Vector Bundle construction via Monads on multiprojective Spaces
Damian Maingi
TL;DR
The paper develops monad-based constructions of indecomposable low-rank vector bundles on multiprojective varieties, addressing $X=\mathbb{P}^{2n+1}\times\cdots\times\mathbb{P}^{2n+1}$ and more generally $X=\mathbb{P}^{a_1}\times\cdots\times\mathbb{P}^{a_n}$. It introduces Type I monads on the Segre-polarized product and proves the kernel bundle $T$ is $\mathscr{L}$-stable while the cohomology bundle $E$ is simple, generalizing Schwarzenberger/instanton constructions; it then extends the framework to Type II monads on arbitrary multiprojective spaces with a given polarization, yielding stable kernels $F$ and simple cohomology bundles $E$ of rank $2n$. Finally, it provides an explicit morphism-based construction on $(\mathbb{P}^1)^m$ via Segre embeddings, producing concrete monads with matrices of multidegree-one monomials. Collectively, the results broaden the catalog of stable, simple vector bundles on multiprojective spaces and enhance monad techniques for producing new examples and understanding their moduli.
Abstract
In this paper we establish the existence of monads on multiprojective spaces $X=\mathbb{P}^{2n+1}\times\mathbb{P}^{2n+1}\times\cdots\times\mathbb{P}^{2n+1}$. We prove stability of the kernel bundle which is a dual of a generalized Schwarzenberger bundle associated to the monads and prove that the cohomology vector bundle is simple, a generalization of instanton bundles. Next we construct monads on $\mathbb{P}^{a_1}\times\cdots\times\mathbb{P}^{a_n}$ and prove stability of the kernel bundle and that the cohomology vector bundle is simple. Lastly, we construct the morphisms that establish the existence of monads on $\mathbb{P}^1\times\cdots\times\mathbb{P}^1$.