On Hermitian-Holomorphic Classes Related to Uniformization, the Dilogarithm, and the Liouville Action

Aldrovandi, Ettore

doi:10.1007/s00220-004-1168-6

Cited by 6 publications

(18 citation statements)

References 34 publications

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“…For each morphism f : P → Q in G U a corresponding morphism f * : herm(P ) → herm(Q) of E 0 U,+ -torsors. 1 This map must be compatible with compositions of morphisms in G U and with the restriction functors. For an object P of G U , an automorphism ∈ Aut(P ) is identified with a section of O × X over U.…”

Section: Definition 521mentioning

confidence: 98%

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Hermitian-holomorphic (2)-gerbes and tame symbols

Aldrovandi

2005

Journal of Pure and Applied Algebra

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The tame symbol of two invertible holomorphic functions can be obtained by computing their cup product in Deligne cohomology, and it is geometrically interpreted as a holomorphic line bundle with connection. In a similar vein, certain higher tame symbols later considered by Brylinski and McLaughlin are geometrically interpreted as holomorphic gerbes and 2-gerbes with abelian band and a suitable connective structure.In this paper we observe that the line bundle associated to the tame symbol of two invertible holomorphic functions also carries a fairly canonical hermitian metric, hence it represents a class in a Hermitian holomorphic Deligne cohomology group.We put forward an alternative definition of hermitian holomorphic structure on a gerbe which is closer to the familiar one for line bundles and does not rely on an explicit "reduction of the structure group". Analogously to the case of holomorphic line bundles, a uniqueness property for the connective structure compatible with the hermitian-holomorphic structure on a gerbe is also proven. Similar results are proved for 2-gerbes as well.We then show the hermitian structures so defined propagate to a class of higher tame symbols previously considered by Brylinski and McLaughlin, which are thus found to carry corresponding hermitian-holomorphic structures. Therefore we obtain an alternative characterization for certain higher Hermitian holomorphic Deligne cohomology groups.

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Section: Definition 521mentioning

confidence: 98%

“…where E 1 (X) (1) are the global sections of E 1 X (1). Thus compatible connections are necessarily flat.…”

Section: Comparisonsmentioning

confidence: 99%

Hermitian-holomorphic (2)-gerbes and tame symbols

Aldrovandi

2005

Journal of Pure and Applied Algebra

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“…A remarkable fact is that the metrized line ( L, M , · ) can be obtained by entirely cohomological means, as a cup product in Hermitian-holomorphic Deligne cohomology between the classes of the metrized line bundlesL = (L, · ) andM = (M, · ), [1,7].…”

Section: Preliminaries and Statement Of The Resultsmentioning

confidence: 99%

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Aldrovandi¹

2005

Internat. Math. Res. Notices

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at University of Iowa Libraries/Serials Acquisitions on June 1, 2015 http://imrn.oxfordjournals.org/ Downloaded from 1016 Ettore Aldrovandi if Pic(X) denotes the group of isomorphism classes of line bundles on X equipped with a smooth Hermitian metric, it is shown that Pic(X) ∼ = H 2 D (X; 1). (1.1) The groups H • D (X; p) have cup products behaving in the standard way: H k D (X; p) ⊗ H l D (X; q) ∪ − − → H k+l D (X; p + q). (1.2) Thus, given the two metrized line bundles as above we can multiply their classes in Pic(X) using this product to obtain a class [L] ∪ [M] ∈ H 4 D (X; 2). If X is a curve, the latter group can be replaced by H 2 (X, R), which yields the sought-after pairing after evaluating the resulting class against the fundamental class of X. (See [1] and Section 4 below. We have neglected the Tate twists by 2π √ −1 here.) The number so obtained is the length of a generator of L, M associated to a specific choice of l and m, see Section 5.1 below. The case L = M = T X turns out to be quite interesting on its own and relevant to the hyperbolic geometry of X considered as a Riemann surface. A Hermitian fiber metric is now just a conformal metric on X. If the genus is greater than or equal to 2, we have shown in [1, Theorem 5.1] that the cup square [T X ] ∪ [T X ], supplemented by the area form, determines a functional whose extremum is precisely the hyperbolic metric of constant curvature equal to −1. In fact, we have shown it coincides with the Liouville functional studied in [16, 18, 19], in relation to Weil-Petersson geometry. Returning to the general case of the pairing L, M where L and M are not necessarily equal, it has been observed-also by Deligne [10]-that the metric on the pairing could be defined also when the fiber metrics on both L and M are allowed to be singular, provided the corresponding loci are disjoint. In a different direction, if X/ Z is an arithmetic surface such that X is the corresponding fiber at infinity, Kühn [14] has formulated a pairing (generalizing Deligne's one) for line bundles where the metric at the infinite places-the connected components of X-has logarithmic singularities at a finite set of points. (This is shown in the same work to be compatible with an earlier version by Bost, cf. [6], based on Green's currents.) This leads to the interesting question of whether pairings with singular metrics can be expressed in terms of a natural cup product, and if the results outlined above extend to singular metrics. We show this is indeed the case. at University of Iowa Libraries/Serials Acquisitions on June 1, 2015 http://imrn.oxfordjournals.org/ Downloaded from Hermitian-Holomorphic Deligne Cohomology 1017In this work, we first present an extension of the framework of Hermitianholomorphic Deligne cohomology to pairs (X, D), where X is a complete complex algebraic manifold and D is a normal crossing divisor. Then, by working on a regular projective curve over C, we use the cup product in this cohomology to obtain the pairing for two Hermitian line bundles with Hermitian metr...

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“…(1) • X provides a characterization of the canonical connection associated with a Hermitian line bundle [10,7]. We will also need a leaner version of the complex (1.2.7) introduced in [11], namely:…”

Section: Various Deligne Complexes and Cohomologiesmentioning

confidence: 99%

“…When X is a complete curve, this map gives a cohomological interpretation of Deligne's determinant of cohomology construction [9], which has been analyzed in various guises in [6,10,11] in the singular case.…”

Section: Background and Motivationsmentioning

confidence: 99%