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Fixed point formula for singular varieties. (English) Zbl 0574.14018
Current trends in algebraic topology, Semin. London/Ont. 1981, CMS Conf. Proc. 2, part 2, 3-22 (1982).
[For the entire collection see Zbl 0538.00016.]
Let X be a projective variety of dimension n and let $$f:X\to X$$ be a holomorphic automorphism of order s. Assume that f has only a finite number of fixed points. $$L_ 0(f)$$ is defined by $$\sum^{n}_{i=0}(- 1)^ iTrace[f^*:H^ i(X,{\mathcal O}_ X)\to H^ i(X,{\mathcal O}_ X)].$$ The author gives a generalization of the holomorphic fixed point formula for a singular variety in two different ways. Let $$X^ f$$ be the set of fixed points of f and let $$P\in X^ f$$. - (I) Let $${\mathfrak O}_ P$$ be the local ring of X at P and let $$f^*:{\mathfrak O}_ P\to {\mathfrak O}_ P$$ be the induced map. Let $$I_ P$$ be the ideal generated by all elements of the form $$u-f^*u.$$ $$f^*$$ induces a linear map $$f^*\!_ r$$ of the finite dimensional vector space $$I^ r/I^{r+1}$$. A rational number $$\epsilon_ 0(P)$$ is defined by $$\epsilon_ 0(P)=Q_ P(1)$$ where $$Q_ P(t)$$ is the rational function $$\sum^{\infty}_{r=0}Trace(f^*\!_ r)\cdot t^ r.$$ Then the holomorphic fixed point formula for $$L_ 0(f)$$ is: $$L_ 0(f)=\sum_{P\in X^ f}\epsilon_ 0(P).$$- (II) The second expression is done through the equivariant intersection number. Let $$T_ PX$$ be the Zariski tangent space of X at P. Let $$G=\{1,f,...,f^{s-1}\}.$$ $$T_ PX$$ is a G-module in a canonical way and there exists an open neighbourhood $$U_ P$$ of P in X which can be equivariantly embedded in $$T_ PX$$. Let $$\phi: U_ P\to T_ PX$$ be the embedding. Let $$W_ P$$ be the subspace of $$T_ PX$$ whose vectors are fixed by G. Let $$X_ P=\phi (U_ P)$$. Then $$\epsilon_ 0(P)$$ can be also defined by $$Trace[f^{-1},\#_ 0(X_ P,W_ P,G)]/\prod^{k}_{i=1}(1-\lambda_ i)$$ where Trace(g,$$\psi)$$ means the trace of $$\psi$$ at g for $$g\in G$$ and $$\psi\in R(G)$$, R(G) the representation ring of G, which is - as an abelian group - the Grothendieck group of the category of G-modules, and where #$${}_ 0(X_ P,W_ P,G)$$ is the equivariant intersection number which is an element of R(G) and where $$\lambda_ 1,...,\lambda_ k$$ are the eigenvalues of $$f_*$$ on $$T_ PX/W_ P$$.
Reviewer: M.Oka
##### MSC:
 14F20 Étale and other Grothendieck topologies and (co)homologies 32Sxx Complex singularities 14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry 14B05 Singularities in algebraic geometry 55M20 Fixed points and coincidences in algebraic topology