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$%P. Jaworski, On Witt rings of function fields of real analytic surfaces and
curves,
Rev. Mat. Univ. Complut. Madrid 10 (1997), supplementary, 153–171.
%$
Let be a paracompact connected real analytic manifold of dimension 1 or
2, i.e. a smooth curve or surface. We consider it as a subset of some complex
analytic manifold
of the same dimension. Moreover by a prime divisor
of
we shall mean the irreducible germ along
of a codimension one
subvariety of
which is an invariant of the complex conjugation. This notion
is independent of the choice of the complexification
. In the one-dimensional
case prime divisors are just points, in the two-dimensional analytic curves or
elliptic points (intersections of two conjugated complex analytic curves). Every
such divisor induces a discrete valuation on the field
of meromorphic
functions on
--the order of the zero or minus the order of the pole of the
function. Therefore it induces the so called residue homomorphisms (first and
second) of the Witt group of the field
to the Witt group of the residue field
--the function field of the divisor.
The main goal of this paper is to show that the intersection of kernels of all
second residue homomorphisms associated to prime divisors is isomorphic to the
Witt group of the Riemannian bundles on .
As an example of an application of this result we provide the new proof of the Artin-Lang property for one and two dimensional real analytic manifolds (both compact and noncompact), which is neither based on the description of all possible orderings of the field of meromorphic functions nor on the compactification of the variety.
1991 Mathematics Subject Classification: 14P15, 11E81, 12D15.