Varieties of Minimal Rational Tangents on the Fano Varieties

The theory of Varieties of Minimal Rational Tangents (VMRT) motivated from the proof of Hartshorne conjecture due to S. Mori. Then developed by Ngaiming Mok and Junmuk Hwang in 80s to 00s. Now this is a important area to discover the geometric properties of Fano varieties of Picard number $1$. Here is our main topics:

(a) First we introduce the basic theory of rational curves, including Hilbert schemes of morphisms, the Chow scheme of rational components $\mathrm{RatCurves}^n_d(X)$ and their useful properties. Then we will give some applications, such as Bend-and-Break, boundedness of smooth Fano varieties and the proof of Hartshorne’s conjecture using our theory.

(b) Then we will introduce some more general facts about Fano manifolds. Moreover we will also introduce some special Fano manifolds such as Gushel-Mukai varieties, rational homogeneous varieties, Hermitian symmetric space and Del-Pezzo manifolds.

(c) Then we will construct the theory of VMRT over smooth uniruled varieties and mainly focused on the Fano varieties of Picard number $1$. We will dicuss the properties of distributions $\mathcal{D}\subset T_X$ and Cartan-Fubini type extension theorem.

(d) Then we will discuss some basic applications of VMRT, such as stability of the tangent bundles and rigidity of generically finite morphisms.

(e) We will consider the VMRT of rational homogeneous varieties which is very important We will consider the properties of minimal sections and dual VMRT.

(g) We will talk about Campana-Peternell conjecture and prove the case $T_X$ is nef, big and $1$-ample.

(h) Consider the most recent work of the conjecture of Fano manifolds with non-isomorphic surjective endomorphism and the most recent work of the conjecture of Fano manifolds with big automorphism group.

(i) Remmert-Van de Ven/Lazarsfeld problem which is also fundamental.

(j) We will again talk about Campana-Peternell conjecture and prove the case $G/B$ and some special cases, such as lower dimension case and large Picard number case.

Here is my notes: VMRT-BasicTheory, updated at 2024-03-25.