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Principal Bundles and Bundles with Parabolic Structure

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Principal Bundles and Bundles with Parabolic Structure Principal Bundles and Bundles with parabolic structure Arijit Dey January 22, 2013 Abstract In this thesis we prove HN reduction for HIggs G-bundles and nonemptyness of parabolic stable bundles Contents 1 Introduction 5 1.0.1 why and how Parabolic Bundle, little bit of history . 8 1.0.2 Balaji’swork ....................... 8 1.0.3 HintofΓ-Balaji ..................... 8 1.0.4 WorkDoneinshort ................... 8 1.1 Introductiontosecondproblem . 8 1.1.1 MainResults ....................... 12 2 Preliminaries 13 2.1 Higgsbundle ........................... 13 2.2 NonemptynessofParabolicModuli . 19 2.2.1 The category of bundles with parabolic structures . 19 2.2.2 The Kawamata Covering lemma ............ 21 2.2.3 The category of Γ–bundles ................ 22 2.2.4 OnlocaltypesofΓ–bundles . 23 2.3 Qin-Li’s proof for rank r case .................. 29 3 Harder-Narasimhan reduction for Higgs G bundle 30 3.1 ProofofexistenceofaHiggsH-Nreduction . 30 3.2 The Higgs structure on the adjoint bundle EG(g) ...... 35 3.3 Proof of the uniqueness of Higgs H-N reduction . 40 4 Nonnemptyness 45 4.0.1 Preliminaries ....................... 46 4.0.2 Proof of Theorem I ................... 53 4.0.3 ProofofTheoremII . 54 5 holonomy 57 2 6 Ramified G-bundle 61 6.1 H-N reduction for ramified G bundles over a smooth curve . 61 6.1.1 H-N reduction for [Γ,G,N]bundle ........... 63 6.1.2 Proof of H-N reduction for ramified G bundle . 64 3 Acknowledgments I would like to thank Prof. V. Balaji , for suggesting me this problem, for many discussions and for his encouragement. I am very grateful to him for introducing me to the study of moduli spaces of vector bundles. I consider myself very fortunate for having found his generous help. I would like to thank Prof. D. S. Nagaraj who introduced me to algebraic geometry. I would also like to thank Shamindra Kumar Ghosh, Paramita Das and T. Muthukumar for their generous help during my course work of Ph.D program. I have also benefited from discussions with R. Parthasarathi, A. Laddha, S. Nayak and many other people as well as from many seminars and courses at Institute of Mathematical Sciences and Chennai Mathematical Institute. I want to thank Institute of Mathematical Sciences for the fellowship that supported my graduate studies. Without their generous support, this thesis couldn’t have been made. Lastly I would like to thank members of IMSc for giving me a very friendly atmosphere during this work. 4 Chapter 1 Introduction In this chapter we will explain the main results of the thesis using as little mathematical background as possible. We will always work over field of complex numbers. Also we assume that varieties are projective, i.e. there is n an embedding in PC. Moduli spaces are fundamental objects of algebraic geometry which have been at the center of interest since the beginnings of the subject. In a nut- shell, moduli spaces are spaces whose points correspond in a natural way with geometric objects that one would like to study, such as Riemann sur- faces, or vector bundles on a fixed space. Understanding the geometry of moduli spaces often helps to understand the underlying geometric objects themselves. For example in [8] by exploiting moduli space of µ-stable bun- dles on a smooth projective surface Donaldson was able to prove some strik- ing new results on smooth classification of certain projective surfaces, for example there is only one differential structure on Rn except when n = 4, in which case there are uncountably many. Moduli are, however, not only of interest to the algebraic geometer. They also play a role in other subjects, like in theoretical Physics. In this thesis we are interested with holomorphic parabolic vector bun- dles (E,D,α∗,r∗) (often denoted as (E∗) if there is no scope of confusion) with normal crossing parabolic divisor D over X, which is a vector bundle E together with a weighted (α∗) flag structure (r∗) on fibres of E|D (for precise definition of parabolic bundle see chapter 2). To distinguish non-isomorphic parabolic vector bundles of fixed quasi-parabolic structure and parabolic weights, we can define certain invariants called parabolic Chern classes. 2i These are cohomology classes parci(E) ∈ H (X, R), 1 ≤ i ≤ dimC X. We have parci(E)=0if i > rank(E). Even after fixing these discrete invari- 5 ants, we can have continuous families of non-isomorphic parabolic vector bundles. I.e., to specify the isomorphism class of a parabolic vector bundle it is not enough to fix some discrete invariants, but we have to fix also some continuous parameters. For fixed parabolic structure and parabolic Chern classes we would like to define a variety called the moduli space of parabolic vector bundles. Each point of this variety will correspond to a different (at least non-isomorphic) vector bundle. In this thesis first we will study the nonempty-ness of this space for rank 2 parabolic vector bundles over any surfaces and using this we can answer nonempty-ness of higher rank Moduli space of certain type of quasi-parabolic structure. Unfortunately, in order to construct this moduli space we have to restrict our attention to parabolic stable vector bundles (this is not a very strong restriction, since it can be proved that in some sense all parabolic vector bundles can be constructed starting from parabolic stable ones). There are different notions of stability (see 2). Here we will only discuss parabolic Mumford stability (also called parabolic-slope stability). Let X be a projec- tive variety and H an ample divisor. For a parabolic vector bundle E∗ one can define par-degree(E∗)(see chapter 2). We define the parabolic slope of E∗ with respect to H as: par-degree(E ) parµ (E )= ∗ H ∗ rank(E) (the product is the intersection product or cup product in cohomology) E∗ ′ is called parabolic H-stable if for every parabolic subbundle E∗ of E∗ we have ′ parµH (E∗) < parµH (E∗). 0 It can be proved that there is a space PMH (r,D,α∗,pci), called the moduli space of H-stable parabolic vector bundles of rank r and parabolic Chern classes pci (if the quasi-parabolic structure is clear from the context we will drop it from the notation). In many situations it will be a quasi-projective 0 variety (maybe with singularities). In general PMH (r,D,α∗,pci) is not compact but one can compactify this variety in such a way that boundary points correponds to polystable parabolic bundles together with a 0 cycle in X. This compactification is known as Donaldson-Uhlenbeck compacti- fication of parabolic bundles (for details see [4]). Let (X, Θ) be a smooth polarized algebraic surface, this gives rise to a underlying Riemannian mani- fold with metric d. There is an important relationship between gauge theory or Yang-Mills theory and stable holomorphic vector bundles; there is a bi- jection between Θ-stable SU(n) bundles on (X, Θ) and Hermite-Einstein or 6 Anti Self Dual (ASD)1 on a fixed differentiable SU(n) vector bundle over (X,d). This correspondence is known as Hitchin-Kobayashi correspondence. Infact more precisely both these spaces gets a canonical algebraic structure and their underlying topological spacesare homeomorphic. Note that the metric of the manifold appears in the ASD-equation. This is reflected (for bundles) that the stability condition of a bundle depends on the chosen polarization Θ. This relationship has been used, for instance, to calculate Donaldson polynomials for the study of the topology of 4-manifolds [8]. This polynomial denoted as γc2 (X) is of the form : γc (X)(α)= µ(α) ∧··· µα 2 Z Mc2 d 2 2 | {z } where µ : H (X) → H (Mc2 and d = dim(Mc2 )/2 by exploiting the geometry of Mc2 Donaldson was able to prove some strik- ing new results on smooth classification of projective surfaces. By exploiting algebro-geometric description of Mc2 he showed that γc2 (X) is nonzero for large c2. Hence asymptotic nonemptyness is good enough to compute Don- aldson polynomials. While studying the problem of embedding a sphere in a 4-fold Kron- heimer and Mrowka found an analogue of Donaldson polynomial for pairs (X, Σ), where Σ is a surface. They found that obstruction to embedding a sphere into a 4-fold is nonvanishing of this polynomial qk,l. They defined α-twisted anti self dual connections which are actually ASD connection in α X − Σ. let Mk,l be the moduli space of equivalence class of α-twisted connections where α ∈ (0, 1/2) is the holonomy parameter, k is the usual second chern class and l denote the instanton number. In [6] they conjec- tured Hitchin-Kobayashi correspondence i.e. there is a bijection between α-twisted ASD connection and rank 2 stable parabolic bundles (E, L,α) on (X,σ). In their paper they proved this conjecture for some simple case like ruled surfaces and later it was proved in greater generality. for a suitable integer n and is nonzero for large values of c2. In this thesis we prove nonemptiness of rank-2 parabolic stable for ar- bitary c1 and large c2 with suitable bound on α2. The bound on c2 and α2 depends on polarisation. We beleive one should be able to prove Gieseker- Taubes kind of theorem for parabolic bundles for generic choice of parabolic datum, but due to lack of knowledge/patience we could not get such nice theorem. In future we would like to investigate this problem. 1Itoh [11] showed that for SU(n) bundles Hermite-Einstein connection is same as ASD connection 7 1.0.1 why and how Parabolic Bundle, little bit of history 1.0.2 Balaji’s work 1.0.3 Hint of Γ-Balaji 1.0.4 Work Done in short The main strategy used is the categorical correspondence of the category of Γ–bundles of fixed type τ on a certain Kawamata cover Y of the surface X with the category of parabolic bundles on X with fixed parabolic datum (see chapter ?? for definitions and terminology).
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