# Bend and break

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Bhargav Bhatt, Princeton University
Fine Hall 314

Our goal is to explain the following theorem due to Mori: Given a compact complex manifold $X$ whose tangent bundle has lots of (holomorphic) determinantal sections (a Fano manifold), any pair of points on it lie in the image of a holomorphic map from the Riemann sphere $P^1$. Despite being a complex geometric statement, the only known proof of this result is by reduction to characteristic $p$. In this talk, we'll discuss what Fano manifolds are, explain the techniques that go into Mori's proof (reducing mod $p$, deformation theory of maps from curves), and the proof itself.