On the degree of the Gauss mapping of a submanifold of an Abelian variety
Let \(X\) be a complex connected submanifold of \(A\), an abelian variety. Associated to the natural surjection of the restriction of the holomorphic cotangent bundle of \(A\) to \(X\) onto the holomorphic cotangent bundle of \(X\) there is the Gauss mapping \(\Gamma:X\to \mathrm{Gr}(\dim X,\dim A)\); \(\mathrm{Gr}(\dim X,\dim A)\) is the Grassmannian of \(\dim X\) quotient spaces of the \(\dim A\) dimensional complex number space. It is shown as a consequence of a result of \textit{T. Gaffney} and \textit{R. Lazarsfeld} [Invent Math. 59, 53- 58 (1980; Zbl 0422.14010)] that if the normal bundle of \(X\) in \(A\) is ample then the degree of \(\Gamma\) is bounded by the absolute value of the topological Euler characteristic of \(X\) divided by the codimension of \(X\) in \(A\). It should be noted that by a result of \textit{R. Hartshorne} [Nagoya Math. J. 43, 73-89 (1971; Zbl 0205.254)] the above normal bundle is ample if either \(X\) is of dimension one and generates \(A\) as a group or if \(A\) is a simple abelian variety (i.e., \(A\) has no proper sub-abelian varieties). A variety of examples, e.g. when \(X\) is of codimension one in \(A\) or \(X\) is a hyperelliptic curve embedded in its Jacobian \(A\), show that result is sharp (as results based on connectedness theorems often are).
scientific article; zbMATH DE number 3875440