An analogue of the Narasimhan-Seshadri theorem and some applications

Seshadri constants and Grassmann bundles over curves

Let $X$ be a smooth complex projective curve, and let $E$ be a vector bundle on $X$ which is not semistable. For a suitably chosen integer $r$, let $\text{Gr}(E)$ be the Grassmann bundle over $X$ that parametrizes the quotients of the fibers of $E$ of dimension $r$. Assuming some numerical conditions on the Harder-Narasimhan filtration of $E$, we study Seshadri constants of ample line bundles on $\text{Gr}(E)$. In many cases, we give the precise value of Seshadri constant. Our results generalize various known results for ${\rm rank}(E)=2$.

某些旗丛上的 Seshadri 常数

Seshadri constants on some flag bundles

本文研究非半稳定向量丛所诱导旗丛上的几何正性，利用 Harder–Narasimhan 滤过确定旗的商维数，并计算其中 nef 线丛的 Seshadri 常数。

Holonomy groups of stable vector bundles

We define the notion of holonomy group for a stable vector bundle F on a variety in terms of the Narasimhan--Seshadri unitary representation of its restriction to curves. Next we relate the holonomy group to the minimal structure group and to the decomposition of tensor powers of F. Finally we illustrate the principle that either the holonomy is large or there is a clear geometric reason why it should be small.

Desingularizations of the moduli space of rank 2 bundles over a curve

Let $X$ be a smooth projective curve of genus $g\ge 3$ and $M_0$ be the moduli space of rank 2 semistable bundles over $X$ with trivial determinant. There are three desingularizations of this singular moduli space constructed by Narasimhan-Ramanan \cite{NR}, Seshadri \cite{Se1} and Kirwan \cite{k5} respectively. The relationship between them has not been understood so far. The purpose of this paper is to show that there is a morphism from Kirwan's desingularization to Seshadri's, which turns out to be the composition of two blow-downs. In doing so, we will show that the singularities of $M_0$ …

Boundedness of semistable principal bundles on a curve, with classical semisimple structure groups

In characteristic zero, semistable principal bundles on a nonsingular projective curve with a semisimple structure group form a bounded family, as shown by Ramanathan in 1970's using the Narasimhan-Seshadri theorem. This was the first step in his construction of moduli for principal bundles. In this paper we prove boundedness in finite characteristics (other than characteristic 2), when the structure group is a semisimple, simply connected algebraic group of classical type. The main ingredient is an analogue of the Mukai-Sakai theorem (which says that any vector bundle admits a proper subbundl…

Harder-Narasimhan polygons and Laws of Large Numbers

We build on the recent techniques of Codogni and Patakfalvi, from \cite{Codogni:Patakfalvi:2021}, which were used to establish theorems about semi-positivity of the Chow Mumford line bundles for families of $\K$-semistable Fano varieties. Here we apply the Central Limit Theorem to ascertain the asymptotic probabilistic nature of the vertices of the \emph{Harder and Narasimhan polygons}. As an application of our main result, we use it to establish a filtered vector space analogue of the main technical result of \cite{Codogni:Patakfalvi:2021}. In doing so, we expand upon the slope stability theo…

Principal bundles on metric graphs: the $\mathrm{GL}_n$ case

Using the notion of a root datum of a reductive group $G$ we propose a tropical analogue of a principal $G$-bundle on a metric graph. We focus on the case $G=\mathrm{GL}_n$, i.e. the case of vector bundles. Here we give a characterization of vector bundles in terms of multidivisors and use this description to prove analogues of the Weil--Riemann--Roch theorem and the Narasimhan--Seshadri correspondence. We proceed by studying the process of tropicalization. In particular, we show that the non-Archimedean skeleton of the moduli space of semistable vector bundles on a Tate curve is isomorphic to…

关于主丛模空间的非奇异化

On the desingularisation of moduli of principal bundles

本文针对特定单连通几乎单代数群，构造了稳定主丛模空间的光滑模化紧化，从而给出了半稳定主丛模空间的标准非奇异化，推广了Seshadri的构造。

Goldman form, flat connections and stable vector bundles

We consider the moduli space $\mathscr{N}$ of stable vector bundles of degree $0$ over a compact Riemann surface and the affine bundle $\mathscr{A}\to\mathscr{N}$ of flat connections. Following the similarity between the Teichmüller spaces and the moduli of bundles, we introduce the analogue of of the quasi-Fuchsian projective connections - local holomorphic sections of $\mathscr{A}$ - that allow to pull back the Liouville symplectic form on $T^{*}\mathscr{N}$ to $\mathscr{A}$. We prove that the pullback of the Goldman form to $\mathscr{A}$ by the Riemann-Hilbert correspondence coincides with …

Frobenius pull backs of vector bundles in higher dimensions

Here we prove that for a smooth projective variety $X$ of arbitrary dimension and for a vector bundle $E$ over $X$, the Harder-Narasimhan filtration of a Frobenius pull back of $E$ is a refinement of the Frobenius pull-back of the Harder-Narasimhan filtration of $E$, provided there is a lower bound on the characteristic $p$ (in terms of rank of $E$ and the slope of the destabilising sheaf of the cotangent bundle of $X$). We also recall some examples, due to Raynaud and Monsky,to show that some lower bound on $p$ is necessary. We further prove an analogue of this result for principal $G$-bundle…

Parallel transport and the p-adic Simpson correspondence

Deninger and Werner developed an analogue for p-adic curves of the classical correspondence of Narasimhan and Seshadri between stable bundles of degree zero and unitary representations of the topological fundamental group for a complex smooth proper curve. Using parallel transport, they associated functorially to every vector bundle on a p-adic curve whose reduction is strongly semi-stable of degree 0 a p-adic representation of the fundamental group of the curve. They asked several questions: whether their functor is fully faithful; whether the cohomology of the local systems produced by this …

Degenerations of Bundle Moduli

Over a family $\mathbb X$ of genus $g$ complete curves, which gives the degeneration of a smooth curve into one with nodal singularities, we build a moduli space which is the moduli space of ${\rm SL}(n, \mathbb C)$ bundles over the generic smooth curve $X_t$ in the family, and is a moduli space of bundles equipped with extra structure at the nodes for the nodal curves in the family. This moduli space is a quotient by $(\mathbb C^*)^s$ of a moduli space on the desingularisation. Taking a "maximal" degeneration of the curve into a nodal curve built from the glueing of three-pointed spheres, we …

Logarithmic connections, WZNW action, and moduli of parabolic bundles on the sphere

Moduli spaces of stable parabolic bundles of parabolic degree $0$ over the Riemann sphere are stratified according to the Harder--Narasimhan filtration of underlying vector bundles. Over a Zariski open subset $\mathscr{N}_{0}$ of the open stratum depending explicitly on a choice of parabolic weights, a real-valued function $\mathscr{S}$ is defined as the regularized critical value of the non-compact Wess--Zumino--Novikov--Witten action functional. The definition of $\mathscr{S}$ depends on a suitable notion of parabolic bundle `uniformization map' following from the Mehta--Seshadri and Birkhof…

A foliated Hitchin-Kobayashi correspondence

We prove an analogue of the Hitchin-Kobayashi correspondence for compact, oriented, taut Riemannian foliated manifolds with transverse Hermitian structure. In particular, our Hitchin-Kobayashi theorem holds on any compact Sasakian manifold. We define the notion of stability for foliated Hermitian vector bundles with transverse holomorphic structure and prove that such bundles admit a basic Hermitian-Einstein connection if and only if they are polystable. Our proof is obtained by adapting the proof by Uhlenbeck and Yau to the foliated setting. We relate the transverse Hermitian-Einstein equatio…

关于$GL(1|1)$Higgs丛的模空间

On $GL(1|1)$ Higgs bundles

本文研究了$GL(1|1)$Higgs丛的模空间，推导了其显式描述，并探讨了Narasimhan-Seshadri定理的超几何类比及Hitchin方程的解法。