The geometry of algebraic Fermi curves. (English) Zbl 0778.14011

Perspectives in Mathematics. 14. Boston, MA: Academic Press, Inc.. vii, 236 p. (1993).
During the past twenty years, various methods of algebraic geometry have emerged as crucial mathematical tools in theoretical physics. In particular, the geometry of complex algebraic curves and their moduli spaces has turned out as to be closely related to the theory of (some) integrable Hamiltonian systems, nonlinear evolution equations in mathematical physics, and quantum field theory.
One more example in this direction is provided by the topic of the book under review. Fermi curves, the objects of study investigated here, arise from a discrete approximation of a certain model from solid state physics, the so-called independent electron model. This model describes the pattern of a gas of electrons, which move independently through a lattice \(\Gamma\) of ions in \(\mathbb{R}^ d\) \((d\leq 3)\), under the influence of a potential \(q\) which is periodic with respect to the lattice \(\Gamma\). Each electron is described by its wave function, and the independent electron model postulates that this wave function is a superposition of solutions \(\psi\) of the complex, nonlinear Schrödinger equation \((-\Delta+q)\psi=\lambda\cdot\psi\), \(\lambda\in\mathbb{R}\), with periodic boundary conditions regarding the ion lattice \(\Gamma\). Replacing the Laplacian \(\Delta\) by a discretization of it, via shift operators on functions \(\psi:\mathbb{Z}^ d\to\mathbb{C}\), one obtains a discrete Schrödinger equation of the form \[ (-\Delta+q- \lambda)\psi=0,\quad\lambda\in\mathbb{C},\;q\in L^ 2(\mathbb{Z}^ d/\Gamma), \tag{*} \] and boundary conditions \[ S^{a_ j}j\psi=\xi_ j\cdot\psi, \tag{**} \] where \(1\leq j\leq d\), \(\xi_ j\in\mathbb{C}^*\), \(\mathbb{Z}^ d\cap([1,a_ 1]\times\cdots\times[1,a_ d])\) is a fundamental domain for the lattice \(\Gamma\), and \(S_ j^{a_ j}\) is the \(j\)-th shift by \(a_ j\) for complex functions on \(\mathbb{Z}^ d\). To this discrete model the authors associate a complex variety, namely \[ B(q)=\{(\xi_ 1,\ldots,\xi_ d,\lambda)|\;\exists\psi\neq 0,\psi\text{ solves } (*) \text{ with } (**)\}. \] This is a subvariety of \((\mathbb{C}^*)^ d\times\mathbb{C}\), called the Bloch variety for the potential \(q\). It is fibred over the second factor \(\mathbb{C}\) into \((d-1)\)-dimensional varieties, and these fibres are called – referring back to the physical background – the (complex) Fermi varieties associated with \(q\). The book under review is devoted to the special case of dimension \(d=2\), i.e., to the case where the Fermi varieties of a potential \(q\) are complex curves, the so-called Fermi curves.
The central topic for the authors is the purely geometrical aspect in the study of Fermi curves, that is, their algebraic geometry, topology, and deformation theory with respect to the families within they vary. The physical origin just serves for motivation and justification. On the other hand, it turns out that the geometry of Fermi curves is extremely rich and interesting for its own sake, so that it is fair to say that Fermi curves are highly fascinating examples for particular families of complex curves. – The theory of Bloch varieties and Fermi varieties has been invented by the authors of the present book, and their work reviewed here is the first research monograph on the subject, especially devoted to the 2-dimensional case. There are two survey articles which describe the content of the book, the methods and the main results, in a reasonably detailed manner, and the interested reader should be referred to them, all the more as the matter is fairly huge, involved and fruitful. The first survey was given by C. Peters in Sémin. Bourbaki, Vol. 1989/90, 42ème Année, Astérisque 189-190, Exposé 723, 239-258 (1990; Zbl 0749.14027), and the second one was provided by the authors themselves under the title “An overview of the geometry of Fermi curves”, published in Algebraic geometry, Proc. Conf., Sundance/UT 1988, Contemp. Math. 116, 19-46 (1991; Zbl 0751.14021). C. Peters’s survey stresses the physical background and the discussion of the authors’ main results on Fermi curves, whereas the survey by the authors also describes the technical framework for the theory.
May it here suffice to say that the monograph under review provides a comprehensive, complete account of the authors’ theory of Fermi curves, including chapters on the physical origin, one- and two-dimensional Bloch varieties, their compactifications, separable Bloch varieties, the topology of the family of Fermi curves, and the monodromy of Bloch varieties. The presentation of the whole (completely new) material is furthermost careful, precise, systematic, lucid and beautifully geometric. The book may well serve as both the first textbook on low- dimensional Bloch varieties (and their Fermi fibres) and a reference for researchers in the field. – In the meantime, the study of general Bloch varieties (i.e., of Bloch varieties in higher dimension) has undergone a remarkable progress, above all as for the problem of compactifying Bloch varieties and their Fermi fibres. As for further reading, going beyond the content of the present book, the recent work of D. Bättig [“A toroidal compactification of the complex Fermi surface” (Thesis, ETH Zürich 1988)] and T. Kappeler [Adv. Appl. Math. 9, No. 4, 428-438 (1988; Zbl 0675.35023)] is recommended.
Altogether, the book under review is rather self-contained and certainly of great interest for algebraic geometers, analysts, topologists, and physicists with relevant field of research.


14H10 Families, moduli of curves (algebraic)
32J05 Compactification of analytic spaces
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry
14H45 Special algebraic curves and curves of low genus
81-02 Research exposition (monographs, survey articles) pertaining to quantum theory
32-02 Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces