Gersten-Witt Complex of Hirzebruch Surfaces.

摘要

The Witt group is classically an abelian group whose elements are represented by isometry classes of anisotropic symmetric bilinear forms over a field. It can be generalized to Witt group of a ring or a scheme X, hence to the Witt sheaf of X..;The Gersten-Witt complex of a scheme X is a cochain complex of Witt groups of residue class fields at all points of X, where the p-th term is the direct sum of Witt groups of residue class field at all codimension p points in X..;Pardon constructed the Gersten-Witt complex of a Gorenstein scheme X, and showed that it is exact when X is the spectrum of a regular local ring which is essentially of finite type over a field of characteristic diffierent from 2. Using this, he defined a flasque resolution of a Witt sheaf. Although the Witt group of a complex surface is a birational invariant, we show that higher cohomologies of the Witt sheaf can distinguish diffierent birational schemes. Specifically, we show that the cohomologies of the Hirzebruch surface H_n are diffierent depending on whether n is even or odd.;Computing cohomologies of the Gersten-Witt complex of a scheme is difficult in practice, because it involves Witt groups of residue fields at all (possibly infinitely many) points of the scheme. However, if the scheme is a complex toric variety, we show that one can construct a quasi-isomorphic cochain complex which has only a finite number of Witt groups of tori, which are orbits of the torus action. Moreover, the Witt group of a complex torus is a vector space of finite dimension. Hence, the new cochain complex is much simpler than the original Gersten-Witt complex, and its cohomologies can be more easily computed.;Using this quasi-isomorphism, we compute cohomologies of the Witt sheaf on the Hirzebruch surfaces H_n. The Hirzebruch surfaces H_n are projective line bundles over a projective line, one for each integer n. We show that the first and second cohomologies depend on the parity of n..