Properness of the K-moduli space

These notes were taken during talk given by Ziquan Zhuang (MIT)

The main aim of this talk: Is there a “nice” moduli space of Fano varieties.

Nice means separated, proper and even projective &
Fano means complete variety who’s anticanonical bundle $K_x^*$ is ample.

Yau-Tian-Donaldson conjecture:

• For $X$=Fano variety. Existence of K$\ddot{a}$hler-Einstein (KE) metric is equivalent to $X$ is $k-$polystable (algebraic stability condition).

This motivates the following conjecture:

$k-$Moduli conjecture:

• There exists proper moduli space of $k-$polystable Fano varieties and all connected components are projective.

Next theorem (If I understood it correctly, builds a moduli space)
Theorem:(long list of names who contributed)

1. Fix $n$ (dimension) $\in\mathbb{N}^*$, $v$ (volume of Fano variety) $\in\mathbb{Q}_{>0}$. The moduli functor $$\mathcal{M}^{kss}_{n.inv}(S)=\text{groupoid of flat family of k-semi stable Fano varieties with} ~dim=n, volume=(-k)^n=v ~\text{satisfying Koll\'{a}r's condition}$$ is represented by an Artin stack of finite type, where $S$ is a scheme/$\mathbb{C}$.

This means if we take K-semi stable(kss) Fano varieties of this dimension and volume then all embed into projective space of fixed dimension. Then if we look at Hilbert scheme of that projective space, then locus that parametrizes kss Fano varieties is locally closed subset.

2. $\mathcal{M}^{kss}$ admits a seperated good moduli space $\mathcal{M}^{kps}$ which parametrizes $k-$ polystable Fano varieties.
3. Every proper subspace of $\mathcal{M}^{kps}$ that parametrizes (KE) Fanos is projective.

Theorem (Liu, Xu, -):
$\mathcal{M}^{kps}$ is proper and YTD conjecture holds for any Fano variety implies $K-$moduli conjecture.

Rest in the talk it’s about showing that $\mathcal{M}^{kps}$ is proper.

First, Ziquan describes the Langton’s algorithm for Moduli space of vector bundles.
Now coming back, the main theme for rest of the talk was to define k-stability, maximal destabilizing object for Fano varieties and the way to adapt Lagton’s algorithm for $\mathcal{M}^{Fano}$.

Definition:

k-semi stable(kss)= average divisor in $|-k_x|_Q$ (in Q-linear system) is log canonical singularity.
E=divisor/X(Fano) : E$\subset Y\xrightarrow{\pi} X$ $(\pi \text{is birational})$
$A_\text{X}(E) =log$ discrepancy
$S_\text{X}(E)=$ expected vanishing order
kss $\Leftrightarrow$ $A_\text{X}(E)\geq S_\text{X}(E)$ $\forall E/X$

Optimal destabilizing conjecture(ODC):

If $X$ is not k-semi stable, then there exists $E/X$ such that $\delta(X)=\frac{A_\text{x}(E)}{S_\text{x}(E)}$.

Theorem:
ODC implies $\mathcal{M}^{kps}$ is proper.