Positivity of the diagonal (Q1669044)

Let \(X\) be a smooth projective variety. A cycle class is said to be big if it lies in the interior of the closed cone generated by classes of effective cycles. A cycle is called homologically big if it is homologically equivalent to the sum of an effective \(\mathbb{Q}\)-cycle and a complete intersection of ample \(\mathbb{Q}\)-divisors. Furthermore a cycles class is said to be nef if it has non-negative intersection against every subvariety of the complementary dimension. One of the main results of this article is the following: If \(X\) is a smooth projective variety and the diagonal \(\Delta_X\subset X\times X\) is homologically big, then \(H^{i,0}(X)\) vanishes for \(i>0\). In particular, no varieties with trivial canonical bundle can have homologically big diagonal. As an application, the author classifies low-dimensional varieties with big and nef diagonals as follows. A smooth projective surface with nef and big diagonal is either \(\mathbb{P}^2\) or a fake projective plane. A smooth projective threefold with nef and homologically big diagonal is either \(\mathbb{P}^3\), a del Pezzo quintic threefold \(V_5\), or the Fano threefold \(V_{22}\).