Non-symmetric orthogonal geometry of Grothendieck rings of coherent sheaves on projective spaces

Details Non-symmetric orthogonal geometry of Grothendieck rings of coherent sheaves on projective spaces Non-symmetric orthogonal geometry of Grothendieck rings of coherent sheaves on projective spaces

1994-09-27

arxiv.org/abs/alg-geom/9409005v1

Mathematics

Algebraic Geometry

30 pages, LATEX (version 2.01), 12pt, twoside, font families "eufm" and "msbm" are used

Scientific paper

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 $\chi$.

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