Last modified date: <%+ tp.file.last_modified_date() %>

• Tags:
  • #todo/untagged
• Refs:
  • #todo/add-references
• Links:
  • #todo/create-links

2022 Talbot Talk Outline V3

• Todos and questions to ask
  • Are any particular theorems necessary for later talks?
  • Slogans for theorems
  • Clarifications on proofs: need to mark questions to ask Inna.
• Major goals to hit in talk:
  • Discuss Q1, Larsen-Lunts/Gromov question on piecewise isomorphism
  • Discuss Q2, \(\operatorname{Ann}({\mathbb{L}}) =_? 0\) and why we care.
  • Discuss Borisov’s result relating it to \(\psi_n\)
  • State and sketch Thm A: description of \({\mathsf{K}}({\mathcal{V}})\) and the sseq
  • State and prove Thm B: what the sseq measures
  • State and sketch Thm C: how Q1 and Q2 are linked
  • State and sketch Thm D: partially characterize \(\operatorname{Ann}({\mathbb{L}})\)
  • State Thm E: strong link to birational geometry.
  • Discuss unknowns, open questions, conjectures.
• Things to prove
  • Thm A, if time. Just show the calculation if short on time.
  • Thm B, to get a handle of \(d_r\) and \({{\partial}}\).
    • Possibly skip proof of Lem 3.2 if short on time?
  • Thm C, sketch proof (lots of auxiliary objects)
  • Thm D, maybe okay to skip diagram chase? Emphasize how to get elements in \(\ker \psi_n\).

Preliminaries

• Where we are:
  • Yesterday: classical scissors congruence.
  • Today: \(\mathrm{SC}\to {\mathsf{K}}\), i.e. how can we encode/detect scissors congruence in the language of \({\mathsf{K}}\) theory using assemblers.
  • Tomorrow: \({\mathsf{K}}\to \mathrm{SC}\): enriching motivic measures, generalizing assemblers to other cut-and-paste problems, towards a topological approach on a generalized Hilbert’s 3rd problem.
• Conventions:
  • \(k\) is a field.
  • A variety $X_{/ {k}} $ means a reduced separated scheme of finite type over \(\operatorname{Spec}k\).
  • A stratification of a space \(X\) is given by a partition \(X=\biguplus_{i \in I} X_{i}\) into locally closed subsets over a poset \(I\) such that for each \(j \in I\) we have \begin{align*} \overline{X_{j}} \subset \biguplus_{i \leq j} X_{i} \end{align*}
  • The parts \(X_{i}\) are called the strata of the stratification.
  • \(X, Y\) are isomorphic iff they are isomorphic in ${\mathsf{Sch}}_{/ {k}} $. Write this as \(X\cong Y\).
    • Induced by ring morphisms on an open affine cover. Not quite a morphism of ringed spaces!
  • The model for \({\mathsf{Sp}}\) we use is symmetric spectra of simplicial sets, take stable model structure with levelwise cofibrations.
  • \({\mathcal{V}}= {\mathcal{V}}_k\) is the aseembler of varieties over \(k\) and closed inclusions (locally closed embeddings).
  • \({\mathsf{K}}_0({\mathcal{V}})\) is the Grothendieck group of varieties as in Michael’s talk (Talk 7).
  • \({\mathbb{L}}= [{\mathbf{A}}^1_{/ {k}} ]\) is the Lefschetz motive, the class of the affine line.
  • \begin{align*}\operatorname{Ann}({\mathbb{L}}) \coloneqq\ker({\mathsf{K}}_0({\mathcal{V}}) \xrightarrow{\cdot {\mathbb{L}}} {\mathsf{K}}_0({\mathcal{V}}) )\end{align*} where \(\cdot {\mathbb{L}}\) is the map induced by $X\mapsto X \underset{\scriptscriptstyle {k} }{\times} {\mathbf{A}}^1_{/ {k}} $.
    • CA fact: \({\mathbb{L}}\) is a zero divisor \(\iff \operatorname{Ann}({\mathbb{L}}) = 0\).
• Examples of working with \({\mathbb{L}}\).
  • If \({\mathcal{E}}\to X\) is a rank \(n\) vector bundle (Zariski-locally trivial fibration with fibers \({\mathbf{A}}^n\)) then \([{\mathcal{E}}] = [X]\cdot [{\mathbf{A}}^n] = [X]\cdot {\mathbb{L}}^n\).
• \(X, Y\) are birational iff there is an isomorphism \(\phi: U { \, \xrightarrow{\sim}\, }V\) of dense open subschemes. Write this as \(X\overset{\sim}{\dashrightarrow}Y\).
  • So in equations \(\phi\) is given by rational functions.
  • Birational maps: “almost isomorphisms” which allow not just polynomial but rational functions, and are isomorphisms away from an exceptional set of e.g. poles or a branch locus
  • Motivations: MMP!
• \(X, Y\) are stably birational iff \(X\times {\mathbf{P}}^N \overset{\sim}{\dashrightarrow}Y\times {\mathbf{P}}^M\) for some \(N, M\). Write this as \(X\overset{\sim_{ {\operatorname{Stab}}} }{\dashrightarrow}Y\).
  • Lots of interesting aspects of birational geometry: \(h^0(X; \Omega_X), \pi_1(X^{\mathrm{an}}), {\operatorname{CH}}_0(X)\) are stable birational invariants (see recent 2010s work of Claire Voisin)
• \(X,Y\) are piecewise isomorphic if there are stratifications \(X = \biguplus_{i\in I} X_i\) and \(Y = \biguplus_{i\in I} Y_i\) with each \(X_i \cong Y_i\). Write this as \(X\underset{\mathrm{pw}}{\cong}Y\).
  • Think of this as cut-and-paste equivalence for varieties.
  • Note \(X\underset{\mathrm{pw}}{\cong}Y \implies [X] = [Y] \in {\mathsf{K}}_0({\mathcal{V}})\).\
  • If \(X \overset{\sim}{\dashrightarrow}Y\) and additionally \(X\setminus U \cong Y\setminus V\), then \(X \underset{\mathrm{pw}}{\cong}Y\) and \([X] = [Y]\).
  • ## Motivation

Reference: Zak17b, Annihilator of the Lefschetz Motive

• Summary of big questions:
  • When is \({\mathsf{K}}_0({\mathcal{V}}) \to {\mathsf{K}}_0({\mathcal{V}}){ \left[ { \scriptstyle \frac{1}{{\mathbb{L}}} } \right] }\) injective? So are equations in the localization still valid in the original ring?
  • What does equality in \({\mathsf{K}}_0({\mathcal{V}})\) mean geometrically? What does an equation in this ring mean?
• Summary of big structural questions about \({\mathsf{K}}_0({\mathcal{V}})\) we’re looking at in this paper:

Q1: Larsen-Lunts/Gromov, PW Isos

• There is a filtration on \({\mathsf{K}}_0({\mathcal{V}}_k)\) where \({\mathsf{gr}\,}_n\) is induced by the image of \begin{align*} {\mathsf{gr}\,}_n {\mathsf{K}}_0({\mathcal{V}}) = \operatorname{im}\qty{ {\mathbb{Z} { \left[ \scriptstyle {X \mathrel{\Big|}\operatorname{dim} X \leq n} \right] } \over \left\langle{ [X]=[Y]+[X \backslash Y]}\right\rangle} \overset{\psi_n}\longrightarrow {\mathsf{K}}_{0}({\mathcal{V}}_k)} \end{align*}
• Q, Gromov: if \(U,V\hookrightarrow X\) with \(X\setminus U \cong X\setminus V\), how far are \(U\) and \(V\) from being birational?
• Q, Larsen-Lunts: \([X] = [Y] \overset{???}\implies X\underset{\mathrm{pw}}{\cong}Y\)?
• Answer: No! Borisov and Karzhemanov construct counterexamples for \(k\hookrightarrow{\mathbf{C}}\), Inna shows that this fails for convenient fields.
• Conjecture: this is almost true, and the only obstructions come from \(\operatorname{Ann}({\mathbb{L}})\).
• Conjecture: for certain varieties, \([X] = [Y] \implies X,Y\) are stably birational.
• Encode these as injectivity of \(\psi_n\), so \(\ker \psi_n = 0\) – when does \(X\overset{\sim}{\dashrightarrow}Y\) extend to \(X\underset{\mathrm{pw}}{\cong}Y\)?

Q2: \(\operatorname{Ann}({\mathbb{L}}) \overset{?}= 0\)

• When is \(\operatorname{Ann}({\mathbb{L}})\) nonzero?
  • Important for motivic measures, rationality questions.
• Answer (Borisov): \({\mathbb{L}}\) generally is a zero divisor, Borisov and Karzhemanov elements in \(\operatorname{Ann}({\mathbb{L}})\) and seemingly coincidentally constructs elements in \(\ker \psi_n\).
  • In case not covered in previous talk
  • Shows an equality in \({\mathsf{K}}_0\):
  • Shows that certain bundles over \(X, Y\) are birational, so \(X,Y\) are stably birational
  • Picks a special mirror pair where stably birational implies birational
  • Show the bundles are pw-iso, so stably birational.
  • Use that \(X, Y\) are known not to be birational.
• Q: How and why are \(\operatorname{Ann}({\mathbb{L}})\) and \(\ker \psi_n\) related?

Outline of Results

• Slogans for what’s shown in this paper:
  • Thm A: Constructs a stable (filtered) homodtopy type \({\mathsf{K}}({\mathcal{V}})\) where \({\mathsf{gr}\,}{\mathsf{K}}({\mathcal{V}})\) is simpler than \({\mathsf{gr}\,}{\mathsf{K}}_0({\mathcal{V}})\).
  • Thm B: The associated spectral sequence is an obstruction theory for birational auts extending to pw auts (so detects \(\ker \psi_n\) for various \(n\))
  • Thm C: Q1 and Q2 are linked: elements in \(\operatorname{Ann}({\mathbb{L}})\) yield elements in \(\ker(\psi_n)\).
  • Thm D: Partial characterizations of \(\operatorname{Ann}({\mathbb{L}})\).
  • Thm E: Identification of \({\mathsf{K}}_0({\mathcal{V}})/\left\langle{{\mathbb{L}}}\right\rangle\) in terms of stable birational geometry.
• Conclusions:
  • Elements in \(\operatorname{Ann}({\mathbb{L}})\) always produce elements in \(\ker \psi_n\)

Theorems

Thm A: There is a homotopical enrichment of \({\mathsf{K}}_0({\mathcal{V}})\) with a simple associated graded

title: Theorem
collapse: open


Let 

- $\mcv^{(n)}_k$ be the $n$th filtered assembler of $\mcv$ generated by varieties of dimension $d\leq n$.
- $\Aut_k\, k(X)$ be the group of birational automorphisms of the variety $X$.
- $B_n$ be the set of birational isomorphism classes of varieties of dimension $d=n$.

There is a spectrum $\K(\mcv)$ such that $\K_0(\mcv) \da \pi_0 \K(\mcv)$ coincides with the Grothendieck group of varieties discussed previously, and $\mcv^{(n)}$ induces a filtration on the $\K(\mcv)$ such that
$$
\gr_n \K(\mcv) = \bigvee_{[X]\in B_n} \Sigma^\infty_+ \B\Aut_k\, k(X),
$$
with an associated spectral sequence
$$E_{p, q}^1 = \bigvee_{[X]\in B_n} \qty{\pi_p \Sigma_+^\infty \B \Aut_k\, k(X) \oplus \pi_p \SS} \abuts \K_p(\mcv)$$
Note that the $p=0$ column converges to $\K_0(\mcv)$.

title: Proof
collapse: open


- Define $\mcv^{(n. n-1)} = \Var^{\dim = n}\slice k \union \ts{\emptyset}$, the varieties of dimension *exactly* $n$.
- Zak17b Thm. 1.8: extract cofibers in the filtration to see the associated graded:
  ![](attachments/Pasted%20image%2020220614002012.png)

- Finish by a computation:
  
$$\begin{align*}
\K(\mcv^{(n, n-1)}) 
&\homotopic \tilde \K(\mcv^{(n, n-1)})  \\
&\homotopic \K(\cat C) \\
&\homotopic \K\qty{\bigvee_{\alpha\in B_n} \cat{C}_{X_\alpha}} \\ &\homotopic\bigvee_{\alpha\in B_n}\K(\cat C_{X_\alpha}) \\
&\cong \bigvee_{\alpha\in B_n} \Sigma_+^\infty \B\Aut_k k(X_\alpha) 
 \qquad \text{Zak17a}\\
&\da \bigvee_{\alpha\in B_n} \Sigma_+^\infty \B\Aut(\alpha).
\end{align*}
$$

where

- $\tilde \K(\mcv^{(n, n-1)})$: the full subassembler of irreducible varieties.
    - **Why the reduction works:** general theorem (Zak17b Thm. 1.9) on subassemblers with enough disjoint open covers

- $\cat C \leq \mcv^{(n, n-1)}$: subvarieties of some $X_\alpha$ representing some $\alpha$, as $\alpha$ ranges over $B_n$.
    - **Why the reduction works:** apply (Zak17b Thm. 1.9) again

- $\cat{C}_{X_\alpha}$ is the subassembler of only those varieties admitting a (unique) morphism to $X_\alpha$ for a fixed $\alpha$.
    - **Why the reduction works:** each nonempty variety admits a morphism to exactly one $X_\alpha$ representing some $\alpha$ -- otherwise, if $X\mapsto X_\alpha, X_\beta$ then $X_\alpha$ and $X_\beta$ are forced to be birational (the morphisms are inclusions of dense opens) implying $\alpha = \beta$
    - 
- $\Aut(\alpha) \da \Aut_k k(X)$ for any $X$ representing $\alpha\in B_n$.

Thm B: the spectral sequence measures \(\ker \psi_n\) and how birational morphisms can fail to extend to piecewise isomorphisms

title: Theorem
collapse: open

There exists nontrivial differentials $d_r$ from column 1 to column 0 in some page of $E^* \iff \Union_n \ker \psi_n\neq 0$ ($\psi_n$ has a nonzero kernel for some $n$),

More precisely, $\phi \in \Aut_k k(X)$ extends to a piecewise automorphism $\iff d_r[\phi] = 0 \quad \forall r\geq 1$.

Before proving, a look at this spectral sequence:

Compute \begin{align*}\begin{align*} {\mathsf{K}}_p({\mathcal{V}}^{(n, n-1)}) &\coloneqq\pi_p {\mathsf{K}}({\mathcal{V}}^{(n, n-1)}) \\ &\simeq\pi_p \bigvee_{\alpha\in B_n} \Sigma_+^\infty {\mathbf{B}}\mathop{\mathrm{Aut}}(\alpha) \\ &\cong \bigoplus_{\alpha\in B_n} \pi_p \Sigma_+^\infty {\mathbf{B}}\mathop{\mathrm{Aut}}(\alpha) \end{align*},\end{align*} and use \(\pi_p \Sigma_+^\infty {{\mathbf{B}}G}\) is \({\mathbf{Z}}\) for \(p=0\) and \(G^{\operatorname{ab}}\oplus C_2\) for \(p=2\) to identifty

There is a boundary map \({{\partial}}\) coming from the connecting map in the LES in homotopy of a pair for the filtration.

title: Lemma 3.2 (Let's understand $\K_1$!)
collapse: open

If $\phi\in \Aut(\alpha)$ for $\alpha\in B_q$ is represented by $\phi: U\to V$ then 
$$
\bd[\phi] = [X\sm V] - [X\sm U] \quad \in \K_0(\mcv^{(q-1)})
$$

title: Proof of Lemma
collapse: open

- In general, $x\in \K_1(\mcv^{(q, q-1) })$ corresponds to data: $X$ a variety, a dense open subset embedded in two different ways, and the two possible complements:
  ![](attachments/Pasted%20image%2020220612045307.png)
  
- (ZakB Prop 3.13) shows that for this data,
  $$\bd[x] = [Z] - [Y] \in \K_0(\mcv^{(q-1)})$$
- For $\phi$, we can represent it with the data:
  ![](attachments/Pasted%20image%2020220612045448.png)
- Then $\bd[\phi] = [Z] - [Y] = [X\sm V] - [X\sm U]$ as desired.

title: Proof of Theorem
collapse: open

$\implies$: suppose $\phi$ extends to a piecewise automorphism.

- Then $[X\sm U] = [X\sm V]\in \K_0(\mcv^{q-1})$ since $X\sm U\iso X\sm V$ by assumption
- By Lem 3.2 above, $$\bd [\phi] = [X\sm V] - [X\sm U] = 0$$
- (Zak17B Lemma 2.1): $d_1$ and higher $d_r$ are built using $\bd$, so $\bd(x) = 0 \implies d_r(x) = 0$ for all $r\geq 1$ (permanent boundary).

$\impliedby$: suppose $d_r[\phi] = 0$ for all $r\geq 1$.

- Since $d_1[\phi] = 0$ in particular, 
$$[X\sm U] = [X\sm V]\in \K_0(\mcv^{(q, q-1)})$$ 
since $d_1 = \bd \circ p$ for some map $p$.
- An inductive argument allows one to write $X = U_r \uplus X_r' = V_r \uplus Y_r'$ where
$$
U_r \pwiso V_r, \quad \dim X_r',\, \dim Y_r' < n-r, \quad \bd[\phi] = [Y_r'] - [X_r']
$$
- Take $r=n$ to get 
  $$
  \dim X_n', \dim Y_n' < 0 \implies X_n' = Y_n' = \emptyset \quad\text{and}\quad X = U_n = V_n
  $$
- Then 
  $$\bd[\phi] = [\emptyset] - [\emptyset] = 0 \implies \phi \text{extends}.$$

• A general remark on why \({{\partial}}[\phi] =0\) implies it extends:
  • \({{\partial}}[\phi]\) measures the failure of \(\phi\) to extend to a piecewise isomorphism: \begin{align*} {{\partial}}[\phi] = 0 \implies [X\setminus V] = [X\setminus U] \implies \exists \psi: X\setminus V \underset{\mathrm{pw}}{\cong}X\setminus U \end{align*}
  • If additionally \(U\cong V\) then \(\phi \uplus \psi\) assemble to a piecewise automorphism of \(X\).

Thm C: There is a direct link between \(\bigcup_{n\geq 0} \ker \psi_n\) and \(\operatorname{Ann}({\mathbb{L}})\)

title: Theorem C
collapse: open

Let $k$ be a **convenient field**, e.g. $\characteristic k = 0$.
Then $\LL$ is a zero divisor in $\K_0(\mcv)$ $\implies \psi_n$ is not injective for some $n$.

Short: For $k$ convenient
$$\Ann(\LL)\neq 0 \implies \Union_n \ker \psi_n \neq \emptyset.$$

title: Proof
collapse: open


- Strategy: contrapositive. Suppose $\ker \psi_n = 0$ for all $n$. Write $\mcv \da \mcv_k$.
- There is a cofiber sequence 
  $$\K(\mcv) \injectsvia{\cdot \LL} \K(\mcv) \surjectsvia{\ell} \K(\mcv/\LL)
  $$
  where $\mcv/\LL$ is a "cofiber assembler" (Zak17b Def 1.11)
- Take the LES to identify $\ker(\cdot \LL)$ with $\coker(\ell)$:
  ![](attachments/Pasted%20image%2020220612041242.png)
- Reduce to analyzing 
  $$\coker(E_{1, q}^\infty \to \tilde E_{1, q}^\infty )$$
  where $\tilde E$ is an auxiliary sseq.

- Suppose all $\alpha$ extend, then all differentials from column 1 to column 0 are zero.
- The map $E^r \to \tilde E^r$ is surjective for all $r$ on all components that survive to $E^\infty$.
- All differentials out of these componenets are zero, so $E^\infty \surjects \tilde E^\infty$.
- Then $\K_1(\mcv) \surjectsvia{\ell} \K_1(\mcv/\LL)$, making $0 = \coker(\ell) = \ker(\cdot \LL)$ so $\LL$ is not a zero divisor.

Thm D: Equality in \({\mathsf{K}}_0\) doesn’t imply PW iso and elements in \(\operatorname{Ann}({\mathbb{L}})\) give rise to elements in \(\bigcup\ker \psi_n\).

title: Theorem
collapse: open

Suppose that $k$ is a *convenient* field. If $\chi \in \Ann(\LL)$ then $\chi = [X]-[Y]$ where 
$$\left[X \times \mathbb{A}^{1}\right]=\left[Y \times \mathbb{A}^{1}\right] \quad \text{but }
X \times \mathbb{A}^{1}\not \pwiso Y \times \mathbb{A}^{1}
.$$
Thus elements in $\Ann(\LL)$ give rise to elements in $\Union \ker \psi_n$.

title: Proof (can omit)
collapse: open


- Let $\chi \in \ker(\cdot \LL)$ and pullback in the LES to $x \in \K(\mcv^{(n)}/\LL)$ where $n$ is minimal among filtration degrees:
  ![](attachments/Pasted%20image%2020220612042212.png)
- Write $\bd[x] = [X] - [Y]$ with $X,Y$ of minimal dimension.
- By (LS10 Cor 5),
  $$\begin{align*}
  [X\times \AA^1] = [Y\times \AA^1] 
  &\implies \dim X + 1 = \dim Y + 1 \\
  &\implies \dim X = \dim Y = d
  \end{align*}$$
- Claim: $d$ is small: $d < n-1$.
- Done if this claim is true: proceed by showing $X$ and $Y$ are not piecewise isomorphic by showing $\ker \psi_n$ is nontrivial by a diagram chase.

Proving the claim:

- **Claim**: If $\LL([X] - [Y]) \in \ker ?$ then we can produce an element in $\ker \psi_n$.
    - Diagram chase:
      ![](attachments/Pasted%20image%2020220612042531.png)
      1. $[X] - [Y] \not \in \im(\bd)$ by the minimality of $n$ for $x$, noting $\bd [x] = [X] - [Y]$.
      2. By exactness $\im \bd = \ker(\cdot \LL)$, so $\LL([X] - [Y]) \neq 0$.
      3. By choice of $n$, $i_*(\LL([X] - [Y])) \in  \im \bd = \ker(\cdot \LL)$ in bottom row, so $\LL([X] - [Y]) = 0$ in bottom-right.
      4. Commutativity forces $\LL([X] - [Y]) \in \ker i_*^{n-1}$.
  - Thus $\LL([X] - [Y])$ corresponds to an element in $\ker \psi_n$. (???)

Thm E: \({\mathsf{K}}\)-theory \(\operatorname{mod}{\mathbb{L}}\) models stable birational geometry

title: Theorem
collapse: open

There is an isomorphism
$$
\K_0(\mcv_\CC)/\gens{\LL} \iso \ZZ[\SB_\CC] \qquad \in \zmod.
$$

Proof: omitted.

Closing Remarks

• What did we accomplish:
  • Established a precise relationship between Q1 and Q2.
• Unknowns:
  • What fields are convenient?
  • What is the associated graded for the filtration induced by \(\psi_n\)?
  • Is there a characterization of \(\operatorname{Ann}({\mathbb{L}})\)?
  • (Interesting) What is the kernel of the localization \({\mathsf{K}}_0({\mathcal{V}}_k) \to {\mathsf{K}}_0({\mathcal{V}}_k){ \left[ { \scriptstyle \frac{1}{{\mathbb{L}}} } \right] }\)?
  • Does \(\psi_n\) fail to be injective over every field \(k\)?

title: Conjecture (A Correction to Q1 on $\ker \psi_n$)
collapse: open

Conjecture. Suppose that $X$ and $Y$ are varieties over a convenient field $k$ such that $[X]=[Y]$ in $K_{0}\left(\mathcal{V}_{k}\right)$. Then there exist varieties $X^{\prime}$ and $Y^{\prime}$ such that $\left[X^{\prime}\right] \neq\left[Y^{\prime}\right],\left[X^{\prime} \times \mathbb{A}^{1}\right]=\left[Y^{\prime} \times \mathbb{A}^{1}\right]$, and $X \mathrm{I}\left(X^{\prime} \times \mathbb{A}^{1}\right)$ is piecewise isomorphic to $Y \mathrm{I}\left(Y^{\prime} \times \mathbb{A}^{1}\right)$

Short: If $[X] = [Y]$, there exist $X', Y'$ st

- $[X'] \neq [Y']$
- $[X'\times \AA^1] = [X']\LL = [Y']\LL = [Y'\times \AA^1]$
- $X\disjoint X'\times \AA^1 \pwiso Y\disjoint Y'\times \AA^1$

• If the conjecture holds, when \(X, Y\) are not birational but are stably birational, then the error of birationality is measured by a power of \({\mathbb{L}}\).
• Possibly contingent upon conjecture: \begin{align*}[X] \equiv [Y] \operatorname{mod}{\mathbb{L}}\implies X \overset{\sim_{ {\operatorname{Stab}}} }{\dashrightarrow}Y.\end{align*}