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**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 |

### Keywords:

metrics on abelian varieties; hermitian metric; distorsion function; Fubini-Study metric; theta functions
Full Text:
DOI

### References:

[1] | G. R. Kempf, Complex Abelian Varieties and Theta Functions , · Zbl 0752.14040 |

[2] | G. R. Kempf, Metric on invertible sheaves on abelian varieties , preprint, 1988. · Zbl 0866.14028 |

[3] | G. R. Kempf, Multiplication over abelian varieties , Amer. J. Math. 110 (1988), no. 4, 765-773. JSTOR: · Zbl 0681.14023 |

[4] | 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 |

[5] | D. Mumford, On the equations defining abelian varieties. III , Invent. Math. 3 (1967), 215-244. · Zbl 0219.14024 |

[6] | 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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