The Hirzebruch-Riemann-Roch theorem. (Q5954573)

The author gives a new proof of the Hirzebruch-Riemann-Roch theorem (HRR), which is now usually proved by evaluating the Grothendieck-Riemann-Roch theorem (GRR) over a point. Grothendieck's proof used the factorization of a proper morphism into a projection NEWLINEand a closed immersion. Here, a proof of HRR without the factorization method is given, and the author indicates that this would work for a proof of GRR as well. NEWLINENEWLINEThe main tool used is a generalization of the Atiyah-Bott-Lefschetz fixed point formula for periodic self mapsNEWLINE[cf. \textit{D. Toledo} and \textit{Y. L. L. Tong}, Ann. Math. (2) 108, 519--538 (1978; Zbl 0413.32006)], which is slightly stronger than the classical result in the case of positive characteristic. The paper is self-contained: In the first two sections, the Riemann-Roch formalism and the fixed point formula are introduced and proved. The third section is devoted to the proof of the Adams-Riemann-Roch theorem, which is applied in the final section to the proof of HRR.