arXiv subjects

Anubhab Pahari

Publications and source records attributed to Anubhab Pahari.

5 recordsLinked to original sources

arXiv2026

Torelli theorem for the product of moduli spaces of vector bundles over a curve

We prove a Torelli-type theorem for a product of the moduli spaces of semistable vector bundles over smooth projective curves of genus $g\,\ge\, 4$. A similar result is proved for the product of the moduli stacks of semistable vector bundles. This is proved using a decomposition theorem for the product of the normal projective varieties with Picard rank one and discrete Picard group. As an application, we compute the automorphism group of the product of the moduli spaces and moduli stacks.

math.AG↗

arXiv2026

Log-concavity and unimodality of Hodge numbers of Hilbert schemes of points over a surface

Let $S$ be a smooth projective complex surface with irregularity $q=h^{1,0}(S)$ and geometric genus $g=h^{2,0}(S)$, and let $S^{[n]}$ denote its Hilbert scheme of $n$ points. We prove that, for every $n\ge0$, the sequence \[ \left(h^{p,0}\bigl(S^{[n]}\bigr)\right)_{p=0}^{2n} \] is log-concave if and only if $g\le\binom{q+1}{2}$. Moreover, log-concavity of all these sequences is already equivalent to log-concavity of the sequence for $n=2$. We also prove that these sequences are unimodal for every $n\ge0$ if and only if $q\ge1$ or $g=0$, and that this condition is already detected by the sequence for $n=1$.

math.AG↗

arXiv2026

On the classification of products of Hilbert schemes of points over a surface

Let $S$ be a smooth projective surface over $\mathbb{C}$ and $S^{[n]}$ be the Hilbert scheme of $n$ points over $S$, for any positive integer $n$. Under suitable hypotheses, we show that the products of Hilbert schemes $S^{[{\bf a}]}$ and $S^{[{\bf b}]}$ associated to two distinct partitions ${\bf a}$ and ${\bf b}$ of $n$ are non-isomorphic, using Betti numbers, Hodge numbers and Euler characteristics of the individual factors. The classifications obtained using Betti numbers and Euler characteristics extend over $\overline{\F}_q$ as well, where $\F_q$ is the finite field with $q$ elements. We also provide a complete classification of such product spaces for K3 surfaces using their inherent symplectic structure.

math.AG↗

arXiv2026

Zero-sum Inverse Realization and Property~(P) under Join Operations

We introduce zero-sum inverse realization of property (P) of a graph $G$, obtained by imposing an additional condition $ \mathbf{1}^{\top}A^{-1}\mathbf{1}=0, $ on a matrix $A\in S(G)$ realizing property (P), where $\mathbf{1}$ is the all-ones vector. We prove that every graph of order at least three having property (P) admits such a zero-sum inverse realization. As applications, we prove that property~(P) is preserved under the join of two graphs of order at least $3$ and, more generally, under the $H$-join of a family of graphs of order at least $3$, where $H$ is arbitrary. Consequently, we obtain sufficient conditions for cographs and lexicographic products of graphs to possess property~(P). Throughout the paper, many examples are given.

math.CO↗

arXiv2026

Strict Log-concavity of $k$-coloured Partitions

In recent years, there has been extensive work on inequalities among partition functions. In particular, Nicolas, and independently DeSalvo--Pak, proved that the partition function $p(n)$ is eventually log-concave. Inspired by this and other results, Chern--Fu--Tang first conjectured log-concavity of $k$-coloured partitions. Three of the authors and Tripp later proved this conjecture by introducing recursive sequences and a strict inequality for fractional partition functions, giving explicit errors. In this paper, we show that the log-concavity is, in fact, strict for $k\geq 2$. We shed further light on this phenomenon by utilizing Hardy--Littlewood--Pólya's notion of majorizing. We prove that for partitions $\bm{a},\bm{b}$ of $n\in\N$, if $\bm b$ majorizes $\bm a$, then $p_k(\bm{b})>p_k(\bm{a})$. Numerical calculations indicate that our result is sharp.

math.NT↗