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Register a new userMultiple Bernoulli series and volumes of moduli spaces of flat bundles over surfaces
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Added by
Arzu Boysal
Publication date
2013
fields
Informatics Engineering
and research's language is
English
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Using Szenes formula for multiple Bernoulli series we explain how to compute Witten series associated to classical Lie algebras. Particular instances of these series compute volumes of moduli spaces of flat bundles over surfaces, and also certain multiple zeta values.
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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.
We generalize the construction of M. Lieblich for the compactification of the moduli stack of $PGL_r$-bundles on algebraic spaces to the moduli stack of Tanaka-Thomas $PGL_r$-Higgs bundles on algebraic schemes. The method we use is the moduli stack of Higgs version of Azumaya algebras. In the case of smooth surfaces, we obtain a virtual fundamental class on the moduli stack of $PGL_r$-Higgs bundles. An application to the Vafa-Witten invariants is discussed.
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.
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.
In a previous article the second author together with A. Pasquale determined the spectrum of the $Cos^lambda$ transform on smooth functions on the Grassmann manifolds $G_{p,n+1}$. This article extends those results to line bundles over certain Grassmannians. In particular we define the $Cos^lambda$ transform on smooth sections of homogeneous line bundles over$G_{p,n+1}$ and show that it is an intertwining operator between generalized ($chi$-spherical) principal series representations induced from a maximal parabolic subgroup of $mathrm{SL} (n+1, mathbb{K})$. Then we use the spectrum generating method to determine the $K$-spectrum of the $Cos^lambda$ transform.
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