arXiv subjects

Mohit Upmanyu

Publications and source records attributed to Mohit Upmanyu.

4 recordsLinked to original sources

arXiv2026

Generalized Zariski cancellation for Brieskorn--Pham varieties

We establish a generalized Zariski cancellation theorem for Brieskorn--Pham varieties over the field of complex numbers. More precisely, we show that if two complex Brieskorn--Pham varieties become isomorphic after taking a product with an arbitrary separated complex scheme having a smooth point, then they are already isomorphic not merely as complex algebraic varieties but, in fact, as $\mathbf{C}^*$-varieties. The proof combines our general cancellation theorem for complex algebraic varieties with a unique singularity, whose proof relies on the analytic cancellation theorem of Hauser--M\"uller, with an exponent rigidity theorem for Brieskorn--Pham varieties. The latter asserts that, over any field of characteristic zero, the exponent tuple appearing in the defining equation completely determines the isomorphism class of the corresponding Brieskorn--Pham variety.

math.AG↗

arXiv2025

Generators of top cohomology

Let $R$ be a commutative noetherian ring and $f: X \to \mathrm{Spec} R$ a proper smooth morphism, of relative dimension $n$. From Hartshorne, Residues and Duality, Springer, 1966, one knows that the trace map $\mathrm{Tr}_f : \mathrm{H}^n(X, \omega_{X/R}) \to R$ is an isomorphism when $f$ has geometrically connected fibres. We construct an exact sequence that generates $\mathrm{Ext}_X^n(\mathcal{O}_X, \omega_{X/R}) = \mathrm{H}^n(X, \omega_{X/R})$ as an $R$-module in the following cases: (1) when $R$ is a DVR and $f$ has a section; (2) when $R=\mathbb{Z}$ and $X$ is the Grassmannian $G_{2,m}$ for some $m \geq 4$. This partially answers a question raised by Lipman.

math.AC↗

arXiv2024

Generalization of Gurjar's Hyperplane section theorem to arbitrary analytic varieties and A$\mathbb{m}$AC classes

The aim of this paper is to generalize the hyperplane section theorem of Gurjar to arbitrary (local) analytic varieties even if the intersection with of hyperplanes is not necessarily isolated. In case of formal varieties, we generalize the statement to work for different classes of functions than just hyperplanes. We call these classes (which are subsets of formal power series ring) to be algebraic $\mathbb{m}$-adicaly closed (A$\mathbb{m}$AC).

math.AG↗

arXiv2023

Milnor and Tjurina numbers for an isolated complete intersection singularity

This paper aims to prove that given a isolated complete intersection singularity, the Milnor number will be bounded by a bound depending only on Tjurina number and dimension of the singularity. The proof uses A$\mathfrak{m}$AC (introduced in arXiv:2204.05594) and as with such methods, the bound is purely existential.

math.AG↗