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Gamma conjecture via mirror symmetry
P. 55–115.
Galkin S., Iritani H.
The asymptotic behaviour of solutions to the quantum differential equation of a Fano manifold F defines a characteristic class A_F of F, called the principal asymptotic class.
Gamma conjecture of Vasily Golyshev and the present authors claims that the principal asymptotic class A_F equals the Gamma class associated to Euler's Gamma-function.
We illustrate in the case of toric varieties, toric complete intersections and Grassmannians how this conjecture follows from mirror symmetry. We also prove that Gamma conjecture is compatible with taking hyperplane sections, and give a heuristic argument how the mirror oscillatory integral and the Gamma class for the projective space arise from the polynomial loop space.
Keywords: зеркальная симметрияквантовые когомологииLandau-Ginzburg modelquantum cohomologyмодели Ландау-Гинзбургаmirror symmetryderived category of coherent sheavesпроизводная категория когерентных пучковмногообразие Фаноexceptional collectionполные исключительные наборыFano manifoldFano varietiesисключительные наборыDubrovin's conjectureGamma classApery constantгипотеза ДубровинаГамма классконстанта Апери
In book
Tokyo: Mathematical Society of Japan, 2019.
Ovcharenko M., / Series arXiv "math". 2026.
We introduce an explicit class of tempered Laurent polynomials in the sense of Villegas and Doran--Kerr in n⩽4 variables including all Landau--Ginzburg models for smooth Fano threefolds with very ample anticanonical class. We check that it contains Landau--Ginzburg models for various Fano fourfolds which are complete intersections in smooth toric varieties and Grassmannians of planes, ...
Added: April 30, 2026
Basalaev A., Journal of Geometry and Physics 2025 Vol. 215 Article 105538
The results of A.~Chiodo, Y.~Ruan and M.~Krawitz associate the mirror partner Calabi--Yau variety $X$ to a Landau--Ginzburg orbifold $(f,G)$ if $f$ is an invertible polynomial satisfying Calabi--Yau condition and the group $G$ is a diagonal symmetry group of $f$.
In this paper we investigate the Landau--Ginzburg orbifolds with a Klein quartic polynomial $f = x_1^3x_2 + ...
Added: November 27, 2025
Kasprzyk A., Katzarkov Ludmil, Przyjalkowski Victor et al., Taiwanese Journal of Mathematics 2025 Vol. 29 No. 6 P. 1411–1494
A new structure connecting toric degenerations of smooth Fano threefolds by projections was introduced by I. Cheltsov et al. in “Birational geometry via moduli spaces”. Using Mirror Symmetry, these connections were transferred to the side of Landau–Ginzburg models, and a nice way to connect the Picard rank one Fano threefolds was described. We apply this ...
Added: October 30, 2025
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
Varolgunes U., Polishchuk A., Mathematische Annalen 2024 Vol. 388 P. 2331–2386
We consider Takahashi’s categorical interpretation of the Berglund–Hubsch mirror symmetry conjecture for invertible polynomials in the case of chain polynomials. Our strategy is based on a stronger claim that the relevant categories satisfy a recursion of directed -categories, which may be of independent interest. We give a full proof of this claim on the B-side. On ...
Added: December 2, 2024
Belousov G., Loginov K., Annali dell'Universita di Ferrara 2024 Vol. 70 P. 1093–1114
We prove that all general smooth Fano threefolds of Picard rank 3 and degree 14 are K-stable, where the generality condition is stated explicitly. ...
Added: December 2, 2024
Ovcharenko M., Siberian Electronic Mathematical Reports 2023 Vol. 20 No. 2 P. 1405–1419
A nef-partition for a weighted complete intersection is a combinatorial structure on its weights and degrees which is important for Mirror Symmetry. It is known that nef-partitions exist for smooth well-formed Fano weighted complete intersections of small dimension or codimension, and that in these cases they are strong in the sense that they can be ...
Added: September 9, 2024
Ovcharenko M., International Journal of Mathematics 2023 Vol. 34 No. 11 Article 2350064
We show that the set of families of smooth well-formed Fano weighted complete intersections admits a natural partition with respect to the variance var(X) = coind(X) - codim(X). Moreover, we obtain the classification of smooth well-formed Fano weighted complete intersections of small variance. We also prove that the anticanonical linear system on a smooth well-formed ...
Added: September 9, 2024
Horja R. P., Katzarkov Ludmil, Advances in Mathematics 2024 Vol. 453 Article 109831
We discuss a categorical approach to the theory of discriminants in the combinatorial language introduced by Gelfand, Kapranov and Zelevinsky. Our point of view is inspired by homological mirror symmetry and provides K-theoretic evidence for a conjecture presented by Paul Aspinwall in a conference talk in Banff in March 2016 and later in a joint paper ...
Added: August 17, 2024
Guseva L., Математические заметки 2023 Т. 113 № 1 С. 144–148
В работе строится полный исключительный набор в ограниченной производной категории когерентных пучков на грассманиане Кэли ...
Added: November 1, 2022
Guseva L., / Series arXiv "math". 2022.
We construct a full exceptional collection consisting of vector bundles in the derived category of coherent sheaves on the so-called Cayley Grassmannian, the subvariety of the Grassmannian Gr(3,7) parameterizing 3-subspaces that are annihilated by a general 4-form. The main step in the proof of fullness is a construction of two self-dual vector bundles which is obtained from ...
Added: September 12, 2022
Zudilin W., Long L., Advances in Mathematics 2021 Vol. 393 Article 108058
We establish the supercongruences for the fourteen rigid hypergeometric Calabi-Yau threefolds over Q conjectured by Rodriguez-Villegas in 2003. Our first method is based on Dwork's theory of p-adic unit roots and it allows us to establish the supercongruences between the truncated hypergeometric series and the corresponding unit roots for ordinary primes. The other method makes ...
Added: November 30, 2021
Polishchuk A., Lekili Y., / Series arXiv "math". 2021.
For an appropriate choice of a ℤ-grading structure, we prove that the wrapped Fukaya category of the symmetric square of a (k+3)-punctured sphere, i.e. the Weinstein manifold given as the complement of (k+3) generic lines in ℂP2 is quasi-equivalent to the derived category of coherent sheaves on a singular surface Z2,k constructed as the boundary of a toric Landau-Ginzburg model (X2,k,w2,k). We do this ...
Added: November 28, 2021
Basalaev A., Ionov A., Theoretical and Mathematical Physics 2021 Vol. 209 No. 2 P. 1491–1506
We study Landau-Ginzburg orbifolds (f,G) with f=xn1+…+xnN and G=S⋉Gd, where S⊆SN and Gd is either the maximal group of scalar symmetries of f or the intersection of the maximal diagonal symmetries of f with SLN(ℂ). We construct a mirror map between the corresponding phase spaces and prove that it is an isomorphism restricted to a ...
Added: November 23, 2021
Cheltsov I., Park J., Prokhorov Y. et al., EMS Surveys in Mathematical Sciences 2021 Vol. 8 No. 1-2 P. 39–105
This paper is a survey about cylinders in Fano varieties and related problems ...
Added: November 16, 2021
Przyjalkowski V., Shramov K., Математические заметки 2021 Т. 109 № 4 С. 590–596
We prove that a smooth well-formed Picard rank-one Fano complete intersection of dimension at least 2 in a toric variety is a weighted complete intersection. ...
Added: November 14, 2021
Fonarev A., International Mathematics Research Notices 2020
We show fullness of the exceptional collections of maximal length constructed by Kuznetsov and Polishchuk in the bounded derived categories of coherent sheaves on Lagrangian Grassmannians. ...
Added: October 10, 2021