 8, 2018
Working paper
Cerulli I., Feigin E., Reineke M. arxiv.org. math. Cornell University, 2013. No. 1302.529.
Added: Mar 4, 2013
Working paper
Pirkovskii A. Y. arxiv.org. math. Cornell University, 2012. No. 1201.2828.
Let X be a Stein manifold, and let Y be a closed complex submanifold of X. Denote by O(X) the algebra of holomorphic functions on X. We show that the weak (i.e., flat) homological dimension of O(Y) as a Fr'echet O(X)-module equals the codimension of Y in X. In the case where X and Y are of Liouville type, the same formula is proved for the projective homological dimension of O(Y) over O(X). On the other hand, we show that if X is of Liouville type and Y is hyperconvex, then the projective homological dimension of O(Y) over O(X) equals the dimension of X.
Added: Mar 12, 2013