--- date: 2022-06-12 18:21 modification date: Sunday 12th June 2022 18:21:43 title: "2022 Talbot Talk Outline V3" aliases: [2022 Talbot Talk Outline V3] tags: projects/talbot-talk status: in-progress🐸 --- 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, $\Ann(\LL) =_? 0$ and why we care. - Discuss Borisov's result relating it to $\psi_n$ - State and sketch Thm A: description of $\K(\mcv)$ 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 $\Ann(\LL)$ - 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 $\bd$. - 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: $\SC \to \K$, i.e. how can we encode/detect scissors congruence in the language of $\K$ theory using assemblers. - Tomorrow: $\K\to \SC$: 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** $X\slice k$ means a reduced separated scheme of finite type over $\spec k$. - 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 $$ \overline{X_{j}} \subset \biguplus_{i \leq j} X_{i} $$ - The parts $X_{i}$ are called the *strata* of the stratification. - $X, Y$ are isomorphic iff they are isomorphic in $\Sch\slice k$. **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 $\Sp$ we use is symmetric spectra of simplicial sets, take stable model structure with levelwise cofibrations. - $\mcv = \mcv_k$ is the aseembler of varieties over $k$ and closed inclusions (locally closed embeddings). - $\K_0(\mcv)$ is the **Grothendieck group of varieties** as in Michael's talk (Talk 7). - $\LL = [\AA^1\slice k]$ is the **Lefschetz motive**, the class of the affine line. - $$\Ann(\LL) \da \ker(\K_0(\mcv) \mapsvia{\cdot \LL} \K_0(\mcv) )$$ where $\cdot \LL$ is the map induced by $X\mapsto X\fiberprod{k}\AA^1\slice k$. - CA fact: $\LL$ is a zero divisor $\iff \Ann(\LL) = 0$. - Examples of working with $\LL$. - If $\mce \to X$ is a rank $n$ vector bundle (Zariski-locally trivial fibration with fibers $\AA^n$) then $[\mce] = [X]\cdot [\AA^n] = [X]\cdot \LL^n$. - $X, Y$ are **birational** iff there is an isomorphism $\phi: U \iso V$ of dense open subschemes. **Write this as $X\birational Y$.** - 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 $X\times \PP^N \birational Y\times \PP^M$ for some $N, M$. **Write this as $X\sbirational Y$.** - Lots of interesting aspects of birational geometry: $h^0(X; \Omega_X), \pi_1(X^\an), \CH_0(X)$ 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 $X\pwiso Y$.** - Think of this as cut-and-paste equivalence for varieties. - Note $X\pwiso Y \implies [X] = [Y] \in \K_0(\mcv)$. - If $X \birational Y$ and additionally $X\sm U \cong Y\sm V$, then $X \pwiso Y$ and $[X] = [Y]$. - ![](attachments/Pasted%20image%2020220612045700.png) ## Motivation Reference: Zak17b, Annihilator of the Lefschetz Motive - Summary of big questions: - When is $\K_0(\mcv) \to \K_0(\mcv)\invert{\LL}$ injective? So are equations in the localization still valid in the original ring? - What does equality in $\K_0(\mcv)$ mean geometrically? What does an equation in this ring mean? - Summary of big structural questions about $\K_0(\mcv)$ we're looking at in this paper: ## Q1: Larsen-Lunts/Gromov, PW Isos - There is a filtration on $\K_0(\mcv_k)$ where $\gr_n$ is induced by the image of $$ \gr_n \K_0(\mcv) = \im\qty{ {\mathbb{Z}\adjoin{X \mid \operatorname{dim} X \leq n} \over \gens{ [X]=[Y]+[X \backslash Y]}} \overset{\psi_n}\longrightarrow \K_{0}(\mcv_k)} $$ - Q, Gromov: if $U,V\injects X$ with $X\sm U \cong X\sm V$, how far are $U$ and $V$ from being birational? - Q, Larsen-Lunts: $[X] = [Y] \overset{???}\implies X\pwiso Y$? - **Answer**: No! Borisov and Karzhemanov construct counterexamples for $k\injects \CC$, Inna shows that this fails for *convenient* fields. - **Conjecture**: this is almost true, and the only obstructions come from $\Ann(\LL)$. - **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 $X\birational Y$ extend to $X\pwiso Y$? ## Q2: $\Ann(\LL) \overset{?}= 0$ - When is $\Ann(\LL)$ nonzero? - Important for motivic measures, rationality questions. - Answer (Borisov): $\LL$ generally **is** a zero divisor, Borisov and Karzhemanov elements in $\Ann(\LL)$ and seemingly coincidentally constructs elements in $\ker \psi_n$. - In case not covered in previous talk - Shows an equality in $\K_0$: ![](attachments/Pasted%20image%2020220612183231.png) - 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 $\Ann(\LL)$ and $\ker \psi_n$ related? ## Outline of Results - Slogans for what's shown in this paper: - **Thm A:** Constructs a stable (filtered) homodtopy type $\K(\mcv)$ where $\gr \K(\mcv)$ is simpler than $\gr \K_0(\mcv)$. - **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 $\Ann(\LL)$ yield elements in $\ker(\psi_n)$. - **Thm D**: Partial characterizations of $\Ann(\LL)$. - **Thm E**: Identification of $\K_0(\mcv)/\gens{\LL}$ in terms of stable birational geometry. - Conclusions: - Elements in $\Ann(\LL)$ always produce elements in $\ker \psi_n$ # Theorems ## Thm A: There is a homotopical enrichment of $\K_0(\mcv)$ with a simple associated graded ```ad-note 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 $\K_0(\mcv)$. ``` ```ad-info 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 ```ad-note 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: ![](attachments/Pasted%20image%2020220614005140.png) Compute $$\begin{align*} \K_p(\mcv^{(n, n-1)}) &\da \pi_p \K(\mcv^{(n, n-1)}) \\ &\homotopic \pi_p \bigvee_{\alpha\in B_n} \Sigma_+^\infty \B \Aut(\alpha) \\ &\cong \bigoplus_{\alpha\in B_n} \pi_p \Sigma_+^\infty \B \Aut(\alpha) \end{align*},$$ and use $\pi_p \Sigma_+^\infty \BG$ is $\ZZ$ for $p=0$ and $G^\ab \oplus C_2$ for $p=2$ to identifty ![](attachments/Pasted%20image%2020220614005528.png) There is a boundary map $\bd$ coming from the connecting map in the LES in homotopy of a pair for the filtration. ```ad-note 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 $$ \bd[\phi] = [X\sm V] - [X\sm U] \quad \in \K_0(\mcv^{(q-1)}) $$ ``` ```ad-info 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. ``` ```ad-info 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 $\bd[\phi] =0$ implies it extends: - $\bd[\phi]$ measures the failure of $\phi$ to extend to a piecewise isomorphism: $$ \bd[\phi] = 0 \implies [X\sm V] = [X\sm U] \implies \exists \psi: X\sm V \pwiso X\sm U $$ - If additionally $U\cong V$ then $\phi \uplus \psi$ assemble to a piecewise automorphism of $X$. ## Thm C: There is a direct link between $\Union_{n\geq 0} \ker \psi_n$ and $\Ann(\LL)$ ```ad-note 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 $$\Ann(\LL)\neq 0 \implies \Union_n \ker \psi_n \neq \emptyset.$$ ``` ```ad-info 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 $\K_0$ doesn't imply PW iso and elements in $\Ann(\LL)$ give rise to elements in $\Union \ker \psi_n$. ```ad-note 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 $\Union \ker \psi_n$. ``` ```ad-info 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: $\K$-theory $\mod \LL$ models stable birational geometry ```ad-note 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 $\Ann(\LL)$? - (Interesting) What is the kernel of the localization $\K_0(\mcv_k) \to \K_0(\mcv_k)\invert{\LL}$? - Does $\psi_n$ fail to be injective over every field $k$? ```ad-question 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 $\LL$. - Possibly contingent upon conjecture: $$[X] \equiv [Y] \mod \LL \implies X \sbirational Y.$$