Smooth surfaces in smooth fourfolds with vanishing first Chern class (Q1689596)

The article deals with the geography of non singular surfaces \(S\) on a smooth fourfold \(X\) with vanishing first integral Chern class. The author notes that the function \(f(S):= \) \(\text{deg} \left( c_1^2(S) - c_2(S) \right)\) depends only on the numerical class of \(S\) in \(X\). Fix an integer \(a\), the question is to prove that only a finite number of numerical classes can satisfy the bound \(f(S) \leq a \). If this is so then the same result holds true when the bound is given to \(\chi(S, \mathcal{O}_S)\), because the classification of surfaces yields \(f(S) \leq 6\chi(S, \mathcal{O}_S)\). The author produces conditions under which \(f(S)\) can be computed by means of a quadratic function on the group of codimension-2 cycles in \(V\) up to numerical equivalence, where \(V\) is some convenient variety in which \(X\) lives. Under suitable hypotheses the associated quadratic form is positive definite, and therefore only a finite number of numerical classes contain a surface \(S\) with \(f(S) \leq a \). For the following cases: (i) \(X\) a sextic hypersurface in \(\mathbb P^5\) or (ii) \(X\) the Fano variety of lines in a cubic fourfold, explicit inequalities on the number of numerical classes of smooth surfaces \(S\) on \(X\) with \(\chi(S) \leq r \) are found. The computation requires the further assumption that \(S\) is given by intersection with \(X\) of a codimension two subvariety inside the natural ambient space in which \(X\) lives.