On the holomorphic length of a complex projective variety
For any projective complex \(n\)-dimensional smooth variety \(X\) we define the holomorphic length \(\hbox{hl}(X)\) as the maximal number of homogeneous elements of the \(\mathbb{C}\)-algebra: \(H^{1,0}(X)\oplus H^{2,0}(X)\oplus\cdots\oplus H^{n,0}(X)\) such that their product is non zero. --- We show that if \(f:M\to N\) is a rational map, generically finite to one (for instance a branched) covering between two projective complex smooth varieties, then: \(\deg(f)[\hbox{hl}(N)+1]\geq\hbox{hl}(M)+1\). As an application of our theorem we show that if \(f\) is a map as above, between an \(n\)-dimensional Abelian variety and \(\mathbb{P}^ n(\mathbb{C})\) then \(\deg(f)\geq n+1\). Our result is a generalization of a previous work (Bernstein-Edmonds- Sommese), in which \(f\) was a branched covering between orientable compact manifolds, the \(\mathbb{C}\)-algebra was \(H^*(X,\mathbb{C})\), and \(\deg(f) \hbox{length}(N)\geq\hbox{length}(M)\). --- Moreover, by using our theorem, we show the existence of an upper bound for the dimension of some rational equivalence classes of zero cycles in the \(k\)-symmetric product of an Abelian variety, and some other minor results.
scientific article; zbMATH DE number 4216212