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Symplectic geometry of unbiasedness and critical points of a potential
P. 1–18.
Bondal A. I., Zhdanovskiy I.
The goal of these notes is to show that the classification problem of algebraically unbiased system of projectors has an interpretation in symplectic geometry. This leads us to a description of the moduli space of algebraically unbiased bases as critical points of a potential functions, which is a Laurent polynomial in suitable coordinates. The Newton polytope of the Laurent polynomial is the classical Birkhoff polytope, the set of double stochastic matrices. Mirror symmetry interprets the polynomial as a Landau-Ginzburg potential for corresponding Fano variety and relates the symplectic geometry of the variety with systems of unbiased projectors
Publication based on the results of:
In book
Tokyo: Mathematical Society of Japan, 2019.
Loginov K., Przyjalkowski V., Trepalin A., Труды Математического института им. В.А. Стеклова РАН 2025 Т. 329 С. 132–164
We introduce and study the notion of G-coregularity of algebraic varieties endowed with an action of a finite group G. We compute the G-coregularity of smooth del Pezzo surfaces of degree at least 6, and give a characterization of groups that can act on conic bundles with G-coregularity 0. We describe the relations between the notions of G-coregularity, G-log ...
Added: September 4, 2025
V. Gorbounov, Guterman L., Bychkov B. et al., Journal of Geometry and Physics 2025 Vol. 207 P. 0
In this paper we relate a well-known in symplectic geometry compactification of the space of symmetric bilinear forms considered as a chart of the Lagrangian Grassmannian to the specific compactifications of the space of electrical networks in the disc obtained in \cite{L}, \cite{CGS} and \cite{BGKT}. In particular, we state an explicit connection between these works ...
Added: October 4, 2024
Prokhorov Y., Zaidenberg M., , in: The Art of Doing Algebraic Geometry.: Birkhäuser, 2023. P. 363–383.
We show that the affine cones over any Fano–Mukai fourfold of genus 10 are flexible in the sense of [1]. In particular, the automorphism group of such a cone acts highly transitively outside the vertex. Furthermore, any Fano–Mukai fourfold of genus 10, with one exception, admits a covering by open charts isomorphic to ...
Added: November 13, 2023
Kuznetsov A., Prokhorov Y., American Journal of Mathematics 2023 Vol. 145 No. 2 P. 335–411
We give necessary and sufficient conditions for unirationality and rationality of Fano threefolds of geometric Picard rank-1 over an arbitrary field of zero characteristic. ...
Added: September 1, 2023
Prokhorov Y., Rendiconti del Circolo Matematico di Palermo 2023 Vol. 2 No. 72 P. 1797–1821
We classify nonrational Fano threefolds X with terminal Gorenstein singularities such that $\mathrm{\rk}\, \mathrm{\Pic}(X)=1$, (−KX)3≥8, and $\mathrm{\rk}\, \mathrm{\Cl}(X)\le 2$. ...
Added: September 1, 2023
Cham: Springer, 2023.
This book collects the proceedings of a series of conferences dedicated to birational geometry of Fano varieties held in Moscow, Shanghai and Pohang
The conferences were focused on the following two related problems:
• existence of Kähler–Einstein metrics on Fano varieties
• degenerations of Fano varieties
on which two famous conjectures were recently proved. The first is the famous ...
Added: May 24, 2023
Alexander Kuznetsov, Prokhorov Y., Journal of the Institute of Mathematics of Jussieu 2024 Vol. 23 No. 1 P. 207–247
We prove rationality criteria over nonclosed fields of characteristic 00 for five out of six types of geometrically rational Fano threefolds of Picard number 11 and geometric Picard number bigger than 11 . For the last type of such threefolds, we provide a unirationality criterion and construct examples of unirational but not stably rational varieties of this type. ...
Added: November 30, 2022
Prokhorov Y., ELECTRONIC RESEARCH ARCHIVE 2022 Vol. 30 No. 5 P. 1881–1897
We obtain a sufficient condition for a Fano threefold with terminal singularities to have a conic bundle structure. ...
Added: November 28, 2022
Vikulova A., / Series arXiv "math". 2022.
In this paper we prove that for n-dimensional smooth l-Fano well formed weighted complete intersections, which is not isomorphic to a usual projective space, the upper bound for l is equal to ⌈log2(n+2)⌉−1. We also prove that the only l-Fano of dimension n among such manifolds with inequalities ⌈log3(n+2)⌉⩽l⩽⌈log2(n+2)⌉−1 is a complete intersection of quadrics in a usual projective space. ...
Added: November 27, 2022
Loginov K., European Journal of Mathematics 2021 Vol. 8 No. 3 P. 991–1005
Consider a family of Fano varieties π:X→B∋o over a curve germ with a smooth total space X. Assume that the generic fiber is smooth and the special fiber F=π−1(o) has simple normal crossings. Then F is called a semistable degeneration of Fano varieties. We show that the dual complex of F is a simplex of dimension not greater than dim F. Simplices ...
Added: September 3, 2021
Przyjalkowski V., Shramov K., Communications in Number Theory and Physics 2020 Vol. 14 No. 3 P. 511–553
We prove that if a smooth variety with non-positive canonical class can be embedded into a weighted projective space of dimension n as a well formed complete intersection and it is not an intersection with a linear cone therein, then the weights of the weighted projective space do not exceed n+1. Based on this bound ...
Added: October 13, 2020
Prokhorov Y., Cheltsov I., Zaidenberg M. et al., / Series arXiv "math". 2020.
This paper is a survey about cylinders in Fano varieties and related problems. ...
Added: August 19, 2020
Lekili Y., Polishchuk A., Compositio Mathematica 2020 Vol. 156 No. 7 P. 1310–1347
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 ...
Added: August 12, 2020
Cheltsov I., Zhang K., European Journal of Mathematics 2019 Vol. 5 P. 729–762
We prove that 𝛿δ-invariants of smooth cubic surfaces are at least 6/5. ...
Added: May 10, 2020
Alekseeva D., Shevchishin V., , in: Сборник научно-исследовательских работ по итогам конкурса НИРС НИУ ВШЭ – 2016.: [б.и.], 2017. P. 171–200.
The goal of this diploma thesis is to understand symplectic mapping class group of rational 4-manifolds and how it changes, when the cohomology class of the symplectic form varies. For this aims, we will show that in some cases this group admits a realization as the fundamental group of the complement to certain divisor on ...
Added: April 20, 2020
M.: National Research University Higher School of Economics, 2017.
The goal of this thesis is to understand symplectic mapping class group of rational 4-manifolds and how it changes, when the cohomology class of the symplectic form varies. For this aims, we will show that in some cases this group admits a realization as the fundamental group of the complement to certain divisor on Hilbert scheme ...
Added: December 11, 2019
Loginov K., / Series arXiv "math". 2019.
Consider a family of Fano varieties π:X⟶B∋o over a curve germ with a smooth total space X. Assume that the generic fiber is smooth and the special fiber F=π^{−1}(o) has simple normal crossings. Then F is called a semistable degeneration of Fano varieties. We show that the dual complex of F is a simplex of dimension ≤dim F. Simplices of any admissible dimension can be realized ...
Added: October 11, 2019
Aleksei Golota, / Series arXiv "math". 2019.
For a polarized variety (X,L) and a closed connected subgroup G⊂Aut(X,L) we define a G-invariant version of the δ-threshold. We prove that for a Fano variety (X,−KX) and a connected subgroup G⊂Aut(X) this invariant characterizes G-equivariant uniform K-stability. We also use this invariant to investigate G-equivariant K-stability of some Fano varieties with large groups of ...
Added: October 7, 2019
Cheltsov I., Dubouloz A., Park J., Compositio Mathematica 2018 Vol. 154 No. 11 P. 2462–2484
We study a wide class of affine varieties, which we call affine Fano varieties. By analogy with birationally super-rigid Fano varieties, we define super-rigidity for affine Fano varieties, and provide many examples and non-examples of super-rigid affine Fano varieties. ...
Added: October 17, 2018
Prokhorov Y., Zaidenberg M., European Journal of Mathematics 2018 Vol. 4 No. 3 P. 1197–1263
It is known that the moduli space of smooth Fano–Mukai fourfolds V18 of genus 10 has dimension one. We show that any such fourfold is a completion of ℂ4 in two different ways. Up to isomorphism, there is a unique fourfold Vs18 acted upon by SL2(ℂ). The group Open image in new window is a ...
Added: September 6, 2018