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2015 | OriginalPaper | Chapter

Introduction to Nonabelian Hodge Theory

Flat connections, Higgs bundles and complex variations of Hodge structure

Authors: Alberto García-Raboso, Steven Rayan

Published in: Calabi-Yau Varieties: Arithmetic, Geometry and Physics

Publisher: Springer New York

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Abstract

Hodge theory bridges the topological, smooth and holomorphic worlds. In the abelian case of the preceding chapter, these are embodied by the Betti, de Rham and Dolbeault cohomology groups, respectively, of a smooth compact Kähler manifold, X.
Footnotes
1
The appropriate stability condition in this setting is slope stability; for holomorphic vector bundles, this is the same as Definition 6 with ϕ = 0.
 
2
The reader unfamiliar with the language of locally-ringed spaces should think of the pair \((\mathfrak{X},\mathcal{O})\) as selecting a geometric context—the examples below should make this clear. “A locally-free sheaf of coherent \(\mathcal{O}\)-modules on \(\mathfrak{X}\)” is then just a way of saying “a vector bundle of finite rank” in that context.
 
3
In the smooth setting, all of our sheaves are fine, hence acyclic, and this extra complication does not show up; some care must be exercised in the holomorphic category though.
 
4
A \(\mathbb{C}\)-linear dg-category is a \(\mathbb{C}\)-linear additive category in which morphisms assemble into complexes of \(\mathbb{C}\)-vector spaces.
 
5
When we say vector bundle, we will always mean of finite rank.
 
6
We remind the reader that this is the notion of slope stability.
 
7
It is the fact that the category of Higgs bundles is abelian that allows us to talk about extensions.
 
8
The reader might have noticed that Condition 2 implies Condition 1. Condition 2\(^{{\prime}}\) is, however, independent of Condition 1, and the equivalence between Conditions 2 and 2 \(^{{\prime}}\) does not hold in the absence of Condition 1.
 
9
A certain overloading of the notation is difficult to avoid at this point without making it overly cumbersome. Hopefully the reader will be able to tell the differential \(\overline{\partial }\) from the connection \(\overline{\partial }\) from the context. The same will unfortunately happen with other symbols.
 
10
The prefix pseudo- reflects the fact that \(D_{K}^{{\prime\prime}}\) is not a connection, but rather a sum of two connections.
 
11
This is minus the dual of \(\theta\) in the sense of Definition 5.
 
12
The π 1 X-action on \(\mathsf{GL}_{r}(\mathbb{C})/\mathsf{U}(r)\) is that given by the monodromy representation of the flat bundle, (E, D); that on the universal covering space, \(\widetilde{X}\), of X comes from deck transformations.
 
13
We are purposefully avoiding here giving the definition of what extensions in a dg-category are. The interested reader can consult §3 of Simpson’s [67].
 
14
On a Riemann surface, the second Chern character of any bundle vanishes automatically.
 
15
The Morse index is essentially deformation theoretic. The Higgs field in a complex variation of Hodge structure shifts the sequence of \(\varLambda _{j}\) bundles by one step when it acts, i.e. \(\phi:\varLambda _{j} \rightarrow \varLambda _{j+1} \otimes \omega _{X}\). Roughly speaking, the downward flow arises algebraically by considering shifts of weight at least 2.
 
16
We were not able to find an appearance of this exact formula in the literature. In §7 of [35], the rank 2 case of this formula is derived. It is shown in [62] how to extract a formula for all ranks from Gothen’s calculation of the Morse index, in the case of co-called “co-Higgs bundles” on the projective line. Superficial modifications to the procedure will produce the formula presented above. See also [23] for a similar formula in the parabolic Higgs setting.
 
17
The \(\mathbb{C}^{\times }\)-action commutes with the action of Jac2 0(X) [32].
 
18
Hitchin’s g = 2 generating function in [35] for the \(\mathsf{SL}_{2}(\mathbb{C})\)-Higgs bundle moduli space is \(1 + y^{2} + 4y^{3} + 2y^{4} + 34y^{5} + 2y^{6}\), which is the same as the second factor of our presentation of \(\mathcal{P}_{y}(2, 1)\) save for a difference in the coefficient of y 5.
 
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Metadata
Title
Introduction to Nonabelian Hodge Theory
Authors
Alberto García-Raboso
Steven Rayan
Copyright Year
2015
Publisher
Springer New York
DOI
https://doi.org/10.1007/978-1-4939-2830-9_5

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