Let $X$ be a positive-dimensional, smooth, connected projective variety (say over $\Bbb{C}$), and let $\sigma $ be an involution of $X$ with a finite number of fixed points; then this number is even. The proof I have is somewhat artificial: blow up the fixed points, take the quotient variety, observe that the image of the exceptional divisor in the quotient is divisible by 2 in the Picard group, and compute its self-intersection. Does someone know a more natural proof?



Number of fixed points of an involution
Indeed, the number of fixed points is divisible by $2^{\dim X}$. This is actually an old result of Conner and Floyd, *Periodic maps which preserve a complex structure*, Bull. Amer. Math. Soc. 70, no. 4 (1964), 574-579. As @SashaP mentions, Atiyah and Bott observed that it is also a consequence of the holomorphic Lefschetz formula (*Notes on the Lefschetz fixed point theorem for elliptic complexes*, Matematika 10, no. 4 (1966), 101-139). 


