## Birational self-maps and piecewise algebraic geometry

J. Math. Sci. Univ. Tokyo
Vol. 19 (2012), No. 3, Page 325--357.

Lamy, St\'ephane; Sebag, Julien
Birational self-maps and piecewise algebraic geometry
Let $X$ be a smooth projective complex variety, of dimension 3, whose Hodge numbers $h^{3,0}(X), h^{1,0}(X)$ both vanish. Let $f\colon X\dasharrow X$ be a birational map that induces an isomorphism $U\cong V$ on (dense) open subvarieties $U,V$ of $X$. Then we show that the complex varieties $(X\setminus U)_{\mathrm{red}},(X\setminus V)_{\mathrm{red}}$ are piecewise isomorphic.