[124] Report on the Woods Hole Fixed Point Theorem Seminar

  • Loring W. Tu
Part of the Contemporary Mathematicians book series (CM)

1) Introduction

This seminar was devoted to the discussion of a beautiful extension of the Lefschetz flxed point theorem which was proposed to the conference by Shimura. Shimura also noted that for curves this extension was a consequence of a result of Eichler.

Through the considerable advertising abilities of the authors a large number of the participants of the conference were drawn into the consideration of this formula and as a consequence of this intervention, especially that of Verdier, Mumford and Hartshorne, it was found that in the algebraic case the Shimura formula was correct and followed along more or less classical lines from the Grothendieck version of Serre duality.

The formula in question is the following one. Suppose that X is a nonsingular projective algebraic variety over an algebraically closed field k, and that f: X → X is a morphism of X into itself. Suppose further that E is a vector bundle over X, and that f admits a lifting ϕ to E - that is, a vector bundle map ϕ: f−1(E) → E. Such a lifting then defines in a natural way an endomorphism (f, ϕ)* of the cohomology vectorspaces H*(X;E), of X with coefficients in the locally free sheaf E of germs of sections of E, and we may therefore form the “Lefschetz number” of this endomorphism:

$${\rm{(1}}{\rm{.}}\,{\rm{1)}}\,{\rm{\chi (f,}}\,\phi {\rm{,}}\,{\rm{E)}}\,{\rm{ = }}\,\sum\limits_{\rm{q}} {{{{\rm{(}} - {\rm{)}}}^{\rm{q}}}} \,{\rm{trace}}\,\left\{ {{\rm{(f,}}\,{\rm{f)*}}\,{\rm{|}}\,{\rm{H*(X}}\,{\rm{;}}\,{\rm{E)}}} \right\}{\rm{.}}$$

Suppose next that f is nondegenerate in the sense that the graph of f intersects the diagonal transversally in X × X. This implies that at each fixed point p of f, the differential dfp : XP → Xp has no eigenvalue equal to 1 so that det(1 - df)p ≠ 0.

Finally note that at a fixed point p, the lifting ϕ determines an endomorphism ϕp of Ep = Ef(p) and so has a well determined trace.

With this understood the Shimura conjecture which we now propose to call the Woods Hole Fixed Point Theorem, is given by the relation:

$${\rm{(1}}{\rm{.}}\,{\rm{2)}}\,{\rm{\chi (f,}}\,\phi {\rm{,}}\,{\rm{E)}}\,{\rm{ = }}\,\sum\limits_{\rm{q}} {{\rm{trace}}\,{\phi _{\rm{p}}}/\det \,(1\, - \,{\rm{d}}{{\rm{f}}_p})} $$

where p runs over the fixed points of f.

2) Some examples

(2. 1) As a first application of (1. 2) we derive the usual Lefschetz formula for f when X is defined over the complex number field ℂ. For this purpose let T* be the cotangent bundle of X, and let λqT* be its qth exterior power. The qth exterior power of the differential of f then defines a natural lifting λqdf: f−1qT*) → λqT* of f so that (1. 2) is applicable and yields the identity:

$${\rm{(2}}{\rm{.}}\,{\rm{2)}}\,{\rm{\chi (f,}}\,{\lambda ^{\rm{q}}}{\rm{df,}}\,{\lambda ^{\rm{q}}}{\rm{T}}*{\rm{)}}\,{\rm{ = }}\,\sum\limits_{\rm{p}} {{\rm{trace}}\,({\lambda ^{\rm{q}}}{\rm{d}}{{\rm{f}}_{\rm{p}}})/\det \,(1\, - \,{\rm{d}}{{\rm{f}}_{\rm{p}}}).} $$

One now takes the alternating sum with respect to q. By virtue of the identity

$$(2.\,3)\,\Sigma {\left( { - 1} \right)^q}\,trace\,{\lambda ^q}df\, = \,\det \,(1\, - \,df).$$

The right-hand side then counts the number of fixed points of f, each with multiplicity + 1, as indeed they should be counted in this nondegenerate and orientation preserving situation. The left hand side becomes Σ(−1)q trace { f* | Hq(X ; C) } by virtue of the Dolboux isomorphisms. In short (2. 2) implies the usual Lefschetz formula.

(2. 4) Let P be projective n-space over K, with homogeneous coordinates (x0, ......., xn). Let f: P → P be the linear map, which sends xi into \({{\rm{\lambda }}_{\rm{i}}}{{\rm{x}}_{\rm{i}}}\,{\rm{;}}\,{{\rm{\lambda }}_{\rm{i}}}\,{\rm{}}\,{\rm{0,}}\,{{\rm{\lambda }}_{\rm{i}}}\,{\rm{}}\,{{\rm{\lambda }}_{\rm{j}}}\,{\rm{if}}\,{\rm{i}}\,{\rm{}}\,{\rm{j}}{\rm{.}}\)

The fixed points of f then correspond to the coordinate axes and are represented by pk = (0, ....., 1, ...., 0); 1 = 0, ...., n; where the 1 occurs at the kth place. Now det (1 - dfp) is easily computed to be

$$\prod\limits_{{\rm{j}} \ne {\rm{k}}} {{\rm{(1}} - {{\rm{\lambda }}_{\rm{j}}}{\rm{/}}{{\rm{\lambda }}_{\rm{k}}}{\rm{)}}{\rm{.}}} $$

Thus for instance, if we take the trivial bundle for E, and lift f to E by means of the constant section, then (1. 2) takes the form:

$$1\, = \,\sum\limits_{{\rm{k}}\, = \,0}^{\rm{n}} {{{\lambda _{\rm{k}}^{\rm{n}}} \over {\mathop {\Pi ({\lambda _{\rm{k}}}\, - \,{\lambda _{\rm{i}}})}\limits_{{\rm{i}}\, \ne \,{\rm{k}}} }}} $$

which is a well-known interpolation formula.

If one takes for E the k power of the Hyperplane bundle, then may be lifted to E, in such a manner that the action of (f, ϕ) on Γ(P; E) = H*(P; E) is precisely the action induced on the polynomials of degree K in K[x0, ......., xn] by the substitution xi → λi xi.

The formula (1. 2) applied to this situation simultaneously for all k then yields the identity of formal power series in t:

$$(2.\,5)\,\Pi {1 \over {(1\, - \,{\rm{t}}{\lambda _{\rm{j}}})}}\, = \,\sum {{{\lambda _{\rm{k}}^{\rm{n}}} \over {\mathop {\Pi ({\lambda _{\rm{i}}}\, - \,{\lambda _{\rm{k}}})}\limits_{{\rm{i}}\, \ne \,{\rm{k}}} }}} \, \cdot \,{1 \over {1\, - \,{\rm{t}}{\lambda _{\rm{k}}}}}.$$

This partial fraction expansin or the left-hand side is useful in the discussion of the characters of the irreducible representations of the full linear group, and indeed if one follows this lead, then (1. 2) is seen to imply the formula of Herman Weyl for the character of an irreducible representation of a semi-simple Lie group in a most natural manner.

Our last example deals with the case when X is defined over a finite field of characteristic p. One may then use the Frobenius endomorphism for f (which is always nondegenerate!), and using the constant lifting of f to the structure sheaf, \({0_{{{\rm{X}}_{\rm{i}}}}} = 1\), one concludes directly from (1. 2) that if X is "regular" in the sense that H (X, OX) = 0; for i > 0, then X must have at least one rational point.

3) Remarks

It is not difficult to propose generalizations of (1. 2). One may drop the nondegeneracy assumption on f, or remove the nonsingularity hypothesis on X; the vector bundle E may by replaced by a coherent sheaf, and finally– alas –with all this generality one may seek a statement relative to any proper morphism, rather then the projection onto a point.

The first step already leads to an interesting framework of ideas, and should shed new light on the problem of Riemann-Roch which corresponds to a highly degenerate f–namely the identity.

For a possible singular X one would at least hope to find a weak version of (1. 2), i.e., that χ((f, ϕ, E) = 0 if f has no fixed points. A straightforward proof of this fact, that is, one not involving duality, would be highly desirable.

The authors' main personal concern was an extension of (1. 2) along different lines. We consider an elliptic complex

$${\rm{\varepsilon }}\,{\rm{:}}\,{\rm{0}}\, \to \,\mathop E\limits_{ - 0} \,\buildrel d \over \longrightarrow \,\mathop E\limits_{ - 1} \buildrel d \over \longrightarrow ...\,\mathop E\limits_{ - m} \, \to \,0$$

of C vector bundles Ei over a compact C manifold X, with differential operators \({\rm{d}}\, = \,\mathop {\rm{E}}\limits_{ - {\rm{i}}} \, \to \,\mathop {\rm{E}}\limits_{ - {\rm{i}} + {\rm{i}}} \) subject to d2 = 0, and the ellipticity condition that the associated symbol sequence :

$$0 \to {{\rm{E}}_0}\,\buildrel {\sigma (d,\xi )} \over \longrightarrow \,{{\rm{E}}_1}\,\buildrel {\sigma (d,\xi )} \over \longrightarrow \,...\,{{\rm{E}}_{\rm{m}}}\, \to \,0$$

should be exact for every nonzero cotangent vector.

Under this hypothesis the complex Γ (ε) formed by the C -sections, Γ (Ei) of Ei with differential operator Γ (d), has finite-dimensional homology and a formula which specialized to (1. 2) when ε is the \(\bar \partial \) resolution of E can be found. Details of this, and other developments will appear elsewhere.

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© Springer International Publishing AG, part of Springer Nature 2017

Authors and Affiliations

  • Loring W. Tu
    • 1
  1. 1.Department of MathematicsTufts UniversityMedfordUSA

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