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Register a new userOn invariant rank two vector bundles on $mathbb{P}^2$
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In this paper we characterize the rank two vector bundles on $mathbb{P}^2$ which are invariant under the actions of the parabolic subgroups $G_p:=mathrm{Stab}_p(mathrm{PGL}(3))$ fixing a point in the projective plane, $G_L:=mathrm{Stab}_L(mathrm{PGL}(3))$ fixing a line, and when $pin L$, the Borel subgroup $mathbf{B} = G_p cap G_L$ of $mathrm{PGL}(3)$. Moreover, we prove that the geometrical configuration of the jumping locus induced by the invariance does not, on the other hand, characterize the invariance itself. Indeed, we find infinite families that are almost uniform but not almost homogeneous.
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In this article we study the Gieseker-Maruyama moduli spaces $mathcal{B}(e,n)$ of stable rank 2 algebraic vector bundles with Chern classes $c_1=ein{-1,0}, c_2=nge1$ on the projective space $mathbb{P}^3$. We construct two new infinite series $Sigma_0$ and $Sigma_1$ of irreducible components of the spaces $mathcal{B}(e,n)$, for $e=0$ and $e=-1$, respectively. General bundles of these components are obtained as cohomology sheaves of monads, the middle term of which is a rank 4 symplectic instanton bundle in case $e=0$, respectively, twisted symplectic bundle in case $e=-1$. We show that the series $Sigma_0$ contains components for all big enough values of $n$ (more precisely, at least for $nge146$). $Sigma_0$ yields the next example, after the series of instanton components, of an infinite series of components of $mathcal{B}(0,n)$ satisfying this property.
We study the problem of rationality of an infinite series of components, the so-called Ein components, of the Gieseker-Maruyama moduli space $M(e,n)$ of rank 2 stable vector bundles with the first Chern class $e=0$ or -1 and all possible values of the second Chern class $n$ on the projective 3-space. The generalized null correlation bundles constituting open dense subsets of these components are defined as cohomology bundles of monads whose members are direct sums of line bundles of degrees depending on nonnegative integers $a,b,c$, where $bge a$ and $c>a+b$. We show that, in the wide range when $c>2a+b-e, b>a, (e,a) e(0,0)$, the Ein components are rational, and in the remaining cases they are at least stably rational. As a consequence, the union of the spaces $M(e,n)$ over all $nge1$ contains an infinite series of rational components for both $e=0$ and $e=-1$. Explicit constructions of rationality of Ein components under the above conditions on $e,a,b,c$ and, respectively, of their stable rationality in the remaining cases, are given. In the case of rationality, we construct universal families of generalized null correlation bundles over certain open subsets of Ein components showing that these subsets are fine moduli spaces. As a by-product of our construction, for $c_1=0$ and $n$ even, they provide, perhaps the first known, examples of fine moduli spaces not satisfying the condition $n$ is odd, which is a usual sufficient condition for fineness.
We describe new components of the Gieseker--Maruyama moduli scheme $mathcal{M}(n)$ of semistable rank 2 sheaves $E$ on $mathbb{P}^3$ with $c_1(E)=0$, $c_2(E)=n$ and $c_3(E)=0$ whose generic point corresponds to non locally free sheaves. We show that such components grow in number as $n$ grows, and discuss how they intersect the instanton component. As an application, we prove that $mathcal{M}(2)$ is connected, and identify a connected subscheme of $mathcal{M}(3)$ consisting of 7 irreducible components.
We use Scholzes framework of diamonds to gain new insights in correspondences between $p$-adic vector bundles and local systems. Such correspondences arise in the context of $p$-adic Simpson theory in the case of vanishing Higgs fields. In the present paper we provide a detailed analysis of local systems on diamonds for the etale, pro-etale, and the $v$-topology, and study the structure sheaves for all three topologies in question. Applied to proper adic spaces of finite type over $mathbb{C}_p$ this enables us to prove a category equivalence between $mathbb{C}_p$-local systems with integral models, and modules under the $v$-structure sheaf which modulo each $p^n$ can be trivialized on a proper cover. The flexibility of the $v$-topology together with a descent result on integral models of local systems allows us to prove that the trivializability condition in the module category may be checked on any normal proper cover. This result leads to an extension of the parallel transport theory by Deninger and the second author to vector bundles with numerically flat reduction on a proper normal cover.
We develop a theory of etale parallel transport for vector bundles with numerically flat reduction on a $p$-adic variety. This construction is compatible with natural operations on vector bundles, Galois equivariant and functorial with respect to morphisms of varieties. In particular, it provides a continuous $p$-adic representation of the etale fundamental group for every vector bundle with numerically flat reduction. The results in the present paper generalize previous work by the authors on curves. They can be seen as a $p$-adic analog of higher-dimensional generalizations of the classical Narasimhan-Seshadri correspondence on complex varieties. Moreover, they provide new insights into Faltings $p$-adic Simpson correspondence between small Higgs bundles and small generalized representations by establishing a class of vector bundles with vanishing Higgs field giving rise to actual (not only generalized) representations.
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