arXiv subjects

Parsa Bakhtary

Publications and source records attributed to Parsa Bakhtary.

6 recordsLinked to original sources

arXiv2026

Basis-Canonical Projectivization for Smooth Complete Toric Varieties

We give an explicit projectivization algorithm for smooth complete toric varieties in arbitrary dimension $n\ge 2$. After fixing an ordered lattice basis, every smooth complete fan~$\Sig$ admits a basis-canonical refinement~$\wSig=\Gam(\Sig)$ that is smooth, complete, projective, and obtained from~$\Sig$ by star subdivisions of two-dimensional cones. Equivalently, $X_{\wSig}\to X_\Sig$ is a finite sequence of ordinary toric blow-ups along smooth invariant centers of codimension two. The algorithm first constructs a projective wall-arrangement fan by extending the spans of the codimension-one cones of~$\Sig$ to central hyperplanes. It then sign-adapts~$\Sig$ to this arrangement by repeatedly subdividing bad two-cones of maximal weight. A lexicographic badness profile gives termination, while projectivity follows from a wall-bend sandwich argument combining a support function pulled back from the arrangement with a relatively ample perturbation. The construction is canonical relative to the chosen ordered basis and requires no additional projectivizing refinement after wall-adaptation. We illustrate the procedure on Oda's non-projective threefold and compare its deterministic length with a separate threefold whose minimal ordinary invariant projectivization length is exactly two.

math.AG↗

arXiv2013

Weights on cohomology, invariants of singularities, and dual complexes

In this paper, we extract natural invariants of a singularity by using the Deligne weight filtration on the cohomology of an exceptional fibre of a resolution, and also on the intersection cohomology of the link. Our primary goal is to study and give natural bounds on the weights in terms of direct images of differential forms. These bounds can be made explicit for various standard classes such as rational, isolated normal Cohen-Macaulay and toroidal singularities, and lead to strong restrictions on the topology of these singularities. A secondary goal of this paper is to make the weight filtration, and related constructions, more widely accessible. So we have tried to make the presentation somewhat self contained. This is supersedes our earlier preprint arXiv:0902.4234.

math.AG↗

arXiv2012

On the minimality of Hamming compatible metrics

A Hamming compatible metric is an integer-valued metric on the words of a finite alphabet which agrees with the usual Hamming distance for words of equal length. We define a new Hamming compatible metric, compute the cardinality of a sphere with respect to this metric, and show this metric is minimal in the class of all "well-behaved" Hamming compatible metrics.

cs.IT↗

arXiv2011

Splitting criteria for vector bundles on higher dimensional varieties

We generalize Horrocks' criterion for the splitting of vector bundles on projective space. We establish an analogous splitting criterion for vector bundles on a class of smooth complex projective varieties of dimension at least four, over which every extension of line bundles splits.

math.AG↗

arXiv2009

The combinatorial part of the cohomology of a singular variety

We study the first step of the weight filtration on the cohomology of a proper complex algebraic variety, which we call the combinatorial part. We obtain a natural upper bound on its size, which gives rather strong information about the topology of rational singularities.

math.AG↗

arXiv2008

On the cohomology of a simple normal crossings divisor

We establish a formula which decomposes the cohomologies of various sheaves on a simple normal crossings divisor (SNC) $D$ in terms of the simplicial cohomologies of the dual complex $Δ(D)$ with coefficients in a presheaf of vector spaces. This presheaf consists precisely of the corresponding cohomology data on the components of $D$ and on their intersections. We use this formula to give a Hodge decomposition for SNC divisors and investigate the toric setting. We also conjecture the existence of such a formula for effective non-reduced divisors with SNC support, and show that this would imply the vanishing of the higher simplicial cohomologies of the dual complex associated to a resolution of an isolated rational singularity.

math.AG↗