## Inequality for the distortion function of invertible sheaves on abelian varieties.(English)Zbl 0711.14024

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 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.
Reviewer: H.Lange

### MSC:

 14K05 Algebraic theory of abelian varieties
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### References:

  G. R. Kempf, Complex Abelian Varieties and Theta Functions , · Zbl 0752.14040  G. R. Kempf, Metric on invertible sheaves on abelian varieties , preprint, 1988. · Zbl 0866.14028  G. R. Kempf, Multiplication over abelian varieties , Amer. J. Math. 110 (1988), no. 4, 765-773. JSTOR: · Zbl 0681.14023  D. Mumford, Abelian varieties , Tata Institute of Fundamental Research Studies in Mathematics, No. 5, Published for the Tata Institute of Fundamental Research, Bombay, 1970. · Zbl 0223.14022  D. Mumford, On the equations defining abelian varieties. III , Invent. Math. 3 (1967), 215-244. · Zbl 0219.14024  B. Shiffman and A. J. Sommese, Vanishing theorems on complex manifolds , Progress in Mathematics, vol. 56, Birkhäuser Boston Inc., Boston, MA, 1985. · Zbl 0578.32055
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