Inequality for the distortion function of invertible sheaves on abelian varieties
Let \(X={\mathbb{C}}^ g/\Lambda\) be an abelian variety over the field of complex numbers, \(\Lambda\) a lattice in \({\mathbb{C}}^ g\), and L an ample bundle on X with first Chern class H, a positive definite hermitian form of \({\mathbb{C}}^ g\). Using H one defines a hermitian metric (,)\({}_ 1\) on L. On the other hand one has the well-known Fubini-Study metric (,)\({}_ 2\) on L. The distorsion function is by definition a positive real analytic function \(b_ L\) on X with \(b_ L(x)\cdot (,)_ 1(x)=(,)_ 2(x)\) for all \(x\in X\). It was proved by \textit{Kempf} [``Metrics on invertible sheaves on abelian varieties'' (Preprint 1988)] that there are positive constants \(C_ 1\) and \(C_ 2\) such that \(C_ 1m^{2g}\leq b_{L^ m}(x)\leq C_ 2m^{2g}.\) Roughly speaking this means that the Fubini-Study metric for \(L^ m\) eventually flattens out if \(m\to \infty\). The main result of the present paper is an improvement of this: It is shown that for any \(\epsilon >0\) there are positive constants \(C_ 1(\epsilon)\) and \(C_ 2(\epsilon)\) such that \(C_ 1(\epsilon)m^{- (3g+2)(2\sqrt{m}+1)-\epsilon}\leq b_{L^ m}(x)\leq C_ 2(\epsilon)m^{7g+3+\epsilon}\) for any \(x\in X\) and any positive integer m. Consequently one has \(\lim_{m\to \infty}(b_{L^ m}(x))^{1/2}=1,\) as was conjectured by Kempf. The method of proof is to use the multiplication formula for finite theta functions.
scientific article; zbMATH DE number 4168831