Higher Fano Manifolds

These notes were taken during the talk given by Enrica Mazzon (University of Michigan)

Notation:

$X$: Smooth projective variety
$T_X$: Tangent bundle of $X$
$-K_X$: $c_1(T_X)$ (first Chern class) $\in H^2(X,\mathbb{Z})$, anticanonical class of $X$.

Definition: $X$ is Fano if $-K_X$ is ample.
Remark: $X$ is Fano implies $-K_X\cdot C>0$ for all $C\subseteq X$.

The goal of this talk to explore conditions of positivity on $T_X$ and geometric properties on $X$.

Properties of Fano manifold:

1. $X_d$ (smooth hypersurface of degree d) $\subset \mathbb{P}^n$ and
   $-K_{X_{d}}=(n+1-d)[H|_{X_{d}}]$,

$X_d$ is Fano if and only if $d\leq n$ if and only if $X_d$ is covered by rational curves.

$d\leq n-1$: then $X_d$ is covered by lines
$d\leq n$: then $X_d$ is covered by conics
(Didn’t exactly get the meaning of covering by lines and conics)
$d\geq n+1$: then there does not exist rational curve through general point of $X_d$.

Theorem (Mori): Fano manifolds are covered by rational curves.
Theorem: Fano manifolds are rationally connected.
(meaning for a pair of general points, we ccan find rational curves passing through both the points).

2. Theorem (Tsen, Lang): $\pi: \mathcal{X}\rightarrow B$ family of hypersurfaces of degree $d$ in $C\mathbb{P}^n$ over B smooth variety of dimension $k$.
   $k=1$: $d\leq n$ implies $\pi \ \text{has a section}$
   $k>1$: $d^k\leq n$ implies $\pi \ \text{has a rational section}$

Next is the idea and definition of $k-$Fano manifold.

Definition:
$X$ is Fano is $K-$ Fano if $ch_i(T_X)\cdot Z>0$ where $Z (variety)\subset X$ ($dim Z=i$ where $i=1,2,...,k$)

Remark:

1. $ch_i$ is the $i$th Chern character
2. $X,Y$ are Fano then $X\times Y$ is never 2-Fano.

Main result:

1. Classification of 2-Fano and 3-Fano of
   $\text{index}(i_X)\geq (dimension-2)$ where
   $i_X=max\{m \in \mathbb{Z}| K_X=mA, A\in H^2 \}$

2. Classification of 2-Fano and 3-Fano among rational homogeneous spaces (just Picard rank 1)

Contrapositive statement of the following two theorems
can be used to check if $X$ is 2-Fano or 3-Fano.
$X$ is smooth Fano and $x\in X$ be general point.
Space $\text{Rat Curves}^n(X,x)$- parametrizing rational curves on $X$ through $x$.
$H_x$-proper irreducible component. i.e.curves of minimal degree w.r.t $-K_X$

Theorem:

1. $X$ is 2-Fano, $\text{dim} H_x\geq1$ implies $H_x$ is Fano.
2. $X$ is 3-Fano, $\text{dim}H_x\geq2$ implies $H_x$ is 2-Fano.

Remark:

1. All known examples of 2-Fano have Picard rank 1.
2. All known examples of 3-Fano are complete intersection in weighted projective space.
3. All known examples of 3-Fano rational homogeneous space are $\mathbb{P}^n$ with $n\geq 3$ or quadric $(Q^n)\subset \mathbb{P}^{n+1}$ with $n\geq 7$.
   Following conjectures (if proved) could give characterization of $\mathbb{P}^n$.

Conjecture:

1. Only n-dimensional K-Fano for K=$\lceil{log_2(n+1)}\rceil+1$ is $\mathbb{P}^n$.

2. Only n-dimensional 2-Fano toric manifold is $\mathbb{P}^n$.
   Proved till 8-dimension.