# Noncommutative minimal model program 

:::{.remark}
Setup:

- $X$ smooth projective over $\CC$,
- $\cat C \da \dbcoh(X)$.

Note that $\cat C$ often has "emergent" structure, e.g. 
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:::{.theorem title="Beilinson"}
There is a full exceptional collection
\[
\dbcoh(\PP^n) = \gens{\OO, \OO(1),\cdots, \OO(n)}
,\]
i.e. these generate the derived category and no $\RHom$ in the reverse direction.
As a result, the cohomology and $K\dash$theory decompose according to these pieces.
:::

:::{.remark}
There isn't much known about the class of varieties that might admit FECs.
The only conjecture in this direction:
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:::{.conjecture title="Dubrovin"}
If $X$ is Fano, $X$ admits a FEC iff the big quantum cohomology $\QH^*(X)$ is generically semisimple.
This is evidenced/motivated by e.g. homological mirror symmetry, but this is not a viable avenue of proof.
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:::{.conjecture title="$D\dash$equivalence"}
If $X\sim X'$ are birationally equivalent Calabi-Yau manifolds, then $\db(X) \iso \db(X')$.
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:::{.remark}
Progress:

- 2000 (Bridgeland): this holds in dimension 3.
- 2020 (HL): this holds if $X$ is a moduli space of twisted sheaves on a K3.  

Can reduce to a single flop, investigate a Fourier-Mukai kernel, variation of Bridgeland stability conditions.
Very moduli-theoretic, so not clear how this might generalize.
Goal: provide a mechanism for these conjectures.
Some key points:

1. Semiorthogonal decompositions of $\db(X)$ arise from paths in the space of Bridgeland stability conditions $\Stab(X) \da \Stab(\db(X))$.
2. Paths converge in a partial compactification of $\Stab(X)/\GG_a$.
3. The noncommutative MMP yields conjectures existence of canonical paths.
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:::{.remark}
Setup: fix $v: \K_0(V) \to \Lambda\da H_{\alg}(X) \subseteq H(X; \CC)$ where $v = (2\pi i)^{\deg\over 2}\chern$.
Compare:

| Semiorthogonal decompositions | Stability condition |
|-------------------------------|---------------------|
| Subcategories $\cat C_1,\cdots, \cat C_n \leq \cat C$ | $\rho_\phi \leq \cat C$ with $\phi\in \RR$ |
| Same | Semiorthogonality of homs |
| Same with $\gr_i E \in \cat C_i$ | Every $E\in \cat C$ has a filtration with $\gr_\phi E \in \rho_\phi$ |
| $\cat C_i[1] = \cat C_i$ | $\rho_\phi[1] = \rho_{\phi + 1}$.

Add additional data of a charge morphism $z: \Lambda\to \CC$ such that $z(\rho_\phi) \subseteq \RR_{\geq 0} e^\phi$.
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:::{.theorem title="?"}
There is a local homeomorphism
\[
\Stab(X) &\to \Hom( \Lambda, \CC) \\
(\rho_\bullet, z) &\mapsto z
,\]
so paths in the latter vector that *do* lift, lift uniquely.
Moreover, quantum cohomology generates paths that might lift.
:::

:::{.lemma title="Key"}
Let $\sigma_t$ be a path in $\Stab(X)$ such that 

1. $\forall E\in \cat C$, the HN filtration is constant for $t\gg 0$.
2. For all eventually semistable $E$, $\log z_t(E) = \alpha_E t + \beta_E + \bigo(1)$.
3. If $\Im(\alpha_E) = \Im( \alpha_F )$ then \( \alpha_E = \alpha_F \).

Then there exists a semiorthogonal decomposition $\cat C = \gens{\cat C_1,\cdots, \cat C_n}$ and $\tl \alpha 1 n \in \CC$ with $\Im(\alpha_i)$ increasing with $i$ such that $\cat C_i$ is generated by eventually semistable objects $E$ with \( \alpha_E = \alpha_i \).
Moreover each $\cat C_i$ has stability condition with $z_i(E) = e^{\beta_E}$.
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:::{.remark}
Look far enough out in $\gr E$ under the HN filtration, set $G_i \da \gr_i E$, then $\phi_{G_i} = \Im( \alpha_{G_i}t + \beta_{G_i} )/\pi$ and the imaginary parts of $\alpha_{G_j}$ are weakly decreasing.
The filtration for the SOD comes from deleting steps in the filtration with the same $\alpha$s.
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:::{.proposition title="Collins-Polishchuk"}
Any SOD of $\cat{C}$ in which all factors admit a stability condition arise from this lemma.
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:::{.slogan}
Categorical birational geometry should be equivalent to the study of "polarizable SODs": SODs of $\db(X)$ where each factor admits stability conditions.
Kähler structures are like ample Néron-Severi classes, and stability conditions are like Kähler structures.
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:::{.proposition title="?"}
Let $\cat C = \gens{\cat C_1,\cdots, \cat C_n}$ be a polarizable SOD and $\dim \K_0(\cat C_i)\tensor \QQ = 1$, then it comes from a FEC.
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:::{.example title="?"}
The Barlow surface $X$ has $\db(X) = \gens{L_1,\cdots L_{10}, \cat A_{L_11}}$ where the last term is an orthogonal complement containing some $L_{11}$, but there are phantoms (invisible to $\HH$ and $\K$) so does not yield a FEC and thus can't admit Bridgeland stability conditions.
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:::{.remark}
Proving: for $E\in \cat {C}$, look at $\log z_t(\gr_j(E))$ as $t\to\infty$. 
Degenerates to some limiting object, *generalized stability conditions*:

1. Marked genus 0 meromorphic differentials,
2. SOD $\cat C = \gens{\tl {\cat C} n }$ labeled by terminal components,
3. $\sigma_i \in \Stab(\cat C_i)/ \GG_a$ for all $i$.

Denote this $\mathrm{GStab}(X)$.
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:::{.conjecture}
$\mathrm{GStab}(X)$ is a manifold with corners.
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:::{.conjecture title="Noncommutative minimal model program"}
A series of conjectures:

1. There exist canonical paths $\sigma_t^\psi \in \Stab(X)/\GG_a$ that satisfy the key lemma for generate $\psi$, and different $\psi$ give mutation-equivalent SODs.
2. $X'\to X$ birational for suitable parameterizations, yielding an SOD for $\db(X)$, where $\db(X') = \gens{ \ker \pi_*, \pi^* \db(X) }$ where the second factor is combined with the canonical SOD of $\db(X)$.

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:::{.proposition title="Existence of a categorical minimal models"}
Assuming these conjectures and that $h^0(K_X) > 0$, there exists a subcategory $M_X \leq \db(X)$ such that if $X' \birationaliso X$ then $M_X \leq \db(X')$.
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:::{.example title="?"}
The Atiyah flop: take blowups/blowdowns along centers which are $\PP^1$, $\roof{X}{Y}{X^+}$, then $\db(Y) = \gens{E_1 , E_2, \db(X)}$.
Then take a mutation to get $\gens{E_2^+, E_1^+, \db(X^+)}$.
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:::{.remark}
Paths satisfy a quantum differential equation.
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