--- date: 2022-06-12 03:03 modification date: Sunday 12th June 2022 03:03:14 title: "2022 Talbot Talk Outline V2" aliases: [2022 Talbot Talk Outline V2] tags: projects/talbot-talk status: completešŸ¤  --- --- Last modified date: <%+ tp.file.last_modified_date() %> --- - Tags: - #todo/untagged - Refs: - #todo/add-references - Links: - #todo/create-links --- # 2022 Talbot Talk Outline V2 ## 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 $\spec k$. - $\mcv_k$ denotes the **assembler** whose objects are $\Var\slice k$ and whose morphisms are locally closed embeddings - $\K_0(\mcv_k)$ is the **Grothendieck group of varieties** as in previous talks. - $\LL = [\AA^1\slice k]$ is the **Lefschetz motive**, the class of the affine line. - $\Ann(\LL) \da \ker(\K_0(\mcv_k) \mapsvia{\cdot \LL} \K_0(\mcv_k) )$. Note that $\LL$ is a zero divisor $\iff \Ann(\LL) = 0$. - Examples of working with $\LL$. - $[\AA^n] = \LL^n$ - $[\PP^n] = 1 + \LL + \cdots + \LL^n$. - If $\mce \to X$ is a rank $n$ vector bundle (Zariski-locally trivial fibration with fibers $\AA^n$) then $[\mce] = [X]\cdot [\AA^n] = [X]\cdot \LL^n$. ## Q1: Larsen-Lunts and piecewise isomorphism - Quasiprojective varieties $X,Y$ are **piecewise isomorphic** if there are stratifications $X = \disjoint_{i\in I} X_i$ and $Y = \disjoint_{i\in I} Y_i$ with each $X_i \cong Y_i$. Write this as $X\sim Y$. - Think of this as cut-and-paste equivalence for varieties. - $X\sim Y \implies [X] = [Y] \in \K_0(\mcv_k)$. - **Question** (Larsen-Lunts): Is the converse true? What can generally be said if $[X] = [Y]$? - Applications: rationality of motivic zeta functions (motivic versions of Weil conjectures?) - Answer: No! Borisov and Karzhemanov construct counterexamples for $k\injects \CC$, Inna shows for a certain class of fields including $\characteristic k = 0$. - **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**: - $X \birational Y$ are **birational** iff there is an isomorphism $\phi: U \iso V$ of dense open subschemes, so in equations $\phi$ is given by rational functions. Note that if $X, Y$ are birational and additionally $X\sm U \cong Y\sm V$, then $X, Y$ are piecewise isomorphic. - $X, Y$ are **stably birational** iff $X\times \PP^N \birational Y\times \PP^M$ for some $N, M$. - If $X, Y$ are not birational but are stably birational, then the error of birationality is measured by a power of $\LL$. - 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 - Lots of interesting aspects of birational geometry: $h^0(X; \Omega_X), \pi_1(X^\an), \CH_0(X)$ are stable birational invariants, birational classification of e.g. surfaces, MMP, etc... ## Q2: When is $\Ann(\LL) = 0$? I.e. when (if ever) is $\LL$ a zero divisor? - There is a filtration on $\K_0(\mcv_k)$ where $\gr_n$ is induced by the image of $$\psi_{n}: {\mathbb{Z}\adjoin{X \mid \operatorname{dim} X \leq n} \over \gens{ [X]=[Y]+[X \backslash Y]}} \longrightarrow \K_{0}(\mcv_k)$$ - Gromov, Larsen-Lunts ask if $\psi_n$ is injective, which is equivalent to Q1 - Answer (Borisov): $\LL$ generally **is** a zero divisor, Borisov constructs elements in $\Ann(\LL)$ and seemingly coincidentally constructs elements in $\ker \psi_n$. - How and why are $\Ann(\LL)$ and $\ker \psi_n$ related? - $\Ann(\LL)$ interesting for other reasons: Kontsevich's motivic integral takes values in $\K_o(\mcv_k)$ but is only well-defined up to powers of $\LL$, so in $\K_0(\mcv_k)\invert{\LL}$. - Commutative algebra fact: $R\to S\inv R$ is injective iff $S$ contains no zero divisors! ## This Paper - Summary of big questions: - When is $\K(\mcv_k) \to \K(\mcv_k)\invert{\LL}$ injective? - What does equality in $\K(\mcv_k)$ mean geometrically? - Summary of big questions we're looking at in this paper: - When $\psi_n$ injective, so that we can understand the filtration + grading on $\K_0(\mcv_k)$? - Important for detecting piecewise isomorphisms and for stable birational geometry. - When is $\Ann(\LL)$ nonzero? - Important for motivic measures, rationality questions. - How are $\psi_n$ and $\Ann(\LL)$ related? - What Inna shows: - **Thm A:** Constructs a stable (filtered) homotopy type $\K(\mcv)$ where $\gr_n$ is simpler. - **Thm B**: The natural spectral sequence arising from this filtered spectrum characterizes when some $\ker \psi_n$ is nonzero. - **Thm C:** Q1 and Q2 are linked: elements in $\Ann(\LL)$ always 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. - Unknowns: - What is the associated graded for the filtration induced by $\psi_n$? - What is the kernel of the localization $\K_0(\mcv_k) \to \K_0(\mcv_k)\invert{\LL}$? - Is there a characterization of $\Ann(\LL)$? - What fields are convenient? # The Work! ## Theorem A: The Splitting Let - $\mcv_k$ be the category of varieties over $k$ and closed inclusions. - $\mcv^{(n)}_k$ be the $n$th filtered subcategory of $\mcv_k$ 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 $\K(\mcv_k)$ such that $\K_0(\mcv_k) \da \pi_0 \K(\mcv_k)$ coincides with the Grothendieck group of varieties discussed previously, and $\mcv_k^{(n)}$ induces a filtration on the $\K(\mcv_k)$ such that $$ \gr^n = \bigvee_{[X]\in B_n} \Sigma^\infty_+ \B\Aut_k\, k(X), $$ with an associated spectral sequence $$E_{p, q}^1 = \bigoplus_{[X]\in B_n} \qty{\pi_p \Sigma_+^\infty \B \Aut_k\, k(X) \oplus \pi_p \SS} \abuts \K_p(\mcv_k)$$ Note that the $p=0$ column converges to $\K_0(\mcv_K)$. #### Proof - Define $\mcv^{(n. n-1)} = \Var^{\dim = n}\slice k \union \ts{\emptyset}$, the varieties of dimension *exactly* $n$. - Thm. 1.8: $\cofib\qty{\K(\mcv^{(n-1)}) \injectsvia{\K(\iota_n)} \K(\mcv^{(n)})} = \K(\mcv^{(n, n-1)})$. - Reduction: STS $\K(\mcv^{(n)}) \homotopic \bigvee_{\alpha\in B_n} \Sigma_+^\infty \B\Aut(\alpha)$ where $\Aut(\alpha) \da \Aut_k k(X)$ for any $X$ representing $\alpha\in B_n$. - Todo: why STS? - Define $\tilde \K(\mcv^{(n, n-1)})$ to be the full subassembler of *irreducible* varieties. - Thm. 1.9: If $D\leq C$ is a subassembler st every object in $C$ admits a finite disjoint covering family by objects in $D$, then $D\injects C$ induces a homotopy equivalence $\K(D) \homotopic \K(C)$. - Applies here since varieties can be covered by irreducibles. - So $\tilde \K(\mcv^{(n, n-1)}) \homotopic \K(\mcv^{(n, n-1)})$ - Reduce further: $\tilde \K(\mcv^{(n, n-1)}) \homotopic \K(\cat C)$ where $\cat C \leq \mcv^{(n, n-1)}$ are only the subvarieties of some $X_\alpha$ representing some $\alpha$, as $\alpha$ ranges over $B_n$. - Decompose: 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 $\alpha = \beta$ - So $\cat C = \bigvee_{\alpha\in B_n} \cat{C}_{X_\alpha}$ where $\cat{C}_{X_\alpha}$ is the subassembler of only those varieties admitting a (unique) morphism to $X_\alpha$ - Now just a computation: $$ \K(\cat C) \homotopic \K\qty{\bigvee_{\alpha\in B_n} \cat{C}_{X_\alpha}} \homotopic\bigoplus_{\alpha\in B_n}\K(\cat C_{X_\alpha}) \cong \bigoplus_{\alpha\in B_n} \Sigma_+^\infty \B\Aut_k k(X_\alpha) \cong \bigoplus_{\alpha\in B_n} \Sigma_+^\infty \B\Aut(\alpha). $$ ### Setup for Thm B - The cofiber sequence $\K(\mcv^{q-1}) \to \K(\mcv^{q}) \to \K(\mcv^{q, q-1})$ yields a LES with a boundary map $\bd$: ![](attachments/Pasted%20image%2020220612045043.png) ##### Lemma 3.2: Representing $\bd$ ##### Proof of Lemma 3.2 (useful for understanding $\K_1$) - Informally, $X\in \K_1(\mcv^{(q, q-1) })$ corresponds to data: ![](attachments/Pasted%20image%2020220612045307.png) - By ZakB (2015, Prop 3.13), $\bd[X] = [Z] - [Y] \in \K_0(\mcv^{(q-1)})$ - 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. ![](attachments/Pasted%20image%2020220612045700.png) ### Theorem B: The spectral sequence and $\psi_n$ There exists nontrivial differentials from column 1 to column 0 in some page of $E^*$ iff $\psi_n$ has a nonzer kernel for some $n$. More precisely, $\phi \in \Aut_k k(X)$ extends to a piecewise automorphism $\iff$ $d_r[\phi] = 0$ for all $r\geq 1$. > Todo: why are these related? #### Proof - Notation: write $A \uplus B$ for disjoint unions to distinguish from taking a coproduct. - Let $X \in \Var_k^{\dim = q, \irr}$, then $X$ is represented by a class in $B_q$ - Let $\phi: X\birational X$ be defined by $\phi: U\iso V$ and write $X = U \uplus (X\sm U) = V \uplus (X\sm V)$. - $\implies$: 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 **???,** $\bd [\phi] = [X\sm V] - [X\sm U] = 0$ by the prev step. - By Lemma 2.1, $d_r[\phi] = 0$ for all $r\geq 1$. - **Todo: state Lemma 2.1**. - $\impliedby$: 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 ???. - An inductive argument allows one to write $X = U_r \uplus X_r' = V_r \uplus Y_r'$ where - $U_r \cong V_r$ are piecewise isomorphic - $\dim X_r'$ and $\dim Y_r' < n-r$. - $\bd[\phi] = [Y_r'] - [X_r']$ - Take $r=n$ to get $\dim X_n', \dim Y_n' < 0 \implies X_n' = Y_n' = \emptyset$ and $X = U_n = V_n$ - Then $\bd[\phi] = [\emptyset] - [\emptyset] = 0$ and $\phi$ extends - A general remark on why: $\bd[\phi]$ measures the failure of $\phi$ to extend to a piecewise isomorphism: - $\bd[\phi] = 0 \implies [X\sm V] = [X\sm U]$ - $\implies X\sm V \cong X\sm U$ are piecewise isomorphic via some map $\psi$. - If additionally $U\cong V$ then $\phi \uplus \psi$ assemble to a piecewise automorphism of $X$. ### Theorem C Let $k$ be a **convenient field**, e.g. $\characteristic k = 0$. Then $\LL$ is a zero divisor in $\K_0(\mcv_k)$ $\implies \psi_n$ is not injective for some $n$. #### Proof - Strategy: contrapositive. Suppose $\ker \psi_n = 0$ for all $n$. Write $\mcv \da \mcv_k$. - There is a cofiber sequence $\K(\mcv) \mapsvia{\cdot \LL} \K(\mcv) \mapsvia{\ell} \K(\mcv/\LL)$. - **Todo: what is $\mcv/\LL$?** - Take the LES to identify $\ker(\cdot \LL)$ with $\coker(\ell)$: ![](attachments/Pasted%20image%2020220612041242.png) - Reduce to analyzing $\coker(E_{1, q}^\infty \to \tilde E_{1, q}^\infty )$ - **Todo: what is $\tilde E_{1, q}^\infty$?** - Lemma 5.4 shows $\tilde E_{1, q}^\infty$ is a quotient of $\qty{\bigoplus_{\beta\in B_n} \pi_1 \tilde C_\beta} \bigoplus_{B_n\sm\ell(B_n)} C_2$ - Cor. 5.5 shows $E_{1, n}^1 \to \tilde E_{1, n}^1$ is surjective. - 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.prove ### Theorem D Let $k$ be a convenient field, then if $\chi \in \Ann(\LL)$ then $\chi = [X] - [Y]$ where $[X\times \AA^1] = [Y\times \AA^1]$ *but* these are not piecewise isomorphic. Moreover, ??? is in $\ker(\psi_n)$ ??? #### Proof - 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. - Note $[X\times \AA^1] = [Y\times \AA^1] \implies \dim X + 1 = \dim Y + 1 \implies \dim X = \dim Y = d$ (See LS10 Cor 5). - Claim: $d < n-1$. - 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. - **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 $\ker \psi_n$. (???) ### Theorem E There is an isomorphism $$ \K_0(\mcv_\CC)/\gens{\LL} \iso \ZZ[\SB_\CC] \qquad \in \zmod $$ Proof: omitted. Remark: so $[X] \equiv [B] \mod \LL \implies$ $X$ and $Y$ are stably birational?