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Register a new userSingular principal bundles over higher dimensional manifolds and their moduli spaces
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Authors
Alexander Schmitt
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In this note, we introduce the notion of a singular principal G-bundle, associated to a reductive algebraic group G over the complex numbers by means of a faithful representation $varrho^pcolon Glra SL(V)$. This concept is meant to provide an analogon to the notion of a torsion free sheaf as a generalization of the notion of a vector bundle. We will construct moduli spaces for these singular principal bundles which compactify the moduli spaces of stable principal bundles.
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In this article, we solve the problem of constructing moduli spaces of semistable principal bundles (and singul
It is a longstanding problem in Algebraic Geometry to determine whether the syzygy bundle $E_{d_1,...,d_n}$ on $mathbb{P}^N$ defined as the kernel of a general epimorphism [phi:mathcal{O}(-d_1)oplus...oplusmathcal{O}(-d_n) tomathcal{O}] is (semi)stable. In this thesis, attention is restricted to the case of syzygy bundles $mathrm{Syz}(f_1,...,f_n)$ on $mathbb{P}^N$ associated to $n$ generic forms $f_1,...,f_nin K[X_0,...,X_N]$ of the same degree $d$, for ${Nge2}$. The first goal is to prove that $mathrm{Syz}(f_1,...,f_n)$ is stable if [N+1le nletbinom{d+N}{N},] except for the case ${(N,n,d)=(2,5,2)}$. The second is to study moduli spaces of stable rank ${n-1}$ vector bundles on $mathbb{P}^N$ containing syzygy bundles. In a joint work with Laura Costa and Rosa Mar{i}a Miro-Roig, we prove that $N$, $d$ and $n$ are as above, then the syzygy bundle $mathrm{Syz}(f_1,...,f_n)$ is unobstructed and it belongs to a generically smooth irreducible component of dimension ${ntbinom{d+N}{N}-n^2}$, if ${Nge3}$, and ${ntbinom{d+2}{2}+ntbinom{d-1}{2}-n^2}$, if ${N=2}$. The results in chapter 3, for $Nge3$, were obtained independently by Iustin Coandu{a} in arXiv:0909.4435.
In this paper, we obtain parametrizations of the moduli space of principal bundles over a compact Riemann surface using spaces of Hecke modifications in several cases. We begin with a discussion of Hecke modifications for principal bundles and give constructions of universal Hecke modifications of a fixed bundle of fixed type. This is followed by an overview of the construction of the wonderful, or De Concini--Procesi, compactification of a semi-simple algebraic group of adjoint type. The compactification plays an important role in the deformation theory used in constructing the parametrizations. A general outline to construct parametrizations is given and verifications for specific structure groups are carried out.
Studying degenerations of moduli spaces of semistable principal bundles on smooth curves leads to the problem of constructing and studying moduli spaces on singular curves. In this note, we will see that the moduli spaces of $delta$-semistable pseudo bundles on a nodal curve constructed by the first author become, for large values of $delta$, the moduli spaces for semistable singular principal bundles. The latter are reasonable candidates for degenerations and a potential basis of further developments as on irreducible nodal curves. In particular, we find a notion of semistability for principal bundles on reducible nodal curves. The understanding of the asymptotic behavior of $delta$-semistability rests on a lemma from geometric invariant theory. The results will allow the construction of a universal moduli space of semistable singular principal bundles relative to the moduli space $overline{mathcal M}_g$ of stable curves of genus $g$.
Let $X$ be a smooth projective curve over the complex numbers. To every representation $rhocolon GL(r)lra GL(V)$ of the complex general linear group on the finite dimensional complex vector space $V$ which satisfies the assumption that there be an integer $alpha$ with $rho(z id_{C^r})=z^alpha id_V$ for all $zinC^*$ we associate the problem of classifying triples $(E,L,phi)$ where $E$ is a vector bundle of rank $r$ on $X$, $L$ is a line bundle on $X$, and $phicolon E_rholra L$ is a non trivial homomorphism. Here, $E_rho$ is the vector bundle of rank $dim V$ associated to $E$ via $rho$. If we take, for example, the standard representation of $GL(r)$ on $C^r$ we have to classify triples $(E,L,phi)$ consisting of $E$ as before and a non-zero homomorphism $phicolon Elra L$ which includes the so-called Bradlow pairs. For the representation of $GL(r)$ on $S^2C^3$ we find the conic bundles of Gomez and Sols. In the present paper, we will formulate a general semistability concept for the above triples which depends on a rational parameter $delta$ and establish the existence of moduli spaces of $delta$-(semi)stable triples of fixed topological type. The notion of semistability mimics the Hilbert-Mumford criterion for $SL(r)$ which is the main reason that such a general approach becomes feasible. In the known examples (the above, Higgs bundles, extension pairs, oriented framed bundles) we show how to recover the usual semistability concept. This process of simplification can also be formalized. Altogether, our results provide a unifying construction for the moduli spaces of most decorated vector bundle problems together with an automatism for finding the right notion of semistability and should therefore be of some interest.
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