Numerical obstructions to abelian surfaces in toric Fano 4-folds (Q2427455)

The author considers embeddings \(\phi\) of an abelian surface \(Y\) into a smooth Fano fourfold \(X\). \(\phi\) is totally nondegenerate when \(\phi(y)\cap D\neq \emptyset\) for all torus invariant prime divisors \(D\). When \(X=\mathbb P^4\), \(X=\mathbb P^1\times \mathbb P^3\) or \(X\) is a product of del Pezzo surfaces, then clearly one finds abelian surfaces inside \(X\). Forgetting these cases, it is known that \(X\) cannot contain totally nondegenerate abelian surfaces, unless possibly when \(X\) belongs to \(19\) well determined types of Fano fourfolds. The author studies the existence of abelian surfaces in fourfolds of these \(19\) types. His main tool is a clever study of numerical conditions imposed to a class \(\alpha\in A^2(X)\) by the existence of an abelian surface in \(\alpha\). With the aid of some computer based computation, the author uses these conditions to exclude the existence of totally nondegenerate abelian surfaces, in fourfolds of \(3\) of the \(19\) types. For the remaining \(16\) types, the author provides strong numerical restrictions on classes \(\alpha\) that could contain abelian surfaces.