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## Interpretations of Spectra

P. 371-407.

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

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

Cheltsov I., Przyjalkowski V., Fibers over infinity of Landau-Ginzburg models / Cornell University. 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

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

Katzarkov L., 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., 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

Przyjalkowski V., Harder A., Katzarkov L., 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

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

Katzarkov L., Przyjalkowski V., , in : Proceedings of the Gökova Geometry-Topology Conference 2011. : Boston : International Pre, 2012. P. 97-124.

In the last three years a new concept — the concept of wall crossing
has emerged. The current situation with wall crossing phenomena, after pa pers of Seiberg–Witten, Gaiotto–Moore–Neitzke, Vafa–Cecoti and seminal works
by Donaldson–Thomas, Joyce–Song, Maulik–Nekrasov–Okounkov–Pandharipande,
Douglas, Bridgeland, and Kontsevich–Soibelman, is very similar to the situation with
Higgs Bundles after the works of Higgs and Hitchin — it is ...

Added: February 16, 2013

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

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

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

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

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

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

Cheltsov I., Przyjalkowski V., Katzarkov-Kontsevich-Pantev Conjecture for Fano threefolds / Cornell University. Series arXiv "math". 2018.

We verify Katzarkov-Kontsevich-Pantev conjecture for Landau-Ginzburg models of smooth Fano threefolds. ...

Added: December 3, 2018

Basalaev A., Takahashi A., International Mathematical 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

Kalashnikov E. G., Laurent polynomial mirrors for quiver flag zero loci / . 2019.

Added: November 26, 2020

Ionov A., Supersymmetric Semisimple Cardy-Frobenius Algebras / . 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

Katzarkov L., 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

Kalashnikov E. G., Chiodo A., Mirror symmetry and automorphisms / . 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

Ilten N. O., Lewis J., Victor Przyjalkowski, Journal of Algebra 2013 Vol. 374 P. 104-121

We show that every Picard rank one smooth Fano threefold has a weak Landau–Ginzburg model coming from a toric degeneration. The fibers of these Landau–Ginzburg models can be compactified to K3 surfaces with Picard lattice of rank 19. We also show that any smooth Fano variety of arbitrary dimension which is a complete intersection of ...

Added: July 2, 2013

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

Iliev A., Katzarkov L., Victor Przyjalkowski, Proceedings of the Edinburgh Mathematical Society 2014 Vol. 57 P. 145-173

This paper suggests a new approach to questions of rationality of threefolds based on category theory. Following M. Ballard, D. Favero, L. Katzarkov (ArXiv:1012.0864) and D. Favero, L. Katzarkov (Noether--Lefschetz Spectra and Algebraic cycles, in preparation) we enhance constructions from A. Kuznetsov (arXiv:0904.4330) by introducing Noether--Lefschetz spectra --- an interplay between Orlov spectra (C. Oliva, ...

Added: July 2, 2013

Galkin S., Belmans P., Mukhopadhyay S., Graph potentials and moduli spaces of rank two bundles on a curve / Cornell University. 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