Wednesday May 1st, 2024 | |
| Time: | 4:00 PM - 5:00 PM |
| Title: | Higher Fano manifolds |
| Speaker: | Svetlana Makarova, Australian National University |
| Location: | Math P-131 |
| Abstract: | |
| Fano manifolds are projective manifolds whose anticanonical class (determinant of the tangent bundle) is ample. The positivity condition has far-reaching geometric implications, e.g., a Fano manifold over complex numbers is simply connected, which has an analogue on the algebro-geometric side: any Fano manifold is covered by rational curves, and in fact rationally connected, i.e., there are rational curves connecting any two of its points. In a series of papers, De Jong and Starr introduce and investigate possible candidates for the notion of higher rationally connectedness, inspired by the natural analogue in topology, and define that a projective manifold X is 2-Fano if it is Fano and the second Chern character ch2(T_X) is positive (intersects positively with every surface in X). In a similar way, one defines n-Fano manifolds for any n ≥ 2; for instance, P^n is n-Fano. In this talk, I will give evidence for the analogy with higher connectedness and present certain classification results. In the second half of the talk, I will focus on the recent progress towards proving the conjecture that the only toric higher Fano manifolds are projective spaces. | |