Variations of hodge structures of rank three k -Higgs bundles and moduli spaces of holomorphic triples
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Variations of hodge structures of rank three k-Higgs bundles and moduli spaces of holomorphic triples Ronald A. Zúñiga-Rojas1 Received: 27 May 2019 / Accepted: 16 October 2020 © Springer Nature B.V. 2020
Abstract There is an isomorphism between the moduli spaces of σ -stable holomorphic triples and some of the critical submanifolds of the moduli space of k-Higgs bundles of rank three, whose elements (E, ϕ k ) correspond to variations of Hodge structure, VHS. There are special embeddings on the moduli spaces of k-Higgs bundles of rank three. The main objective here is to study the cohomology of the critical submanifolds of such moduli spaces, extending those embeddings to moduli spaces of holomorphic triples. Keywords Higgs bundles · Holomorphic triples · Moduli spaces · Variations of Hodge structure Mathematics Subject Classification Primary 14F45; Secondary 14D07 · 14H60
Introduction Consider a compact connected Riemann surface X of genus g ≥ 2. Algebraically, X is a complete irreducible non-singular curve over C. Let N = N (r , d) be the moduli space of polystable vector bundles of rank r and degree d over X . In this paper, we consider the coprime condition GCD(r , d) = 1, which ensures that polystable implies stable. This space has been widely worked by Atiyah and Bott [2], Desale and Ramanan [8], Earl and Kirwan [9], among other authors. Here, we consider it as the corresponding minimal critical submanifold of M(r , d), the moduli space of polystable Higgs bundles. A Higgs bundle over X is a pair (E, ϕ) where E → X is a holomorphic vector bundle and ϕ : E → E ⊗ K is an
Supported by Universidad de Costa Rica through Escuela de Matemática, specifically through CIMM (Centro de Investigaciones Matemáticas y Metamatemáticas), Project 820-B8-224. This work is partly based on the Ph.D. Project [29] called “Homotopy Groups of the Moduli Space of Higgs Bundles”, supported by FEDER through Programa Operacional Factores de Competitividade-COMPETE, and also supported by FCT (Fundação para a Ciência e a Tecnologia) through the Projects PTDC/MAT-GEO/0675/2012 and PEst-C/MAT/UI0144/2013 with Grant Reference SFRH/BD/51174/2010.
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Ronald A. Zúñiga-Rojas [email protected] Centro de Investigaciones Matemáticas y Metamatemáticas CIMM Escuela de Matemática, Universidad de Costa Rica, San José 11501, Costa Rica
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Geometriae Dedicata
endomorphism twisted by the cotangent bundle K = T ∗ X . Fixing rank r and degree d of the underlying vector bundle E, the isomorphism classes of polystable Higgs bundles are parametrized by a quasiprojective variety: the moduli space of polystable Higgs bundles M ps (r , d). Again, since GCD(r , d) = 1, polystability implies stability and then, the space M ps (r , d) = Mss (r , d) = Ms (r , d) becomes a smooth projective variety. These spaces were first worked by Hitchin [20] and Simpson [27]. Since then, they have been around for more than 30 years, and have been studied extensively by a number of authors: e.g. [7,15–19,28]. Higgs bundles are an interesting topic of research be
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