Luca Tasin

Fano Varieties in Mori Fibre Spaces

We show that being a general fibre of a Mori fibre space (MFS) is a rather restrictive condition for a Fano variety. More specifically, we obtain two criteria (one sufficient and one necessary) for a Q-factorial Fano variety with terminal singularities to be realised as a fibre of a Mori fibre space, which turn into a characterisation in the rigid case. We apply our criteria to figure out this property up to dimension 3 and on rational homogeneous spaces. The smooth toric case is studied and an interesting connection with K-semistability is also investigated.

FANO-MORI CONTRACTIONS OF HIGH LENGTH ON PROJECTIVE VARIETIES WITH TERMINAL SINGULARITIES

Let X be a projective variety with Q-factorial terminal singulari- ties and let L be an ample Cartier divisor on X. We prove that if f is a birational contraction associated to an extremal ray RNE(X) such that R.(KX + (n 2)L) < 0 then f is a weighted blow-up of a smooth point. We then classify divisorial contractions associated to extremal rays R such that R.(KX + rL) < 0, where r is a non-negative integer, and the fibres of f have dimension less or equal to r + 1.

Algebraic structures with unbounded Chern numbers

We determine all Chern numbers of smooth complex projective varieties of dimension at least four which are determined up to finite ambiguity by the underlying smooth manifold. We also give an upper bound on the dimension of the space of linear combinations of Chern numbers with that property and prove its optimality in dimension four.

Kähler structures on spin 6-manifolds

We show that many spin 6-manifolds have the homotopy type but not the homeomorphism type of a Kähler manifold. Moreover, for given Betti numbers, there are only finitely many deformation types and hence topological types of smooth complex projective spin threefolds of general type. Finally, on a fixed spin 6-manifold, the Chern numbers take on only finitely many values on all possible Kähler structures.

A note on the fibres of Mori fibre spaces

We consider the problem of determining which Fano manifolds can be realised as fibres of a Mori fibre space. In particular, we study the case of toric varieties, Fano manifolds with high index and some Fano manifolds with high Picard rank.