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Mikhail Ovcharenko

Publications and source records attributed to Mikhail Ovcharenko.

7 recordsLinked to original sources
arXiv2026

On Arithmetic Mirror Symmetry for smooth Fano fourfolds

We introduce an explicit class of tempered Laurent polynomials in the sense of Villegas and Doran--Kerr in $n \leqslant 4$ 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, or are quiver flag zero loci. We discuss implications to Arithmetic Mirror Symmetry conjecture, a Hodge-theoretic approach to the study of Apéry constants of Fano varieties proposed by Golyshev--Kerr--Sasaki. Using the partial case of Arithmetic Mirror Symmetry conjecture proved by Kerr, we provide three examples of a Mirror Symmetry correspondence between specific algebraic classes.

math.AG↗
arXiv2025

Weighted Grassmannians and their explicit description

We propose an explicit construction of a weighted generalised Grassmannian. For a weighted Grassmannian (i.e., for series A) we obtain an effective parametrisation of possible $\mathbb{Z}$-gradings on Plücker coordinates, and provide the explicit formulae for its dualising sheaf and Hilbert series in terms of this parametrisation. Our approach can be generalised to other irreducible root systems.

math.AG↗
arXiv2025

Modularity of Landau-Ginzburg models

For each Fano threefold, we construct a family of Landau-Ginzburg models which satisfy many expectations coming from different aspects of mirror symmetry; they are log Calabi-Yau varieties with proper potential maps; they admit open algebraic torus charts on which the potential function $w$ restricts to a Laurent polynomial satisfying a deformation of the Minkowski ansatz; the general fibres of $w$ are Dolgachev-Nikulin dual to the anticanonical hypersurfaces in $X$. To do this, we study the deformation theory of Landau-Ginzburg models in arbitrary dimension, following the third-named author, Kontsevich, and Pantev, specializing to the case of Landau-Ginzburg models obtained from Laurent polynomials. Our proof of Dolgachev-Nikulin mirror symmetry is by detailed case-by-case analysis, refining work of Cheltsov and the fifth-named author.

math.AG↗
arXiv2024

Belyi's theorem for smooth complete intersections of general type in generalised Grassmannians and weighted projective spaces

We show that A. Javanpeykar's proof of Belyi's theorem for smooth complete intersections of general type in ordinary projective spaces can be generalised to smooth complete intersections of general type in generalised Grassmannians and weighted projective spaces. We propose an approach to the generalisation of this result to smooth complete intersections of general type in more general Mori dream spaces.

math.AG↗
arXiv2023

On the existence of nef-partitions for smooth well-formed Fano weighted complete intersections

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 realized as fibers of morphisms of weighted simplicial complexes, i.e., finite abstract simplicial complexes equipped with a weight function. It was conjectured that this approach can be extended to the case of arbitrary codimension. We show that in the case of any codimension greater than 3 strong nef-partitions may not exist, and provide a sufficient combinatorial condition for existence of a strong nef-partition in terms of weights. We also show that the combinatorics of smooth well-formed weighted complete intersections can be arbitrarily complicated from the point of view of simplicial geometry.

math.AG↗
arXiv2023

The classification of smooth well-formed Fano weighted complete intersections

We show that the set of families of smooth well-formed Fano weighted complete intersections admits a natural partition with respect to the variance $\mathrm{var}(X) = \mathrm{coind}(X) - \mathrm{codim}(X)$. 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 Fano weighted complete intersection of anticanonical degree one is never base-point free.

math.AG↗