arXiv subjects
Publications and source records attributed to Mykola Pochekai.
We prove that the coordinate Artinian reduction of the Stanley-Reisner ring of a regular unimodular embedded lattice simplicial ball whose boundary is an induced subcomplex admits an explicit top-heavy Lefschetz element.
We construct a natural crepant resolution of the Batyrev-Borisov mirror dual family to the complete intersection of two cubic hypersurfaces in $\mathbb P^5$. It is, similarly to the mirrors of quintic threefolds, a family over $\mathbb{P}^1$ with singular fibers over the set $\{0, \infty\} \cup μ_6$. We compute an explicit height function producing the MPCP desingularization of the $B$-model toric ambient space. We compute the limiting mixed Hodge structures of the singular fibers. We show that the singular fiber over $\infty$ has maximal unipotent monodromy, whereas the singular fiber over $0$ is of a new type compared to the quintic case.
We formulate and prove Parseval-Rayleigh identities for graded Artinian Gorenstein algebras over fields of positive characteristic. Specializing the general result, we provide an alternative proof of the Parseval-Rayleigh identities of generic Artinian reductions of Stanley-Reisner rings of oriented simplicial spheres.
This paper establishes BCOV-type genus one mirror symmetry for the intersections of two cubics in $\mathbb{P}^5$. The proof applies previous constructions of the mirror family by the second author and computations of genus one Gromov-Witten invariants by A. Popa. The approach adapts the strategy used for hypersurfaces, as developed by the first author and collaborators, but addresses the distinct geometry involved. A key feature is a systematic usage of toric techniques and related computer aided calculations to determine seemingly otherwise inaccessible invariants.