Lazzeri’s Jacobian of oriented compact Riemannian manifolds. (English) Zbl 1030.14019

Rubei, Elena

Ark. Mat. 38, No. 2, 381-397 (2000).

Given a compact oriented Riemannian manifold \((M,g)\) of dimension \(2m\), with \(m=2k+1\) odd, one defines a Jacobian \(RJ(M)=H^m(M,\mathbb R)/\big(H^m(M,\mathbb Z)/\text{torsion}\big)\) with complex structure given by the Hodge \(\star\)-operator on harmonic forms and a polarisation \(H\) given by \(\text{Im} H(\alpha,\beta)=-\int_M(\alpha\wedge\beta)\). It is easy to see that this is a principally polarised abelian variety.
This Jacobian was introduced by F. Lazzeri, but his work was not published, and no date is given here. (Others may also have proposed this independently. The reviewer first heard of the construction in 1999, in a report of a lecture by I. M. Singer, who asked for a geometric description of the theta line bundle in the case \(m=3\).)
The author begins the study of this Jacobian in earnest. As a first test case, she examines flat metrics on real tori. She shows that the analogue of the Torelli map is generically locally injective (but not injective) and describes the image in terms of period matrices.
If \(M\) is Kähler, one also has the Weil and Griffith’ s \(k\)-th intermediate Jacobians. These are all the same real torus, with different complex structures: This paper compares the three. The Jacobian \(RJ\), like the Weil Jacobian, does not vary holomorphically (this observation is essentially due to A. Mattuck). The author studies the local behaviour of the analogue of the Torelli map for the Lazzeri Jacobian in this Kähler case: She gives a condition for it to be holomorphic, and in the case of trivial \(K_M\), a local injectivity condition.
Finally, the author considers fibre bundles \(F\to M\) with even-dimensional fibres, and constructs, using suitable data, a good holomorphic map \(RJ(M)\to RJ(F)\).

Reviewer: G.K.Sankaran (Cambridge)

MSC:

14K30	Picard schemes, higher Jacobians
53C25	Special Riemannian manifolds (Einstein, Sasakian, etc.)
14R15	Jacobian problem
32G20	Period matrices, variation of Hodge structure; degenerations

Keywords:

oriented Riemannian manifold; principally polarised abelian variety; theta line bundle; holomorphic Torelli map; Lazzeri Jacobian

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