Higher Fano manifolds

Abstract

We address in this paper Fano manifolds with positive higher Chern characters, which are expected to enjoy stronger versions of several of the nice properties of Fano manifolds. For instance, they should be covered by higher dimensional rational varieties, and families of higher Fano manifolds over higher dimensional bases should admit meromorphic sections (modulo the Brauer obstruction). Aiming at finding new examples of higher Fano manifolds, we investigate positivity of higher Chern characters of rational homogeneous spaces. We determine which rational homogeneous spaces of Picard rank 1 have positive second Chern character, and show that the only rational homogeneous spaces of Picard rank 1 having positive second and third Chern characters are projective spaces and quadric hypersurfaces. We also classify Fano manifolds of large index having positive second and third Chern characters. We conclude by discussing conjectural characterizations of projective spaces and complete intersections in terms of these higher Fano conditions.

Higher order minimal families of rational curves and Fano manifolds with nef Chern characters

“…T −(i − 1)c 1 (L i−1 ) + i−(i−1,1,k) T k ch k (X) c 1 (L i−1 ) +b (i−1,1,i) T i−1 ch i (X) − −(i − 1) + i k=1 b (i−1,1,k) T k ch k (X) − −(i − 1) + (2m + 1) − (i + 1) − 2 > 0 and c 1 (H i ) = −ic 1 (L i ) + i k=1 b (i,1,k) T k ch k (X) c 1 (L i ) + b (i,1,i+1) T i ch i+1 (X) (2m + 1) − (i + 2) c 1 (L i ) > 0.Moreover, Lemma 2.3, Propositions 3.1, and 3.5(2) yield dimH i+1 = T c 1 (H i ) − 2 = dimH i 2 a i+1 + T 2 ch 2 (H i−1 ) − 2 and…”

“…In [4], Araujo and Castravet classified Fano manifolds of large index having positive second Chern character, where the index is defined as the largest integer dividing the anticanonical divisor. Furthermore, in [2], Araujo, Beheshti, Castravet, Jabbusch, Makarova, Mazzon, Taylor, and Viswanathan classified rational homogeneous spaces of Picard number 1 having positive second Chern character and Fano manifolds of large index having positive second and third Chern characters. Their results show that positivity conditions of Chern characters are restrictive, and they proposed the following conjecture:…”