A Positivstellensatz for projective real varieties

Abstract

Given two positive definite forms ${f,\,g\in\mathbb {R}[x_0,\ldots,x_n]}$ , we prove that fg ^N is a sum of squares of forms for all sufficiently large N ≥ 0. We generalize this result to projective ${\mathbb {R}}$ -varieties X as follows. Suppose that X is reduced without one-dimensional irreducible components, and ${X(\mathbb {R})}$ is Zariski dense in X. Given everywhere positive global sections f of ${L^{\otimes2}}$ and g of ${M^{\otimes2}}$ , where L, M are invertible sheaves on X and M is ample, fg ^N is a sum of squares of sections of ${L\otimes M^ {\otimes N}}$ for all large N ≥ 0. In fact we prove a much more general version with semi-algebraic constraints, defined by sections of invertible sheaves. For nonsingular curves and surfaces and sufficiently regular constraints, the result remains true even if f is just nonnegative. The main tools are local-global principles for sums of squares, and on the other hand an existence theorem for totally real global sections of invertible sheaves, which is the second main result of this paper. For this theorem, X may be quasi-projective, but again should not have curve components. In fact, this result is false for curves in general.