Squares of ample line bundles on abelian varieties
Let \(X\) be an abelian variety of dimension \(g\) defined over the complex field and let \(L\) be an ample line bundle. The rational mapping \(\phi =\Phi_{L^2}: X\to\mathbb{P}(H^0(L^2))\) associated with the complete linear system \(| L^2|\) is known to be a morphism. Let \((X,L) = (X_1,L_1)\times\cdots\times (X_s,L_s)\) be the decomposition into a product of irreducible polarized abelian varieties such that \((X_i,L_i)\) is principally polarized for \(1\le i\le r\) and not principally polarized for \(r+1\le i\le s\). Let \(K_i = X_i/(\pm 1)\) be the Kummer variety of \(X_i\) for \(1\le i\le r\) and \(p_i: X_i\to K_i\) the canonical projection. Let \(K=K_1\times\cdots\times K_r\times X_{r+1}\times\cdots\times X_ s\) and \(p=p_ 1\times...\times p_ r\times id_{X_{r+1}}\times\cdots\times id_{X_s}: X\to K\). Then it is shown that the morphism \(\phi\) decomposes as \(\psi \circ p: X\to K\to\mathbb{P}^N\) with an embedding \(\psi: K\hookrightarrow\mathbb{P}^N\). The result is a generalization of the results of \textit{S. Ramanan} [Proc. Lond. Math. Soc. (3) 51, 231--245 (1985; Zbl 0603.14013)], \textit{A. Ohbuchi} [Manuscr. Math. 57, 225--238 (1987; Zbl 0595.14013)], \textit{A. Andreotti} and \textit{A. L. Mayer} [Ann. Sc. Norm. Super. Pisa, Sci. Fis. Mat., III. Ser. 21, 189--238 (1967; Zbl 0222.14024)] and \textit{D. Sasaki} [J. Reine Angew. Math. 345, 111--121 (1983; Zbl 0513.10027)].
scientific article; zbMATH DE number 4163067