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Interpretations of Spectra
P. 371–407.
Katzarkov L. V., Lee K. S., Svoboda J., Petkov A.
The studies of homological mirror symmetry as correspondence of Lefshetz pencils was initiated as part of the general theory of categorical linear systems. In this paper, we look at the monodromy of these linear systems via a new notion of noncommutative spectrum.
Publication based on the results of:
In book
Vol. 409. , Cham: Springer, 2023.
Basalaev A., Rarovskii A., Journal of Singularities 2026 Vol. 30 P. 61–80
Saito theory associates to an isolated singularity rich structure that plays an important role in mirror symmetry. In this note we construct Saito theory for A and D type Landau-Ginzburg orbifolds. Namely, for the pairs (f,G), where f defines an isolated singularity of A and D type and G is a group of symmetries of ...
Added: September 1, 2026
Rarovskii A., Journal of Singularities 2025 Vol. 28 P. 217–233
Based on the classification of quasihomogeneous singularities, any polynomial $f$ defining such a singularity can be decomposed as f = f_\kappa + f_{add}. The polynomial f_\kappa takes a specific form, whereas f_{add} is constrained only by the requirement that the singularity of f should be isolated. The polynomial f_{add} is zero if and only if ...
Added: January 22, 2026
Basalaev A., Takahashi A., International Mathematics Research Notices 2022 Vol. 2022 No. 19 P. 14865–14922
For any triple of positive integers A′=(a′1,a′2,a′3) and c∈C∗, cusp polynomial fA′=xa′11+xa′22+xa′33−c−1x1x2x3 is known to be mirror to Geigle–Lenzing orbifold projective line P1a′1,a′2,a′3. More precisely, with a suitable choice of a primitive form, the Frobenius manifold of a cusp polynomial fA′ turns out to be isomorphic to the Frobenius manifold of the Gromov–Witten theory of ...
Added: September 9, 2022
Katzarkov L. V., Lupercio E., Meersseman L. et al., Advances in Mathematics 2021 Vol. 391 Article 107945
n this paper, we will introduce Quantum Toric Varieties which are (non-commutative) generalizations of ordinary toric varieties where all the tori of the classical theory are replaced by quantum tori. Quantum toric geometry is the non-commutative version of the classical theory; it generalizes non-trivially most of the theorems and properties of toric geometry. By considering ...
Added: September 24, 2021
Galkin S., Belmans P., Mukhopadhyay S., / Series math "arxiv.org". 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 ...
Added: April 15, 2021
Ebeling W., Gusein-Zade S., Pure and Applied Mathematics Quarterly 2020 Vol. 16 No. 4 P. 1099–1113
In the framework of constructing mirror symmetric pairs of Calabi-Yau manifolds, P.Berglund, T.Hubsch and M.Henningson considered a pair (f,G) consisting of an invertible polynomial f and a finite abelian group G of its diagonal symmetries and associated to this pair a dual pair (f~, G~). A.Takahashi suggested a generalization of this construction to pairs (f, ...
Added: February 3, 2021
Kalashnikov E. G., Chiodo A., / Series arXiv "arXiv". 2019.
We show that there is an extra dimension to the mirror duality discovered in the early nineties by Greene-Plesser and Berglund-Hübsch. Their duality matches cohomology classes of two Calabi--Yau orbifolds. When both orbifolds are equipped with an automorphism s of the same order, our mirror duality involves the weight of the action of s∗ on cohomology. In particular, it ...
Added: November 26, 2020
Kalashnikov E. G., / Series arXiv "arXiv". 2019.
Added: November 26, 2020
Kalashnikov E. G., Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 2019 Vol. 475 No. 2225 P. 1–23
Quiver flag zero loci are subvarieties of quiver flag varieties cut out by sections of representation theoretic vector bundles. We prove the Abelian/non-Abelian correspondence in this context: this allows us to compute genus zero Gromov–Witten invariants of quiver flag zero loci. We determine the ample cone of a quiver flag variety, and disprove a conjecture ...
Added: November 18, 2020
Przyjalkowski V., Shramov K., Collectanea Mathematica 2020 Vol. 71 P. 549–574
We give lower bounds for Hodge numbers of smooth well formed Fano weighted complete intersections. In particular, we compute their Hodge level, that is, the maximal distance between non-trivial Hodge numbers in the same row of the Hodge diamond. This allows us to classify varieties whose Hodge numbers are like that of a projective space, ...
Added: November 13, 2020
Przyjalkowski V., Harder A., Katzarkov L. V., Mathematical notes 2020 Vol. 108 No. 1 P. 33–46
In this paper, we describe recent work towards the mirror P=W conjecture, which relates the weight filtration on the cohomology of a log Calabi–Yau manifold to the perverse Leray filtration on the cohomology of the homological mirror dual log Calabi–Yau manifold taken with respect to the affinization map. This conjecture extends the classical relationship between Hodge numbers ...
Added: November 2, 2020
Cheltsov I., Przyjalkowski V., / Series arXiv "math". 2020.
We conjecture that the number of components of the fiber over infinity of Landau--Ginzburg model for a smooth Fano variety X equals the dimension of the anticanonical system of X. We verify this conjecture for log Calabi--Yau compactifications of toric Landau--Ginzburg models for smooth Fano threefolds, complete intersections, and some toric varieties. ...
Added: August 19, 2020
Genz V., Koshevoy Gleb, Schumann B., Advances in Mathematics 2020 Vol. 369 P. 107178
We establish the relation of Berenstein–Kazhdan’s decoration function and Gross–Hacking–Keel–Kontsevich’s potential on the open double Bruhat cell in the base affine space G/N of a simple, simply connected, simply laced algebraic group G. As a byproduct we derive explicit identifications of polyhedral parametrization of canonical bases of the ring of regular functions on G/N arising ...
Added: May 19, 2020
Cheltsov I., Przyjalkowski V., / Series arXiv "math". 2018.
We verify Katzarkov-Kontsevich-Pantev conjecture for Landau-Ginzburg models of smooth Fano threefolds. ...
Added: December 3, 2018
Lunts V., Przyjalkowski V., Advances in Mathematics 2018 Vol. 329 P. 189–216
We consider the conjectures of Katzarkov, Kontsevich, and Pantev
about Landau--Ginzburg Hodge numbers associated to tamely compactifiable Landau--Ginzburg models. We test these conjectures
in case of dimension two, verifying some and giving a counterexample to the other. ...
Added: February 23, 2018
Katzarkov L. V., Kontsevich M., Pantev T., Journal of Differential Geometry 2017 Vol. 105 No. 1 P. 55–117
In this paper we prove the smoothness of the moduli space of Landau–Ginzburg models. We formulate and prove a Bogomolov–Tian–Todorov theorem for the deformations of Landau–Ginzburg models, develop the necessary Hodge theory for varieties with potentials, and prove a double degeneration statement needed for the unobstructedness result. We discuss the various definitions of Hodge numbers ...
Added: October 23, 2017
Katzarkov L. V., Gross M., Ruddat H., Advances in Mathematics 2017 Vol. 308 P. 208–275
The goal of this paper is to propose a theory of mirror symmetry for varieties of general type. Using Landau–Ginzburg mirrors as motivation, we describe the mirror of a hypersurface of general type (and more generally varieties of non-negative Kodaira dimension) as the critical locus of the zero fibre of a certain Landau–Ginzburg potential. The ...
Added: October 23, 2017
Ballard M., Deliu D., Favero D. et al., Journal of the European Mathematical Society 2017 Vol. 19 No. 4 P. 1127–1158
We provide a geometric approach to constructing Lefschetz collections and Landau–Ginzburg homological projective duals from a variation of Geometric Invariant Theory quotients. This approach yields homological projective duals for Veronese embeddings in the setting of Landau–Ginzburg models. Our results also extend to a relative homological projective duality framework. ...
Added: October 23, 2017
Kulichkov S. N., Tsybulskaya N. D., Chunchuzov I. P. et al., Izvestia, Atmospheric and Oceanic Physic 2017 Vol. 53 No. 4 P. 402–412
Internal gravity wave (IGW) data obtained during the passage of atmospheric fronts over the Moscow region in June–July 2015 is analyzed. IGWs were recorded using a group of four microbarographs (developed at the Obukhov Institute of Atmospheric Physics, Russian Academy of Sciences) located at distances of 7 to 54 km between them. Regularities of variations ...
Added: September 28, 2017
Куличков С. Н., Цыбульская Н. Д., Чунчузов И. П. et al., Известия РАН. Физика атмосферы и океана 2017 Т. 53 № 4 С. 455–469
⎯Internal gravity wave (IGW) data obtained during the passage of atmospheric fronts over the Moscow region in June–July 2015 is analyzed. IGWs were recorded using a group of four microbarographs (developed at the Obukhov Institute of Atmospheric Physics, Russian Academy of Sciences) located at distances of 7 to 54 km between them. Regularities of variations ...
Added: December 15, 2016
Ionov A., / Series arXiv:1504.07930 "math.arxiv". 2015.
Cardy-Frobenius algebra is the algebraic structure on the space of states in open-closed topological field theory. We prove that every semisimple super Cardy-Frobenius algebras is the direct sum of the super Cardy-Frobenius algebras of three simple types. We also apply our results to singularity theory via Landau-Ginzburg models and matrix factorizations. ...
Added: November 8, 2016
Przyjalkowski V., Shramov K., International Mathematics Research Notices 2015 Vol. 21 P. 11302–11332
We prove that the Hodge number h1,N−1(X) of an N-dimensional (N 3) Fano complete intersection X is less by one then the number of irreducible components of the central fiber of (any) Calabi–Yau compactification of Givental’s Landau–Ginzburg model for X. ...
Added: October 12, 2015
Coates T., Corti A., Galkin S. et al., , in: European Congress of Mathematics Kraków, 2 – 7 July, 2012.: Zürich: European Mathematical Society Publishing house, 2014. Ch. 16 P. 285–300.
We consider mirror symmetry for Fano manifolds, and describe how one can recover the classification of 3-dimensional Fano manifolds from the study of their mirrors. We sketch a program to classify 4-dimensional Fano manifolds using these ideas. ...
Added: February 19, 2014