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 Title: Holomorphic chains on the projective line Author(s): To, Jin Hyung Director of Research: Bradlow, Steven B. Doctoral Committee Chair(s): Haboush, William J. Doctoral Committee Member(s): Bradlow, Steven B.; Schenck, Henry K.; Nevins, Thomas A. Department / Program: Mathematics Discipline: Mathematics Degree Granting Institution: University of Illinois at Urbana-Champaign Degree: Ph.D. Genre: Dissertation Subject(s): Holomorphic chains α-stability Chamber Geometric Invariant Theory (GIT) Nonreductive GIT Symplectic quotient Co-Higgs bundles. Abstract: Holomorphic chains on a smooth algebraic curve are tuples of vector bundles on the curve together with the homomorphisms between them. A type of a holomorphic chain is a tuple t of integers consisting of the ranks and degrees of the underlying vector bundles. The definition of holomorphic chains was introduced and their stability were defined by L. ´Alvarez-C´onsul and O. Garc´ıa-Prada. The moduli spaces of holomorphic chains were constructed using the Geometric Invariant Theory (GIT) by A. H. W. Schmitt. They studied holomorphic chains on a smooth algebraic curve of genus g ≥ 2. In general it is difficult to describe moduli spaces. The stability of holomorphic chains depends on a real vector parameter in R^n called α-stability (α ∈ R^n). The region R^s(t) (R(t)) is the set of α for which there exists an α-(semi)stable holomorphic chain of type t. Thus the moduli spaces depend on the parameter. A rational vector parameter corresponds to a given linearization of the GIT quotient. R^n is partitioned into locally closed subsets called chambers where the stability does not change. The moduli spaces in each chamber do not change. The case in which the underlying bundles are all line bundles is simple. The chamber structure for this case is classified. Line bundles on a smooth algebraic curve can be parameterized by a Poincar´e line bundle. Considering this, holomorphic chains composed of line bundles with fixed degrees are parameterized by the direct sum of vector bundles when the gaps of degrees between consecutive line bundles are sufficiently large, and the moduli space is identified as the product of the corresponding projective space bundles. Finally, the automorphism group of the holomorphic chains composed of line bundles is (C^∗)^n. The variation of α relates to the variation of the GIT quotient for the action of (C^∗)^n. The next step is to look at holomorphic chains on P^1. Vector bundles on P^1 are splitting, by A. Grothendieck. The first case to look at is the holomorphic chains of type t = (1, 2; 0,−s). The chamber structure for holomorphic chains of type t = (1, 2; 0,−s) is identified and that for its dual chain of type t = (2, 1; s, 0) is the same as for those of type t^∨ = (1, 2; 0,−s). The chamber is determined by partitioning R. Moreover, the moduli spaces of type t = (1, 2; 0,−s) can be identified as those of type t^∨ = (1, 2; 0,−s) by sending a chain to its dual chain. The stability region R^s(t) is a bounded open interval which is partitioned into subinterval chambers. It is relatively easy to describe the moduli spaces corresponding to the leftmost and rightmost intervals. They are Grassmannian varieties for s even and projective spaces, respectively. The same results are found in different guises. A coherent system on a smooth algebraic curve is the pair of a vector bundle and a vector subspace of holomorphic sections. A coherent system can be described as a holomorphic chain. The isomorphism class of a coherent system of type (2, s, 1) can be identified as the isomorphism class of the associated holomorphic chain of type t = (2, 1; s, 0). A holomorphic pair is the pair of a vector bundle and a holomorphic section of it. A holomorphic pair can be described as a holomorphic chain. The isomorphism class of a holomorphic pair of rank two can be identified as the isomorphism class of the associated holomorphic chain of type t = (2, 1, ; s, 0). Moreover, their stabilities coincide. The stabilities for coherent systems and holomorphic pairs involve real parameters. The parameter α relates to these parameters. P. E. Newstead and H. Lange studied coherent systems on P^1. M. Thaddeus studied holomorphic pairs of rank 2 with a fixed determinant on a smooth algebraic curve of genus g ≥ 2. His description of the moduli spaces are applicable for any genus. On P^1, if the degree of a vector bundle is fixed, then its determinant is automatically fixed. The moduli spaces of the holomorphic chains of type t = (2, 1, ; s, 0) on P1 can be identified as those of the associated coherent systems and holomorphic pairs. The chamber structure for the holomorphic chains of type t = (2, 1, 2; d_0, d_1, d_2) is identified. The stability region R^s(t) is an open subset of R^2 bounded by a parallelogram. The stability region is partitioned into sub-parallelogram chambers. Each edge of the parallelogram has a nonzero slope. Analogous to the leftmost and rightmost interval chambers are bottommost and topmost ones. The corresponding moduli spaces are the product of two Grassimannian varieties for d_0 and d_2 even, and the product of two projective spaces. A co-Higgs bundle on a smooth algebraic curve is a vector bundle with a Higgs field. A Higgs field is a holomorphic section of the tensor product of the endomorphism bundle of the vector bundle and the dual of the canonical line bundle. A co-Higgs bundle can be described as a holomorphic chain. If two co-Higgs bundles are isomorphic then the associated holomorphic chains are isomorphic. On a smooth algebraic curve, interesting co-Higgs bundle are found only on P^1. S. Rayan classified stable co-Higgs bundles on P1. He characterized the moduli space of stable co-Higgs bundles of rank 2 and degree odd as a universal elliptic curve with a globally defined equation. The stability of co-Higgs bundles of rank 2 and degree odd is compared with the α-stability of the associated holomorphic chains. The α-stable holomorphic chains associated with co-Higgs bundles of rank 2 and degree −1 are classified. A co-Higgs bundle of rank 2 and degree −1 is stable if and only if the associated holomorphic chain is 3-semistable. A co-Higgs bundle of rank 2 and degree −1 is stable if the associated holomorphic chain is α-stable for α > 3. The moduli spaces of holomorphic chains of type t = (1, 2; 0,−s) with s > 2 on P1 have natural subspaces with fixed underlying bundles. The underlying bundles are determined by splitting types denoted by (0, (−d,−e)) with s = d + e. The automorphism group of a chain is non-reductive unless the underlying bundles are semistable. In the non-reductive case, Dr´ezet-Trautmann’s non-reductive GIT method applies to the subspaces. Their method involves a tuple of rational parameters Λ = (λ_1, λ_2, μ_1) called a polarization. Given a splitting type (0, (−d,−e)) with 1 ≤ e < d ≤ ⌈s/2⌉ and d + e = s, if d ̸= e, a chain (O,O(−d) ⊕ O(−e); ϕ) is α-stable for some α if and only if the map ϕ is stable with respect to some polarization Λ. The subspace of the fixed splitting type (0, (−d,−e)) can be identified as Dr´ezet and Trautmann’s non-reductive GIT quotient for some polarization. The parameter α relates to the polarization Λ. For the dual type t^∨ = (2, 1; s, 0), Doran and Kirwn’s non-reductive GIT method applies to the subspaces. Their non-reductive GIT quotients involve a rational parameter δ. Given a splitting type ((d, e), 0) with 1 ≤ e < d ≤ ⌈s/2⌉ and d + e = s, if d − e > 1, then a chain (O(d) ⊕ O(e),O, ; ϕ) is α-stable for some α if and only if the map ϕ is δ-stable for some δ. The subspace of the fixed splitting type ((d, e), 0) can be identified as Doran and Kirwan’s non-reductive GIT quotient for some δ. In the paper it is explained how the parameter α relates to the parameter δ. Moreover, by a symplectic description, Doran and Kirwan’s non-reductive GIT quotient is a P^e bundle over P^{e−1}. The subspace of fixed splitting type (0, (−d,−e)) can be identified as the subspace of fixed splitting type ((d, e), 0) by mapping dual chains. If d − e > 1 then Dr´ezet-Trautmann’s non-reductive GIT quotient is identical to Doran and Kirwan’s non-reductive GIT quotient. It is explained how the polarization Λ relates to the parameter δ. Issue Date: 2012-05-22 URI: http://hdl.handle.net/2142/31129 Rights Information: Copyright 2012 Jin Hyung To Date Available in IDEALS: 2012-05-22 Date Deposited: 2012-05
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