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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
Galkin S., Golyshev V., Iritani H., / Cornell University. Series math "arxiv.org". 2014. No. 1404.6407.
We propose Gamma Conjectures for Fano manifolds which can be thought of as a square root of the index theorem. Studying the exponential asymptotics of solutions to the quantum differential equation, we associate a principal asymptotic class A_F to a Fano manifold F. We say that F satisfies Gamma Conjecture I if A_F equals the ...
Added: May 4, 2014
Springer, 2013
Vladimir Arnold was one of the great mathematical scientists of our time. He is famous for both the breadth and the depth of his work. At the same time he is one of the most prolific and outstanding mathematical authors. This second volume of his Collected Works focuses on hydrodynamics, bifurcation theory, and algebraic geometry. ...
Added: February 20, 2013
Basel : Birkhauser/Springer, 2020
We study the problem of description of the symplectic mapping class groups π0 Symp(X, ω) (SyMCG) of rational 4-manifolds X = CP #lCP . We specify certain class of symplectic forms ω on such X for which we give a finite presentation of the SyMCG with generators symplectic Dehn twists along Lagrangian spheres. ...
Added: December 11, 2019
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
Prokhorov Y., Zaidenberg M., European Journal of Mathematics 2016 Vol. 2 No. 1 P. 262-282
We construct four different families of smooth Fano fourfolds with Picard rank 1, which contain cylinders, i.e., Zariski open subsets of the form Z ×A1, where Z is a quasiprojective variety. The affine cones over such a fourfold admit effective Ga-actions. Similar constructions of cylindrical Fano threefolds were done previously in the papers by Kishimoto ...
Added: November 27, 2015
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
Pushkar P. E., / Cornell University. Series arXiv "math". 2016. No. arXiv:1602.07948.
We construct counterexamples to lifting properties of Hamiltonian and contact isotopies ...
Added: December 7, 2016
V. V. Shevchishin, Izvestiya. Mathematics 2009 Vol. 73 No. 4 P. 797-859
In this paper we prove the non-existence of Lagrangian embeddings of the Klein bottle K in R4 and CP2. We exploit the existence of a special embedding of K in a symplectic Lefschetz pencil pr:X→S2 and study its monodromy. As the main technical tool, we develop the combinatorial theory of mapping class groups. The results ...
Added: March 18, 2013
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
Loginov K., / Cornell University. 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
Prokhorov Y., Cheltsov I., Zaidenberg M. et al., / Cornell University. Series arXiv "math". 2020.
This paper is a survey about cylinders in Fano varieties and related problems. ...
Added: August 19, 2020
Slavnov S. A., Theoretical Computer Science 2006 Vol. 357 No. 1-3 P. 215-229
Added: March 4, 2013
Vsevolod Shevchishin, / Cornell University. Series math "arxiv.org". 2010.
We introduce the secondary Stiefel-Whitney class $\tilde w_2$ of homotopically trivial diffeomorphisms and show that a homotopically trivial symplectomorphism of a ruled 4-manifold is isotopic to identity if and only if the class $\tilde w_2$ vanishes.
Using this, we give a detailed description of the combinatorial structure of the diffeotopy group of ruled symplectic 4-manifolds ...
Added: March 18, 2013
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
Kishimoto T., Yuri Prokhorov, Zaidenberg M., Osaka Journal of Mathematics 2014 Vol. 51 No. 4 P. 1093-1113
We address the following question: When an affine cone over a smooth Fano threefold admits an effective action of the additive group? In this paper we deal with Fano threefolds of index 1 and Picard number 1. Our approach is based on a geometric criterion from our previous paper, which relates the existence of an ...
Added: October 10, 2013
Aleksei Golota, / Cornell University. 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
Vikulova A., / Cornell University. 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
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
Ю. Г. Прохоров, Известия РАН. Серия математическая 2013 Т. 77 № 3 С. 199-222
We study elements $\tau$ of order two in the birational automorphism groups of rationally connected three-dimensional algebraic varieties such that there exists a non-uniruled divisorial component of the $\tau$-fixed point locus. Using the equivariant minimal model program, we give a rough classification of such elements. ...
Added: July 1, 2013
Pushkar P. E., / Cornell University. Series arXiv "math". 2016. No. arXiv:1602.08743.
We prove a Chekanov-type theorem for the spherization of the cotangent bundle ST∗B of a closed manifold B. It claims that for Legendrian submanifolds in ST∗B the property "to be given by a generating family quadratic at infinity" persists under Legendrian isotopies. ...
Added: December 7, 2016
Cheltsov Ivan, Park J., Won J., Mathematische Zeitschrift 2014 No. 276 P. 51-79
We study log canonical thresholds on quartic threefolds, quintic fourfolds, and double spaces. As an important application, we show that they have Kähler–Einstein metrics if they are general. ...
Added: November 14, 2013
Prokhorov Y., Advances in Geometry 2013 Vol. 13 No. 3 P. 389-418
We classify Fano threefolds with only terminal singularities whose canonical class is
Cartier and divisible by 2 with the additional assumption that the G-invariant part of the Weil divisor
class group is of rank 1 with respect to an action of some group G. In particular, we find a lot of
examples of Fano 3-folds with “many” symmetries. ...
Added: October 7, 2013
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
Prokhorov Y., / Cornell University. Series arXiv "math". 2015. No. 1508.04371.
We study singular Fano threefolds of type V22. ...
Added: October 9, 2015