  In this paper we consider orthogonal geometry of the free
$\protect\ZZ$-module $K_0(\protect\PP_n)$ with respect to the non-symmetric
unimodular bilinear form $$\chi(E,F)=\sum (-1)^\nu\dim\ext^\nu(E,F). $$ We
calculate the isometry group of this form and describe invariants of its
natural action on $K_0(\protect\PP_n)$. Also we consider some general
constructions with non-symmetric unimodular forms. In particular, we discuss
orthogonal decomposition of such forms and the action of the braid group on a
set of semiorthonormal bases. We formulate a list of natural arithmetical
conjectures about semiorthogonal bases of the form $\chi$.
