arXiv subjects
Publications and source records attributed to Soham Mondal.
Let $A$ be a complex abelian variety of dimension $g\ge 2$ with $\NS(A)\cong\ZZ$ generated by an ample class $L$. We classify, up to twist by powers of $L$, the rank-two arithmetically Cohen--Macaulay bundles on $A$ with empty defect: they are sums of two ACM line bundles or nonsplit self-extensions of a nontrivial degree-zero line bundle. We also show none is Ulrich.
Let \((A, L)\) be a polarized abelian variety of dimension \(g \geq 1\) over an algebraically closed field of characteristic zero. We prove that every nontrivial line bundle \(P\) in the connected component \(\operatorname{Pic}^0(A)\) of the Picard variety is arithmetically Cohen--Macaulay (ACM) with respect to \(L\). For \(g \geq 2\) and any fixed nontrivial \(P \in \operatorname{Pic}^0(A)\), we construct by induction an infinite sequence of indecomposable ACM vector bundles \(E_r\) of every rank \(r \geq 1\). In addition, this paper studies classification questions for ACM line bundles and shows that, for abelian varieties of dimension at least two, the category of ACM bundles is of wild representation type. This paper settles the existence problem for nontrivial ACM bundles on polarized abelian varieties and supply large explicit families of indecomposable examples
Let $X$ be a smooth irreducible projective variety of dimension $n\ge 3$ over an algebraically closed field of characteristic zero, polarized by a very ample line bundle $\OO_X(1)$. Let $\E$ be an Ulrich bundle on $X$. We prove that there exists an explicitly computable integer $M\gg 0$ such that for every $m\ge M$ the global syzygy bundle $S_{\E(m)}$ is slope semistable with respect to $\OO_X(1)$. This confirms Conjecture~3.11 of Miró-Roig.
Let $X$ be a smooth projective scheme and $E$ a vector bundle on $X$. For a relative hypersurface $Y_f \subset \mathbb{P}(E)$ of degree $d$ defined by a global section $f$, we establish a functorial equivalence between the category of relatively Ulrich bundles on $Y_f$ and the category of representations of the associated generalized Clifford algebra $C_f$. This equivalence generalizes the classical Ulrich-Clifford correspondence of Coskun-Kulkarni-Mustopa and provides a purely algebraic framework that bypasses geometric obstructions in the relative setting. As a first application, we prove that relative hypersurfaces are Ulrich-wild: there exist families of indecomposable relatively Ulrich bundles $\{E_N\}$ with \[ \dim \mathrm{Ext}^1_{Y_f}(E_N, E_N) \to \infty \quad \text{as } N \to \infty. \] We further show that relative hyperplanes possess a minimal Ulrich complexity of one. Moving beyond degree one, we illustrate how unavoidable homological obstructions require complex machinery, such as matrix factorizations, equivalently generalized Clifford algebras, to find solutions.
We functorially identify similarity classes of line-bundle-valued quadratic forms on rank two vector bundles with isomorphism classes of pairs consisting of the degree zero and the degree one parts of the associated generalized Clifford algebras. As applications, we generalize the Gauss Composition and explore connections with Picard groups of quadratic algebras.