### ?

## Hodge level for weighted complete intersections

Collectanea Mathematica. 2020. Vol. 71. P. 549-574.

Przyjalkowski V., Shramov K.

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, of a curve, or of a Calabi–Yau variety of low dimension.

Publication based on the results of:

Galkin S., Belmans P., Mukhopadhyay S., / 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

Chebochko N.G., / Cornell University. Series math "arxiv.org". 2017. No. 1712.01810.

The description of global deformations of Lie algebras is important since it is related to unsolved problem of the classification of simple Lie algebras over a field of small characteristic.
In this paper we study global deformations of Lie algebras of type ${D}_{l}$ over an algebraically closed field K of characteristic 2. It is proved that ...

Added: December 8, 2017

Cheltsov I., Przyjalkowski V., / Cornell University. Series arXiv "math". 2018.

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

Added: December 3, 2018

Podkopaev O., Вестник Санкт-Петербургского университета. Серия 1. Математика. Механика. Астрономия 2018 Т. 5(63) № 4 С. 631-636

The goal of this note is to give a proof of the following proposition. Let π be a profinite group and K∗ be a bounded complex of discrete Fp[π]-modules. Assume all Hi (K∗) are finite abelian groups. Then there exists a quasiisomorphism L∗ −→ K∗, where L∗ is a bounded complex of discrete Fp[π]-modules such ...

Added: April 18, 2021

V.A.Vassiliev, Doklady Mathematics 2018 Vol. 98 No. 1 P. 330-333

Stable rational cohomology groups of spaces of non-resultant homogeneous polynomial systems of growing degree in R^n are calculated ...

Added: December 7, 2018

Buryak A., Shadrin S., Zvonkine D., Journal of the European Mathematical Society 2016 Vol. 18 No. 12 P. 2925-2951

We describe the structure of the top tautological group in the cohomology of the moduli space of smooth genus g curves with n marked points. ...

Added: September 27, 2020

Angella D., Tomassini A., Verbitsky M., / Cornell University. Series arXiv "math". 2016.

We study cohomological properties of complex manifolds. In particular, we provide an upper bound for the Bott-Chern cohomology in terms of Betti numbers for compact complex surfaces, according to the dichotomy b1 even or odd. In higher dimension, a similar result is obtained at degree 1 under additional metric conditions (see Theorem 2.4). ...

Added: May 14, 2016

Fedor Bogomolov, Tschinkel Y., Communications on Pure and Applied Mathematics 2013 Vol. 66 No. 9 P. 1335-1359

We explore connections between birational anabelian geometry and abstract projective geometry. One of the applications is a proof of a version of the birational section conjecture. ...

Added: December 27, 2013

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

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

Kurnosov N., Soldatenkov A., Verbitsky M., Advances in Mathematics 2019 Vol. 351 P. 275-295

Let M be a simple hyperkähler manifold. Kuga-Satake
construction gives an embedding of H^2(M, C) into the
second cohomology of a torus, compatible with the Hodge
structure. We construct a torus T and an embedding of the
graded cohomology space H^•(M, C) → H^{•+l}(T, C) for some
l, which is compatible with the Hodge structures and the
Poincaré pairing. Moreover, this ...

Added: June 3, 2019

Cruz Morales J. A., Galkin S., Symmetry, Integrability and Geometry: Methods and Applications (SIGMA) 2013 Vol. 9 No. 005 P. 1-13

In this note we provide a new, algebraic proof of the excessive Laurent phenomenon for mutations of potentials (in the sense of [Galkin S., Usnich A., Preprint IPMU 10-0100, 2010]) by introducing to this theory the analogue of the upper bounds from [Berenstein A., Fomin S., Zelevinsky A., Duke Math. J. 126 (2005), 1–52]. ...

Added: May 27, 2013

Lopatkin V., Kolesnikov P., Alhussein H., / Cornell University. Series arXiv "math". 2022.

We apply discrete algebraic Morse theory to the computation of Hochschild cohomologies of associative conformal algebras. As an example, we evaluate the dimensions of the universal associative conformal envelope U(3) of the Virasoro Lie conformal algebra relative to the associative locality N=3 on the generator with scalar coefficients. ...

Added: May 5, 2022

Cheltsov I., Przyjalkowski V., / 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

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

Ionov A., / Cornell University. 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

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

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

Galkin S., Nagaraj D. S., / Cornell University. Series math "arxiv.org". 2020. No. 2006.12112.

The aim of this note is to investigate the relation between two types of non-singular projective varieties of Picard rank 2, namely the Projective bundles over Projective spaces and certain Blow-up of Projective spaces. ...

Added: April 15, 2021

Guere J., Rossi P., Buryak A., Geometry and Topology 2019 Vol. 23 No. 7 P. 3537-3600

We present a family of conjectural relations in the tautological ring of the moduli spaces of stable curves which implies the strong double ramification/Dubrovin–Zhang equivalence conjecture introduced by the authors with Dubrovin. Our tautological relations have the form of an equality between two different families of tautological classes, only one of which involves the double ...

Added: April 21, 2020

Брудный Ю. А., Зайденберг М. Г., Лин В. Я. et al., Успехи математических наук 2019 Т. 74 № 5 С. 170-180

A detailed review of the scientific activities of the remarkable domestic mathematician E. A. Gorin and his results ...

Added: March 17, 2020

В.А.Васильев, Известия РАН. Серия математическая 2016 Т. 80 № 4 С. 163-184

Rational homology groups of spaces of non-resultant (that is, having only trivial common zeros) systems of homogeneous quadratic polynomial systems in R^3 are calculated ...

Added: March 3, 2018

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