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2022 Talbot Talk Outline V2

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 $\operatorname{Spec}k$.
- ${\mathcal{V}}_k$ denotes the assembler whose objects are ${\mathsf{Var}}_{/ {k}} $ and whose morphisms are locally closed embeddings
- ${\mathsf{K}}_0({\mathcal{V}}_k)$ is the Grothendieck group of varieties as in previous talks.
- ${\mathbb{L}}= [{\mathbf{A}}^1_{/ {k}} ]$ is the Lefschetz motive, the class of the affine line.
- $\operatorname{Ann}({\mathbb{L}}) \coloneqq\ker({\mathsf{K}}_0({\mathcal{V}}_k) \xrightarrow{\cdot {\mathbb{L}}} {\mathsf{K}}_0({\mathcal{V}}_k) )$. Note that ${\mathbb{L}}$ is a zero divisor $\iff \operatorname{Ann}({\mathbb{L}}) = 0$.
Examples of working with ${\mathbb{L}}$.
- $[{\mathbf{A}}^n] = {\mathbb{L}}^n$
- $[{\mathbf{P}}^n] = 1 + {\mathbb{L}}+ \cdots + {\mathbb{L}}^n$.
- If ${\mathcal{E}}\to X$ is a rank $n$ vector bundle (Zariski-locally trivial fibration with fibers ${\mathbf{A}}^n$) then $[{\mathcal{E}}] = [X]\cdot [{\mathbf{A}}^n] = [X]\cdot {\mathbb{L}}^n$.

Q1: Larsen-Lunts and piecewise isomorphism

Quasiprojective varieties $X,Y$ are piecewise isomorphic if there are stratifications $X = {\textstyle\coprod}_{i\in I} X_i$ and $Y = {\textstyle\coprod}_{i\in I} Y_i$ with each $X_i \cong Y_i$. Write this as $X\sim Y$.
- Think of this as cut-and-paste equivalence for varieties.
$X\sim Y \implies [X] = [Y] \in {\mathsf{K}}_0({\mathcal{V}}_k)$.
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 $k\hookrightarrow{\mathbf{C}}$, Inna shows for a certain class of fields including $\operatorname{ch}k = 0$.
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:
- $X \overset{\sim}{\dashrightarrow}Y$ are birational iff there is an isomorphism $\phi: U { \, \xrightarrow{\sim}\, }V$ of dense open subschemes, so in equations $\phi$ is given by rational functions. Note that if $X, Y$ are birational and additionally $X\setminus U \cong Y\setminus V$, then $X, Y$ are piecewise isomorphic.
- $X, Y$ are stably birational iff $X\times {\mathbf{P}}^N \overset{\sim}{\dashrightarrow}Y\times {\mathbf{P}}^M$ 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 ${\mathbb{L}}$.
- 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: $h^0(X; \Omega_X), \pi_1(X^{\mathrm{an}}), {\operatorname{CH}}_0(X)$ are stable birational invariants, birational classification of e.g. surfaces, MMP, etc…

Q2: When is $\operatorname{Ann}({\mathbb{L}}) = 0$? I.e. when (if ever) is ${\mathbb{L}}$ a zero divisor?

There is a filtration on ${\mathsf{K}}_0({\mathcal{V}}_k)$ where ${\mathsf{gr}\,}_n$ is induced by the image of \begin{align*}\psi_{n}: {\mathbb{Z} { \left[ \scriptstyle {X \mathrel{\Big|}\operatorname{dim} X \leq n} \right] } \over \left\langle{ [X]=[Y]+[X \backslash Y]}\right\rangle} \longrightarrow {\mathsf{K}}_{0}({\mathcal{V}}_k)\end{align*}
Gromov, Larsen-Lunts ask if $\psi_n$ is injective, which is equivalent to Q1
Answer (Borisov): ${\mathbb{L}}$ generally is a zero divisor, Borisov constructs elements in $\operatorname{Ann}({\mathbb{L}})$ and seemingly coincidentally constructs elements in $\ker \psi_n$.
- How and why are $\operatorname{Ann}({\mathbb{L}})$ and $\ker \psi_n$ related?
$\operatorname{Ann}({\mathbb{L}})$ interesting for other reasons: Kontsevich’s motivic integral takes values in ${\mathsf{K}}_o({\mathcal{V}}_k)$ but is only well-defined up to powers of ${\mathbb{L}}$, so in ${\mathsf{K}}_0({\mathcal{V}}_k){ \left[ { \scriptstyle \frac{1}{{\mathbb{L}}} } \right] }$.
- Commutative algebra fact: $R\to S^{-1}R$ is injective iff $S$ contains no zero divisors!

This Paper

Summary of big questions:
- When is ${\mathsf{K}}({\mathcal{V}}_k) \to {\mathsf{K}}({\mathcal{V}}_k){ \left[ { \scriptstyle \frac{1}{{\mathbb{L}}} } \right] }$ injective?
- What does equality in ${\mathsf{K}}({\mathcal{V}}_k)$ 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 ${\mathsf{K}}_0({\mathcal{V}}_k)$?
  - Important for detecting piecewise isomorphisms and for stable birational geometry.
- When is $\operatorname{Ann}({\mathbb{L}})$ nonzero?
  - Important for motivic measures, rationality questions.
- How are $\psi_n$ and $\operatorname{Ann}({\mathbb{L}})$ related?
What Inna shows:
- Thm A: Constructs a stable (filtered) homotopy type ${\mathsf{K}}({\mathcal{V}})$ where ${\mathsf{gr}\,}_n$ 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 $\operatorname{Ann}({\mathbb{L}})$ always 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.
Unknowns:
- What is the associated graded for the filtration induced by $\psi_n$?
- What is the kernel of the localization ${\mathsf{K}}_0({\mathcal{V}}_k) \to {\mathsf{K}}_0({\mathcal{V}}_k){ \left[ { \scriptstyle \frac{1}{{\mathbb{L}}} } \right] }$?
- Is there a characterization of $\operatorname{Ann}({\mathbb{L}})$?
- What fields are convenient?

The Work!

Theorem A: The Splitting

Let

${\mathcal{V}}_k$ be the category of varieties over $k$ and closed inclusions.
${\mathcal{V}}^{(n)}_k$ be the $n$th filtered subcategory of ${\mathcal{V}}_k$ generated by varieties of dimension $d\leq n$.
$\mathop{\mathrm{Aut}}_k\, k(X)$ 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 ${\mathsf{K}}({\mathcal{V}}_k)$ such that ${\mathsf{K}}_0({\mathcal{V}}_k) \coloneqq\pi_0 {\mathsf{K}}({\mathcal{V}}_k)$ coincides with the Grothendieck group of varieties discussed previously, and ${\mathcal{V}}_k^{(n)}$ induces a filtration on the ${\mathsf{K}}({\mathcal{V}}_k)$ such that \begin{align*} {\mathsf{gr}\,}^n = \bigvee_{[X]\in B_n} \Sigma^\infty_+ {\mathbf{B}}\mathop{\mathrm{Aut}}_k\, k(X), \end{align*} with an associated spectral sequence \begin{align*}E_{p, q}^1 = \bigoplus_{[X]\in B_n} \qty{\pi_p \Sigma_+^\infty {\mathbf{B}}\mathop{\mathrm{Aut}}_k\, k(X) \oplus \pi_p {\mathbb{S}}} \Rightarrow{\mathsf{K}}_p({\mathcal{V}}_k)\end{align*} Note that the $p=0$ column converges to ${\mathsf{K}}_0({\mathcal{V}}_K)$.

Proof

Define ${\mathcal{V}}^{(n. n-1)} = {\mathsf{Var}}^{\dim = n}_{/ {k}} \cup\left\{{\emptyset}\right\}$, the varieties of dimension exactly $n$.
Thm. 1.8: ${\operatorname{cofib}}\qty{{\mathsf{K}}({\mathcal{V}}^{(n-1)}) \xhookrightarrow{{\mathsf{K}}(\iota_n)} {\mathsf{K}}({\mathcal{V}}^{(n)})} = {\mathsf{K}}({\mathcal{V}}^{(n, n-1)})$.
Reduction: STS ${\mathsf{K}}({\mathcal{V}}^{(n)}) \simeq\bigvee_{\alpha\in B_n} \Sigma_+^\infty {\mathbf{B}}\mathop{\mathrm{Aut}}(\alpha)$ where $\mathop{\mathrm{Aut}}(\alpha) \coloneqq\mathop{\mathrm{Aut}}_k k(X)$ for any $X$ representing $\alpha\in B_n$.
- Todo: why STS?
Define $\tilde {\mathsf{K}}({\mathcal{V}}^{(n, n-1)})$ 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 $D\hookrightarrow C$ induces a homotopy equivalence ${\mathsf{K}}(D) \simeq{\mathsf{K}}(C)$.
- Applies here since varieties can be covered by irreducibles.
- So $\tilde {\mathsf{K}}({\mathcal{V}}^{(n, n-1)}) \simeq{\mathsf{K}}({\mathcal{V}}^{(n, n-1)})$
Reduce further: $\tilde {\mathsf{K}}({\mathcal{V}}^{(n, n-1)}) \simeq{\mathsf{K}}(\mathsf{C})$ where $\mathsf{C} \leq {\mathcal{V}}^{(n, n-1)}$ 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 $\mathsf{C} = \bigvee_{\alpha\in B_n} \mathsf{C}_{X_\alpha}$ where $\mathsf{C}_{X_\alpha}$ is the subassembler of only those varieties admitting a (unique) morphism to $X_\alpha$
Now just a computation: \begin{align*} {\mathsf{K}}(\mathsf{C}) \simeq{\mathsf{K}}\qty{\bigvee_{\alpha\in B_n} \mathsf{C}_{X_\alpha}} \simeq\bigoplus_{\alpha\in B_n}{\mathsf{K}}(\mathsf{C}_{X_\alpha}) \cong \bigoplus_{\alpha\in B_n} \Sigma_+^\infty {\mathbf{B}}\mathop{\mathrm{Aut}}_k k(X_\alpha) \cong \bigoplus_{\alpha\in B_n} \Sigma_+^\infty {\mathbf{B}}\mathop{\mathrm{Aut}}(\alpha). \end{align*}

Setup for Thm B

The cofiber sequence ${\mathsf{K}}({\mathcal{V}}^{q-1}) \to {\mathsf{K}}({\mathcal{V}}^{q}) \to {\mathsf{K}}({\mathcal{V}}^{q, q-1})$ yields a LES with a boundary map ${{\partial}}$:

Lemma 3.2: Representing ${{\partial}}$

Proof of Lemma 3.2 (useful for understanding ${\mathsf{K}}_1$)

Informally, $X\in {\mathsf{K}}_1({\mathcal{V}}^{(q, q-1) })$ corresponds to data:
By ZakB (2015, Prop 3.13), ${{\partial}}[X] = [Z] - [Y] \in {\mathsf{K}}_0({\mathcal{V}}^{(q-1)})$
For $\phi$, we can represent it with the data:
Then ${{\partial}}[\phi] = [Z] - [Y] = [X\setminus V] - [X\setminus U]$ as desired.

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, $\phi \in \mathop{\mathrm{Aut}}_k k(X)$ 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 $X \in {\mathsf{Var}}_k^{\dim = q, {\mathrm{irr}}}$, then $X$ is represented by a class in $B_q$
Let $\phi: X\overset{\sim}{\dashrightarrow}X$ be defined by $\phi: U { \, \xrightarrow{\sim}\, }V$ and write $X = U \uplus (X\setminus U) = V \uplus (X\setminus V)$.
$\implies$: suppose $\phi$ extends to a piecewise automorphism
- Then $[X\setminus U] = [X\setminus V]\in {\mathsf{K}}_0({\mathcal{V}}^{q-1})$ since $X\setminus U { \, \xrightarrow{\sim}\, }X\setminus V$ by assumption
- By ???, ${{\partial}}[\phi] = [X\setminus V] - [X\setminus U] = 0$ 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, $[X\setminus U] = [X\setminus V]\in {\mathsf{K}}_0({\mathcal{V}}^{(q, q-1)})$ 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 $\dim Y_r' < n-r$.
  - ${{\partial}}[\phi] = [Y_r'] - [X_r']$
- 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 ${{\partial}}[\phi] = [\emptyset] - [\emptyset] = 0$ and $\phi$ extends
  - A general remark on why: ${{\partial}}[\phi]$ measures the failure of $\phi$ to extend to a piecewise isomorphism:
  - ${{\partial}}[\phi] = 0 \implies [X\setminus V] = [X\setminus U]$
  - $\implies X\setminus V \cong X\setminus U$ 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. $\operatorname{ch}k = 0$. Then ${\mathbb{L}}$ is a zero divisor in ${\mathsf{K}}_0({\mathcal{V}}_k)$ $\implies \psi_n$ is not injective for some $n$.

Proof

Strategy: contrapositive. Suppose $\ker \psi_n = 0$ for all $n$. Write ${\mathcal{V}}\coloneqq{\mathcal{V}}_k$.
There is a cofiber sequence ${\mathsf{K}}({\mathcal{V}}) \xrightarrow{\cdot {\mathbb{L}}} {\mathsf{K}}({\mathcal{V}}) \xrightarrow{\ell} {\mathsf{K}}({\mathcal{V}}/{\mathbb{L}})$.
- Todo: what is ${\mathcal{V}}/{\mathbb{L}}$?
Take the LES to identify $\ker(\cdot {\mathbb{L}})$ with $\operatorname{coker}(\ell)$:
Reduce to analyzing $\operatorname{coker}(E_{1, q}^\infty \to \tilde E_{1, q}^\infty )$
- Todo: what is $\tilde E_{1, q}^\infty$?
Lemma 5.4 shows $\tilde E_{1, q}^\infty$ is a quotient of $\qty{\bigoplus_{\beta\in B_n} \pi_1 \tilde C_\beta} \bigoplus_{B_n\setminus\ell(B_n)} C_2$
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 $E^\infty \twoheadrightarrow\tilde E^\infty$.
Then ${\mathsf{K}}_1({\mathcal{V}}) \xrightarrow[]{\ell} { \mathrel{\mkern-16mu}\rightarrow }\, {\mathsf{K}}_1({\mathcal{V}}/{\mathbb{L}})$, making $0 = \operatorname{coker}(\ell) = \ker(\cdot {\mathbb{L}})$ so ${\mathbb{L}}$ is not a zero divisor.prove

Theorem D

Let $k$ be a convenient field, then if $\chi \in \operatorname{Ann}({\mathbb{L}})$ then $\chi = [X] - [Y]$ where $[X\times {\mathbf{A}}^1] = [Y\times {\mathbf{A}}^1]$ but these are not piecewise isomorphic. Moreover, ??? is in $\ker(\psi_n)$ ???

Proof

Let $\chi \in \ker(\cdot {\mathbb{L}})$ and pullback in the LES to $x \in {\mathsf{K}}({\mathcal{V}}^{(n)}/{\mathbb{L}})$ where $n$ is minimal among filtration degrees:
Write ${{\partial}}[x] = [X] - [Y]$ with $X,Y$ of minimal dimension.
Note $[X\times {\mathbf{A}}^1] = [Y\times {\mathbf{A}}^1] \implies \dim X + 1 = \dim Y + 1 \implies \dim X = \dim Y = d$ (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 ${\mathbb{L}}([X] - [Y]) \in \ker ?$ then we can produce an element in $\ker \psi_n$.
  - Diagram chase:
    - $[X] - [Y] \not \in \operatorname{im}({{\partial}})$ by the minimality of $n$ for $x$, noting ${{\partial}}[x] = [X] - [Y]$.
    - By exactness $\operatorname{im}{{\partial}}= \ker(\cdot {\mathbb{L}})$, so ${\mathbb{L}}([X] - [Y]) \neq 0$.
    - By choice of $n$, $i_*({\mathbb{L}}([X] - [Y])) \in \operatorname{im}{{\partial}}= \ker(\cdot {\mathbb{L}})$ in bottom row, so ${\mathbb{L}}([X] - [Y]) = 0$ in bottom-right.
    - Commutativity forces ${\mathbb{L}}([X] - [Y]) \in \ker i_*^{n-1}$.
  - Thus ${\mathbb{L}}([X] - [Y])$ corresponds to an element in $\ker \psi_n$. (???)

Theorem E

There is an isomorphism \begin{align*} {\mathsf{K}}_0({\mathcal{V}}_{\mathbf{C}})/\left\langle{{\mathbb{L}}}\right\rangle { \, \xrightarrow{\sim}\, }{\mathbf{Z}}[\mathsf{SB}_{\mathbf{C}}] \qquad \in {}_{{\mathbf{Z}}}{\mathsf{Mod}} \end{align*}

Proof: omitted.

Remark: so $[X] \equiv [B] \operatorname{mod}{\mathbb{L}}\implies$ $X$ and $Y$ are stably birational?

2022 Talbot Talk Outline V2

2022 Talbot Talk Outline V2

Preliminaries

Q1: Larsen-Lunts and piecewise isomorphism

Q2: When is \(\operatorname{Ann}({\mathbb{L}}) = 0\)? I.e. when (if ever) is \({\mathbb{L}}\) a zero divisor?

This Paper

The Work!

Theorem A: The Splitting

Proof

Setup for Thm B

Lemma 3.2: Representing \({{\partial}}\)

Proof of Lemma 3.2 (useful for understanding \({\mathsf{K}}_1\))

Theorem B: The spectral sequence and \(\psi_n\)

Proof

Theorem C

Proof

Theorem D

Proof

Theorem E