Related papers: Brill-Noether-general Limit Root Bundles: Absence …

The study of vector-like spectra in 4-dimensional F-theory compactifications involves root bundles, which are important for understanding the Quadrillion F-theory Standard Models (F-theory QSMs) and their potential implications in physics.…

Motivated by the appearance of fractional powers of line bundles in studies of vector-like spectra in 4d F-theory compactifications, we analyze the structure and origin of these bundles. Fractional powers of line bundles are also known as…

In the largest, currently known, class of one Quadrillion globally consistent F-theory Standard Models with gauge coupling unification and no chiral exotics, the vector-like spectra are counted by cohomologies of root bundles. In this work,…

Much of the analysis of F-theory-based Standard Models boils down to computing cohomologies of line bundles on matter curves. By varying parameters one can degenerate such matter curves to singular ones, typically with many nodes, where the…

Motivated by engineering vector-like (Higgs) pairs in the spectrum of 4d F-theory compactifications, we combine machine learning and algebraic geometry techniques to analyze line bundle cohomologies on families of holomorphic curves. To…

Let C be a smooth projective curve over the field of the complex numbers. We consider Brill-Noether loci over the moduli of maps from C to the Grassmannian G(m,n) and the corresponding Quot schemes of quotients of a trivial vector bundle on…

How likely is it that there are particles in a vector-like pair of representations in low-energy spectrum, when neither symmetry nor anomaly consideration motivates their presence? We address this question in the context of supersymmetric…

F-bundle is a formal/non-archimedean version of variation of nc-Hodge structures which plays a crucial role in the theory of atoms as birational invariants from Gromov-Witten theory. In this paper, we establish the spectral decomposition…

F-theory admits 7-branes with exceptional gauge symmetries, which can be compactified to give phenomenological four-dimensional GUT models. Here we study general supersymmetric compactifications of eight-dimensional Yang-Mills theory. They…

Let $M$ be a complex projective manifold of dimension $n+1$ and $f$ a meromorphic function on $M$ obtained by a generic pencil of hyperplane sections of $M$. The $n$-th cohomology vector bundle of $f_0=f|_{M-\RR}$, where $\RR$ is the set of…

Complex supermanifold structures being deformations of the exterior algebra of a holomorphic vector bundle, have been parametrized by orbits of a group on non-abelian cohomology by P. Green. For the case of odd dimension $4$ and $5$ an…

Here we study vector bundles $E$ on the Hirzebruch surface $F_e$ such that their twists by a spanned, but not ample, line bundle $M = \mathcal {O}_{F_e}(h+ef)$ have natural cohomology, i.e. $h^0(F_e,E(tM)) >0$ implies $h^1(F_e,E(tM)) = 0$.

In this paper we generalise the theory of real vector bundles to a certain class of non-Hausdorff manifolds. In particular, it is shown that every vector bundle fibred over these non-Hausdorff manifolds can be constructed as a colimit of…

In this paper, we study cohomology groups of vector bundles on neighborhoods of a non-pluriharmonic locus in Stein manifolds and in projective manifolds. By using our results, we show variants of the Lefschetz hyperplane theorem.

We analyze exotic matter representations that arise on singular seven-brane configurations in F-theory. We develop a general framework for analyzing such representations, and work out explicit descriptions for models with matter in the…

Tautological bundles of realizations of matroids were introduced in [BEST23] as a unifying geometric model for studying matroids. We compute the cohomologies of exterior and symmetric powers of these vector bundles, and show that they…

For a finite dimensional vector space V of dimension n, we consider the incidence correspondence (or partial flag variety) X in P(V) x P(V*), parametrizing pairs consisting of a point and a hyperplane containing it. We completely…

Using an explicit resolution of the diagonal for the variety V_5, we provide cohomological characterizations of the universal and quotient bundles. A splitting criterion for bundles over V_5 is also proved. The presentation of semistable…

In \cite{Broer1993}, it was shown that certain line bundles on $\widetilde{\mathcal{N}}=T^*G/B$ have vanishing higher cohomology. We prove a generalization of this theorem for real reductive algebraic groups. More specifically, if…

Scheme-theoretic methods are used to classify ternary quadratic forms with values in line bundles over arbitrary schemes and to canonically determine the isomorphisms between them. The association of a quadratic bundle to its even Clifford…