Đến nội dung # Motivic integration: an introduction

### #1 bangbang1412 Đã gửi 13-04-2022 - 18:10

bangbang1412

In this topic, I introduce the notion of the so-called motivic integration, which is an upgrade version of the old version, namely, the p-adic integration. The word motivic literally means the values of this integration is essentially geometric. It was introduced by M. Kontsevich in his lecture in Orsay in 1995 to solve a theorem of Bartyrev stating that two birational Calabi-Yau varieties have the same Betti numbers.

Let $S$ be a scheme. By a $S$-algebraic variety, we mean a $S$-scheme of finite presentation. We denote by $\mathrm{Var}_S$ the isomorphism classes of finite presentation $S$-schemes. When $S = \mathrm{Spec}(k)$ with $k$ a field, we simply write $\mathrm{Var}_k$ instead of $\mathrm{Var}_{\mathrm{Spec}(k)}$.

Jet scheme and arc space

Let $X$ be a $k$-variety.

Proposition 1. For $m \in \mathbb{N}$, there exists an algebraic $k$-variety $J_m(X)$ such that:
\begin{equation*}
\mathrm{Hom}_k(Z \times \mathrm{Spec}(k[t]/(t^{m+1})), X) \simeq \mathrm{Hom}_k(Z,J_m(X))
\end{equation*} for any $k$-scheme $Z$.

Proof. It is sufficient to deal with the case $X, Z$ are affine, i.e., $X = \mathrm{Spec}(R)$ and $Z = \mathrm{Spec}(A)$ for some $k$-algebra $R$ and some finitely generated $k$-algebra $R = k[x_1,...,x_n]/(f_1,...,f_r)$.
\begin{equation*}
\mathrm{Hom}_k(\mathrm{Spec}(A) \times_k \mathrm{Spec}(k[t]/(t^{m+1})), \mathrm{Spec}(R))  \simeq \mathrm{Hom}_k(\mathrm{Spec}(A \otimes k[t]/(t^{m+1})), \mathrm{Spec}(R))
\end{equation*} which is nothing but $\left \{\varphi: k[x_1,...,x_n] \longrightarrow A[t]/(t^{m+1}) \mid \varphi(f_i) = 0 \ \forall \ i = \overline{1,r} \right \}$. For such a $\varphi$, set:
\begin{equation*}
\varphi(x_i) = a_i^0 + a_i^1 t + \cdots + a_i^m t^m \ \forall \ i =\overline{1,n}
\end{equation*} and,
\begin{equation*}
\varphi(f_i) = F^0_i(a^u_v) + F^1_i(a^u_v)t + \cdots + F^m_i(a^u_v) t^m
\end{equation*} where $u = \overline{0,m}, v = \overline{1,n}$ and $F^t_i$'s are polynomials in $a^{u}_v$. Consequently, we see that $\varphi(f_i)=0$ if and only if all $F^t_i(a^u_v) = 0$; and hence
\begin{align*}
\left \{\varphi: k[x_1,...,x_n] \longrightarrow A[t]/(t^{m+1}) \mid \varphi(f_i) = 0 \ \forall \ i = \overline{1,r} \right \} &  = \mathrm{Hom}(k[x_j,x^0_j,...,x^m_j]_{j=\overline{1,n}}/(F_i^l(x^u_j)), A) \\
& = \mathrm{Hom}(\mathrm{Spec}(A),\mathrm{Spec}(R_m))
\end{align*}
where $R_m = k[x_j,x^0_j,...,x^m_j]_{j=\overline{1,n}}/(F_i^l(x^u_j))$; and finally we can define $J_m(X) = \mathrm{Spec}(R_m)$.

Definition 2. For $m \geq n$, the natural surjections:
\begin{equation*}
\end{equation*} induced transition morphisms $\pi_{m,n}: J_m(X) \longrightarrow J_n(X)$, make $(J_m(X), \pi_{m,n})$ a projective system. Define $J_{\infty}(X) = \underset{m \longrightarrow \infty}{\lim} J_m(X)$ and denote by $\pi_m$ the $m^{th}$-canonical projection $\pi_m:J_m(X) \longrightarrow J(X)$.

Remark. It is not trivial that the limit $\underset{m \longrightarrow \infty}{\lim} J_m(X)$ exists in the category of schemes. We must prove that the transition morphisms $\pi_{m,n}$'s are affine.

Proposition 3. For any $k$-scheme $Z$, we have:
\begin{equation*}
\mathrm{Hom}_k(Z \hat{\times_k}  \mathrm{Spec}(k[[t]]), X) \simeq \mathrm{Hom}_k(Z,J_{\infty}(X))
\end{equation*} where $Z \hat{\times} \mathrm{Spec}(k[[t]])$ means the formal completion of $Z \hat{\times} \mathrm{Spec}(k[[t]])$ along the subscheme $Z \times_k \left \{0 \right \}$.

Definition 4. For $m \in \mathbb{N}$, the scheme $J_m(X)$ is called the $m^th$ jet scheme of $X$ and $J_{\infty}(X)$ is called the arc space of $X$. For any $k$-scheme $Z$, elements in $\mathrm{Hom}_k(Z,J_m(X))$ are called $Z$-valued $m$-jets of $X$ and elements in $\mathrm{Hom}_k(Z,J_{\infty}(X))$ are called $Z$-valued arcs of $X$. If $Z = \mathrm{Spec}(k)$, we just say $m$-jets or arcs.

Example 5. Let $X = V(x^3 + y^2) \subset \mathbb{A}^2_k$. View $x,y$ as formal power series in $t$ and consider the equation:
\begin{equation*}
(a_0+a_1t+\cdots)^3 + (b_0+b_1t+\cdots)^2 = 0.
\end{equation*} By truncating the above equation at degree $m+1$, it gives us the defining equations of $J_m(X)$. For instance, $J_0(X)$ is given by $a_0^3+b_0^2=0$; $J_1(X)$ is given by $a_0^3+b_0^2=0$ and $3a_0^2 a_1 + 2b_0 b_1=0$.

Proposition 6. Let $X \longrightarrow Y$ be an étale morphism of $k$-varieties, then $J_m(X) \cong J_m(Y) \times_Y X$ for any $m \in \mathbb{N} \cup \left \{\infty \right \}$.

Proof. We prove that equality on the level of functors of points. We have:
\begin{equation*}
\mathrm{Hom}(-.J_m(X)) \simeq \mathrm{Hom}(- \times_k \mathrm{Spec}(k[[t]]/(t^{m+1})), X)
\end{equation*} and
\begin{equation*}
\mathrm{Hom}(-,J_m(Y) \times_Y X) \simeq\mathrm{Hom}(-,J_m(Y)) \times\mathrm{Hom}(-,X) \simeq \mathrm{Hom}(- \times_k \mathrm{Spec}(k[[t]]/(t^{m+1})), Y) \times \mathrm{Hom}(-,X)
\end{equation*} For a $k$-scheme $Z$ we consider the diagram:

We have to show that  for a $Z$-valued $m$-jet of $Y$ and $Z$-valued $0$-jet of $X$ induce a $Z$-valued $m$-jet of $X$ (the other direction is obvious). Since $X \longrightarrow Y$ is étale, it is formally étale so such a dashed arrow exists.

Corollary 7. Let $U \hookrightarrow X$ be an open immersion, then $J_m(U) \hookrightarrow J_m(X)$ is also an open immersion for any $m \in \mathbb{N} \cup \left \{\infty \right \}$.

By an analogous method, we deduce the following important result:

Proposition 8. Let $X$ be a smooth $k$-scheme of dimension $d$. Then $J_m(X)$ is locally a $\mathbb{A}^{md}$-bundle over $X$. In particular, $J_m(X)$ is smooth of dimension $(m+1)d$. In the same way, $J_{m+1}(X)$ is locally a $\mathbb{A}^d$-bundle over $J_m(X)$.

Bài viết đã được chỉnh sửa nội dung bởi bangbang1412: 13-04-2022 - 18:27

$$[\Psi_f(\mathbb{1}_{X_{\eta}}) ] = \sum_{\varnothing \neq J} (-1)^{\left|J \right|-1} [\mathrm{M}_{X_{\sigma},c}^{\vee}(\widetilde{D}_J^{\circ} \times_k \mathbf{G}_{m,k}^{\left|J \right|-1})] \in K_0(\mathbf{SH}_{\mathfrak{M},ct}(X_{\sigma})).$$

### #2 bangbang1412 Đã gửi 13-04-2022 - 18:17

bangbang1412

Grothendieck ring of varieties

Definition 9. Let $S$ be a ring. A motivic measure $\lambda$ from the category $\mathrm{Var}_k$ with values in $S$, assigns to any $X$ in $\mathrm{Var}_k$ an element $\lambda(X)$ of $S$ such that:

• $\lambda([\mathrm{Spec}(k)]) = 1$.
• $\lambda([X]) = \lambda([Y]) + \lambda([X \setminus Y])$ for $Y$ closed in $X$.
• $\lambda([X][Y]) = \lambda([X])\lambda([Y])$ for $X, Y \in \mathrm{ob}(\mathrm{Var}_k)$.

Remark. Any motivic measure $\lambda$ naturally extend to take its values on constructible subsets of algebraic varieties. Indeed a constructible subset $W$ maybe written as a finite disjoint union of locally closed subvarieties $Z_i$ and hence we can define $\lambda(W)$ to be $\sum \lambda(Z_i)$. By the very axioms, this is independent of the choice of the decomposition into locally closed subvarieties.

Example 10. Let $k$ be a finite field and $K/k$ a finite extension, then the additive invariant $[X] \longmapsto \left | X(K) \right|$ is a motivic measure.

Example 11. Let $l$ be a prime number distinct from the characteristic of $k$. The assignment $$[X] \longmapsto \sum_{i=0}^{2\dim(X)}(-1)^i H_{c}^n(X,\mathbb{Q}_l)$$ where $H_c^i$ denote the $i^{th}$ $l$-adic cohomology with compact support, defines a motivic measure. This also works effectively for every other classic cohomology theories, e.g., Hodge theory, crystalline cohomology.

Example 12. Let us assume $k$ is a field of characterisitc zero. It follows from Deligne's mixed Hodge theory that there is a unique motivic measure $H: \mathrm{Var}_k \longrightarrow \mathbb{Z}[u,v]$, which assigns each smooth projective variety $X$ over $k$ its Hodge polynomial:
\begin{equation*}
H(X,u,v) = \sum_{p,q}(-1)^{p+q}h^{p,q}(X)u^p v^q
\end{equation*} where $h^{p,q}(X) = \dim H^q(X,\Omega^p)$ is the $(p,q)$-Hodge number of $X$. For instance, if $k = \mathbb{C}$ then $H(\mathbb{P}^n,u,v) = \sum_{i=0}^n (uv)^i$. The uniqueness of $H$ is highly nontrivial to be proved; it is a consequence of the fact that $K_0(\mathrm{Var}_k)$ (defined below) is generated by classes of smooth proper $k$-varieties.

Definition 14. In the sequel, $[-]$ always denotes the isomorphism class of some object in a appropriate category. We denote by $\mathbb{Z}[\mathrm{Var}_S]$ the free abelian group on $\mathrm{Var}_S$. The Grothendieck ring over $S$ $K_0(\mathrm{Var}_S)$ is the quotient of $\mathbb{Z}[\mathrm{Var}_S]$ by its subgroup, generated by element of the form $[X] - [Z] - [X \setminus Z]$, where $X$ is a $S$-scheme of finite presentation, and $Z$ a closed subscheme of $X$. The fiber product over $S$ induces, by linearity, a ring structure on it by setting $[X] \cdot [Y] = [X \times_S Y]$. We note $\mathbf{L} = [\mathbb{A}_S^1]$ and $\mathscr{M}_S = K_0(\mathrm{Var}_S)[\mathbf{L}^{-1}]$ the localization of $K_0(\mathrm{Var}_S)$ by the element $\mathbf{L}$. When $S = \mathrm{Spec}(k)$ with $k$ a field, we simply write $\mathscr{M}_k$ instead of $\mathscr{M}_{\mathrm{Spec}(k)}$.

We denote by $\mathscr{M}_k[T]_{loc}$ the subring of $\mathscr{M}_k[[T]]$ generated by $\mathscr{M}_k[T]$ and the series $(1 - \mathbb{L}^a T^b)^{-1}$ with $a \in \mathbb{Z}$ and $b \in \mathbb{N} \setminus \left \{0 \right \}$.

Remark.

• It is evident from the very definition of $K_0(\mathrm{Var}_k)$ that it possesses a universal property: any motivic measure on $\mathrm{Var}_k$ factors through $K_0(\mathrm{Var}_k)$.
• $K_0(\mathrm{Var}_{\mathbb{C}})$ is not a domain. It was showed that there exist two abelian varieties $A, B$ such that $[A] \neq [B]$ but $A \times A \simeq B \times B$, which implies $([A] - [B])([A] + [B]) = [A]^2 - [B]^2 = 0$.
• It is proved that if $k$ has characteristic zero, then $K_0(\mathrm{Var}_k)$ can be generated by isomorphism classes of irreducible smooth projective varieties subject to the blow-up relations.

Example 15. $[\mathbb{P}_k^n] = \mathbb{L}^n + \mathbb{L}^{n-1} + \cdots + \mathbb{L} + 1$.

Example 16. For $k$ is algebraically closed and $X = V(x^3 - y^2)$ in $\mathbb{A}^2_k$, then $[X] = [\mathbb{A}_k^1 \setminus \left \{0\right \}] + [\left \{0 \right \}] = [\mathbb{A}^1_k] = \mathbb{L}$.

Example 17. For any $f: Y \longrightarrow X$ a piecewise trivial fibration with fiber $F$, i.e., $X = \coprod X_i$ locally closed and $f_{\mid f^{-1}(X_i)}: f^{-1}(X_i) \longrightarrow X_i$ is of the form (precisely, isomorphic to) $X_i \times F \longrightarrow X_i$, then:
\begin{equation*}
[Y] = \sum [f^{-1}(X_i)] = [F] \sum [X_i] = [F][X].
\end{equation*}
For instance, for a reasonable $k$-scheme $X$, $[J_m(X)] = \mathbb{L}^{m\dim(X)}[X]$.

Definition 18. Let $X$ be an algebraic $k$-variety of dimension $d$ (maybe singular). A subset $C \subset J_{\infty}(X)$ is called a cylinder set if $C = \pi^{-1}_n(B_n)$ for some $n \in \mathbb{N}$ and $B_n \subset J_n(X)$ is a constructible subset, i.e., a finite disjoint union of locally closed subvarieties.

Remark. In a paper of Denef and Loeser, it has been proven that for any algebraic variety $X$ (not necessarily smooth), $\pi_n(J_{\infty}(X))$ is constructible for any $n \in \mathbb{N}$.

Definition 19. Let $X$ be a smooth $k$-variety of dimension $d$ and $C = \pi^{-1}(B_n)$ a cylinder set where $B_n \subset J_n(X)$ a constructible subset. The function:
\begin{align*}
\widetilde{\mu}: \left \{\text{cylinder sets in} \ J_{\infty}(X) \right \} & \longrightarrow \mathscr{M}_k \\
\pi^{-1}_n(B_n) & \longmapsto \frac{[B_n]}{\mathbb{L}^{(n+1)d}}.
\end{align*}
It is straightforward to show that $\widetilde{\mu}$ is a finitely additive measure.

Bài viết đã được chỉnh sửa nội dung bởi bangbang1412: 13-04-2022 - 18:58

$$[\Psi_f(\mathbb{1}_{X_{\eta}}) ] = \sum_{\varnothing \neq J} (-1)^{\left|J \right|-1} [\mathrm{M}_{X_{\sigma},c}^{\vee}(\widetilde{D}_J^{\circ} \times_k \mathbf{G}_{m,k}^{\left|J \right|-1})] \in K_0(\mathbf{SH}_{\mathfrak{M},ct}(X_{\sigma})).$$

### #3 bangbang1412 Đã gửi 13-04-2022 - 18:25

bangbang1412

Order function associated to an effective divisor

Let $X$ be a smooth $k$-variety of dimension $d$. Let $D$ be an effective divisor, $x$ is a point in $X$ and $g$ is a local defining equation for $D$ on a neighborhood $U$ of $x$ in $X$. For an arc $\gamma_u$ over a point $u \in U$. The intersection number $\gamma_u \cdot D$ is defined to be the order of vanishing of the formal power series $g(\gamma_u(t))=0$ at $t=0$.

Definition 20. Define the function $F_D$ to be:

\begin{align*}
F_D: J_{\infty}(X) & \longrightarrow \mathbb{Z}_{\geq 0} \cup \left \{\infty \right \}\\
\gamma_u & \longmapsto \gamma_u \cdot D.
\end{align*}

We want to integrate the function $F_D$ over $J_{\infty}(X)$ and hence we must understand the level sets $F^{-1}_D(s)$ for $s \in \mathbb{Z}_{\geq 0} \cup \left \{\infty \right \}$.

For an effective divisor $D = \sum_{i=1}^r a_i D_i$ ($D_i$'s are prime divisors) and $J \subset \left \{1,2,...,r \right \}$ a subset, define:
\begin{equation*}
D_J = \begin{cases}
\bigcap_{j \in J} D_j & \text{if} \ J \neq \varnothing, \\
Y & \text{if} \ J = \varnothing
\end{cases}
\ \ \ \ \text{and} \ \ \ \
D^{\circ}_J  = D_J \setminus \bigcup_{j \in \left \{1,2,...,r\right \}\setminus J} D_j.
\end{equation*} Recall that a divisor $D = \sum_{i=1}^r a_i D_i$ on $X$ has only simple normal crossings if at each point $x \in X$, there is a neighborhood $U$ of $x$ with coordinates $x_1,...,x_n$ for which a local defining equation for $D$ is $g = x_1^{a_{j_1}}\cdots x_{j_r}^{a_{j_x}}$ with $0 \leq j_x \leq d$.

Lemma 21. For $D=\sum a_i D_i$ has only simple normal crossings such that all $D_i$'s are smooth, $F^{-1}_D(s)$ is a cylinder set for $s \in \mathbb{Z}_{\geq 0}$.

Lemma 22. $F^{-1}_D(\infty)$ is not a cylinder set but a countable intersection of cylinder sets.

So far, lemma 22 tells us that $F_D$ is not $\widetilde{\mu}$-measurable ($\widetilde{\mu}$ in definition 19) because $F_D^{-1}(\infty)$ is not a cylinder set. To proceed, we have to extend $\widetilde{\mu}$ to a measure $\mu$ such that $F^{-1}_D(\infty)$ is $\mu$-measurable. We first see that $J_{\infty}(X) \setminus F_D^{-1}(\infty)$ is a countable disjoint union of cylinder sets
\begin{equation*}
J_{\infty}(X) \setminus \pi^{-1}_0 \pi_0(F_D^{-1}(\infty)) \sqcup \bigsqcup_{n \in \mathbb{Z}_{\geq 0}} \left(\pi_n^{-1}\pi_n(F_D^{-1}(\infty)) \setminus \pi_{n+1}^{-1}\pi_{n+1}(F_D^{-1}(\infty)) \right).
\end{equation*} The above decomposition suggests us that we should extend $\widetilde{\mu}$ to a measure $\mu$ defined on the colletion of countable disjoint union of cylinder sets so that $J_{\infty}(X)\setminus F_D^{-1}(\infty)$ (and hence $F_D^{-1}(\infty)$) is $\mu$-measurable. However, countable sums are not defined in $\mathscr{M}_k = K_0(\mathrm{Var}_k)[\mathbb{L}^{-1}]$ and nothing warrants that our measure is well-defined in the sense that it is independent of the choice of the decomposition into countable disjoint union of cylinder sets. Kontsevich solved both problems at once! Follow a paper of Loeser, one should proceed by analogy with $p$-adic integration: $K_0(\mathrm{Var}_k)$ plays the role of $\mathbb{Z}$ and $K_0(\mathrm{Var}_k)[\mathbb{L}^{-1}]$ plays the role of $\mathbb{Z}[p^{-1}]$. Since in $\mathbb{R}$, $p^{-i}$ has limit $0$ as $i \longrightarrow \infty$, we should complete $K_0(\mathrm{Var}_k)[\mathbb{L}^{-1}]$ in such a way that $\mathbb{L}^{-i}$ has limit $0$ as $i \longrightarrow \infty$.

Definition 23. Definine $F^m$ to be the subgroup of $\mathscr{M}_k$ generated by element of the form $[S]\mathbb{L}^{-i}$ with $\dim(S)\leq i - m$. We have $F^{m+1} \subset F^m, \ \mathbb{L}^{-m} \in F^m$ and $F^m F^n \subset F^{m+n}$. We denote by $\hat{\mathscr{M}}_k$ the completion with respect to this filtration.

Definition 24. Let $\mathcal{C}$ denote the collection of countable disjoint unions of cylinder sets $\coprod_{i \in \mathbb{N}} C_i$ for which $\widetilde{\mu}(C_i) \longrightarrow 0$ as $i \longrightarrow \infty$, together with their complements. Extend $\widetilde{\mu}$ to a measure $\mu$ on $\mathcal{C}$ given by
\begin{equation*}
\bigsqcup_{i \in \mathbb{N}} C_i \longmapsto \sum_{i \in \mathbb{N}}\widetilde{\mu}(C_i).
\end{equation*} It is nontrivial to show that this definition is independent of the choice of $C_i$'s.

Lemma 25. $\widetilde{\mu}(\pi_n^{-1}\pi_n(F_D^{-1}(\infty))) \in F^{n+1} \subset \mathscr{M}_k$.

Lemma 26. $F_D^{-1}(\infty)$ is $\mu$-measurable and $\mu(F_D^{-1}(\infty))= 0$.

Definition 27. (naive version of motivic integration). Keeping the same hypotheses on $X$ and $D$ ($D$ has only simple normal crossings), we define the motivic integral of the pair $(X,D)$ to be
\begin{equation*}
\int_{J_{\infty}(Y)} \mathbb{L}^{-F_D}d\mu = \sum_{s \in \mathbb{Z}_{\geq 0} \cup \left \{\infty \right \} } \mu(F_D^{-1}(s))\mathbb{L}^{-s}.
\end{equation*} Since $F_D^{-1}(\infty)$ has measure zero so in fact the sum on the right is over $\mathbb{Z}_{\geq 0}$.

Proposition 28. With the same hypotheses as in the previous definition. Then:
\begin{equation*}
\int_{J_{\infty}(X)} \mathbb{L}^{-F_D}d\mu = \sum_{J \subset \left \{1,2,...,r\right \}} [D^0_J]\left( \prod_{j \in J}\frac{\mathbb{L}-1}{\mathbb{L}^{a_j+1}-1} \right) \mathbb{L}^{-d}.
\end{equation*}

Bài viết đã được chỉnh sửa nội dung bởi bangbang1412: 13-04-2022 - 19:56

$$[\Psi_f(\mathbb{1}_{X_{\eta}}) ] = \sum_{\varnothing \neq J} (-1)^{\left|J \right|-1} [\mathrm{M}_{X_{\sigma},c}^{\vee}(\widetilde{D}_J^{\circ} \times_k \mathbf{G}_{m,k}^{\left|J \right|-1})] \in K_0(\mathbf{SH}_{\mathfrak{M},ct}(X_{\sigma})).$$

### #4 bangbang1412 Đã gửi 13-04-2022 - 19:44

bangbang1412

Motivic integration: general definition (without smoothness)

Let $X$ be an algebraic $k$-variety of pure dimension $d$, we want to extend our measure to a broader generality (without smoothness conditions) so that we can integrate simple functions $\alpha: J_{\infty}(X) \longrightarrow \mathbb{Z}$ whose fibers are well-understood.

Definition 29. Let $C$ be a constructible subset of $X$ such that $\pi_n^{-1}(B_n) = C$ for some constructible subset of $J_n(X)$. If furthermore:

• $\pi_n(C) = B_n$,
• $\pi_m(C) \subset J_{m}(X)$ is constructible for any $m \geq n$,
• The truncation morphisms $\pi_{m+1,m}: \pi_{m+1}(C) \longrightarrow \pi_m(C)$ is a piecewise trivial fibration with fiber $\mathbb{A}^d$,

then we say that $C$ is stable at level $n$. We say that $C$ is stable if it is stable at some level. When $X$ is smooth, all cylinders are stable.

Definition 30. Let $C$ be a stable cylinder at level $n$. We set $$\widetilde{\mu}(C) = \frac{[\pi_n(C)]}{\mathbb{L}^{(n+1)d}} \in \mathscr{M}_k$$ The stability condition ensures that this definition is independent of the choice of $n$. By proposition 8, when $X$ is smooth, all cylinder subsets are stable. In particular, $J_{\infty}(X)$ is a stable cylinder, and:
\begin{equation*}
\widetilde{\mu}(J_{\infty}(X)) = \frac{[X]}{\mathbb{L}^{d}}.
\end{equation*} Theorem 31. There exists an algebra $\mathbf{B}_X$ . of subsets of $J_{\infty}(X)$, which contains all stable cylinders nd a unique map $\mu: \mathbf{B}_X \longrightarrow \hat{\mathscr{M}_k}$ satisfying the following conditions:

• If $C$ is a stable cylinder set, then $\mu(C) = \widetilde{\mu}(C)$.
• If $C \in \mathbf{B}_X$ is contained in $J_{\infty}(Z)$ where $Z$ is a closed subvariety of $X$ with $\dim(Z) < \dim(X)$, then $\mu(C) =0$.
• Let $(C_i)_{i \in \mathbb{N}}$ be a sequence in $\mathbf{B}_X$ such that $C_i$'s are mutually disjoint and $C = \bigsqcup C_i$ belongs to $\mathbf{B}_X$, then $\sum \mu(C_i)$ converges to $\mu(C)$ in $\hat{\mathscr{M}}_k$.
• If $C, D$ are in $\mathbf{B}_X$ with $C \subset D$, and if $\mu(D)$ belongs to the closure $\hat{F}_m$ of $F_m$ in $\hat{\mathscr{M}_k}$, then $\mu(C) \in \hat{F}_m$.

Remark. Elements in $\mathbf{B}_X$ are called semi-algebraic sets, but we do not stress to the precise definition here.

Definition 32. Let $C$ be in $\mathbf{B}_X$ and $\alpha: C \longrightarrow \mathbb{Z} \cup \left \{ \infty \right \}$ a function such that $\alpha^{-1}(s) \in \mathbf{B}_X$ for any $s \in \mathbb{Z} \cup \left \{ \infty \right \}$ and $\mu(\alpha^{-1}(\infty)) = 0$, we set:
\begin{equation*}
\int_C \mathbb{L}^{-\alpha}d\mu  = \sum_{s\in \mathbb{Z}} \mu(C \cap \alpha^{-1}(s)) \mathbb{L}^{-s},
\end{equation*} in $\hat{\mathscr{M}}_k$, whenever the right hand side converges in $\hat{\mathscr{M}}_k$, in which case we say $\mathbb{L}^{-\alpha}$ is integrable on $C$.

Theorem 33. (Change of varibles) Let $X, Y$ be algebraic $k$-varieties of pure dimension $d$ and $h : Y \longrightarrow X$ be a proper birational morphism. Let assume $Y$ to be smooth. Let $C \in \mathbf{B}_X$ and $\alpha: J_{\infty}(X) \longrightarrow \mathbb{N}$ be a simple function. Then
\begin{equation*}
\int_C \mathbb{L}^{-\alpha}d\mu = \int_{h^{-1}(C)} \mathbb{L}^{-\alpha \circ h - \mathrm{ord}  h^*(\Omega^d_X)} d\mu.
\end{equation*}
Now we can present a rough proof of Bartyrev's theorem. The following diagram illustrates the construction of $\mathscr{M}_k$:

and it motivates the following proof.

Theorem 34. (Bartyrev) Let $X_1,X_2$ be two birational smooth Calabi-Yau varieties, then they have the same Hodge numbers.

Proof. Let $K$ denote the canonical divisor. We resolve the birational map to a Hironaka hut:

In the change of variables formula, we let $\alpha: J_{\infty}(X_1) \to \mathbb{Z} \cup \left \{ \infty \right \}$ be the zero map. Then

$$\int_{J_{\infty}(X_1)} \mathbb{L}^{-\alpha} d\mu = \mu(F^{-1}(0)) = \frac{[\pi(J_{\infty}(X_1)]}{\mathbb{L}^{(0+1)n}} = \frac{[X_1]}{\mathbb{L}^n},$$ and analogously, let $\alpha': J_{\infty}(X_2) \to \mathbb{Z} \cup \left \{ \infty \right \}.$ be the zero map, we see that $\alpha \circ (\pi_1)_{\infty} = \alpha' \circ (\pi_2)_{\infty}$, both equal the zero map. Applying the change of varibles,

$$\int_{J_{\infty}(Y)} \mathbb{L}^{-\alpha \circ (\pi_1)_{\infty} - \mathrm{ord}\pi_1^*K_{Y/X_1}}d\mu = \int_{J_{\infty}(X_2)} \mathbb{L}^{-\alpha'}d\mu = \frac{[X_2]}{\mathbb{L}^n},$$ which implies that $[X_1]=[X_2]$ in $\hat{\mathscr{M}}_{\mathbb{C}}$, we apply the Hodge polynomial to deduce the theorem.

Example 35. Let $D = \varnothing, X' = \mathrm{Bl}_Y(X)$ be the blow of $X$ along the smooth center $Y$ of codimension $c$ in $X$. The relative canonical divisor is $K_{X'/X}=(c-1)E$ where $E$ is the exceptional divisor of the blowup. Using the previous proposition, we have:
\begin{align*}
\int_{J_{\infty}(X')} \mathbb{L}^{-\mathrm{ord}_{K_{X'/X}}} d\mu_{X'} & = \int_{J_{\infty}(X')} \mathbb{L}^{-\mathrm{ord}_{(c-1)E}} d\mu_{X'} \\
& = [X' \setminus E] + \frac{[E]}{[\mathbb{P}^c]} \\
& = [X \setminus Y ] + [Y] = [X].
\end{align*}

Thanks to the change of variables formula, we deduce the rationality of the motivic zeta function and a proof of Bartyrev's theorem.

Theorem 36. Let $X$ be an algebraic $k$-variety and let $A$ be a semi-algebraic subset ($\in \mathbf{B}_X$) of $J_{\infty}(X)$. The power series:
\begin{equation*}
P_C(T) =  \sum_{n=0}^{\infty}[\pi_n(A)]T^n,
\end{equation*} considered as an element of $\mathscr{M}_k[[T]]$, is rational and belongs to $\mathscr{M}_k[T]_{loc}$.

Motivic zeta function and motivic nearby cycles

Let $k$ be a field of characteristic zero. Let us assume that $X$ is a smooth quasi-projective $k$-scheme of pure dimension $d$ and $f: X \longrightarrow \mathbb{A}_k^1$ is a flat morphism of $k$-schemes. Consider the diagram

where $i$ is the zero section of the structural morphism of the affine line and $j$ its complement. We identify $f$ with the image of $t$ under the ring morphism $k[t] \longrightarrow \Gamma(X,\mathcal{O}_X)$.

Definition 37. The motivic zeta function is defined as follows
\begin{equation*}
Z_f(T) = \sum_{n \geq 1} Z^1_n T^n \in \mathscr{M}_{X_{\sigma}}[[T]],
\end{equation*} where $Z^1_n = \mathbb{L}^{-nd}[\left \{\varphi \in J_n(X) \mid f \circ \varphi = t^n + O(t^{n+1}) \right \}] = \mathbb{L}^{-nd}[\mathscr{X}^1_n] \in \mathscr{M}_{X_{\sigma}}$.

Remark. The expression of $\mathscr{X}^1_n$ requires $X$ to be of pure dimension $d$. Otherwise, one has to work connected components by connected components. We also know from theorem 36 that $Z_f(T) \in \mathscr{M}_{X_{\sigma}}[T]_{loc}$.

Let $h: X' \longrightarrow X$ be an embedded resolution of the singularities of $(X,X_{\sigma})$. By this, we mean a proper morphism $h: Y \longrightarrow X$ with $X'$ smooth such that the restriction
\begin{equation*}
h: X' \setminus h^{-1}(X_{\sigma}) \longrightarrow X \setminus X_{\sigma}
\end{equation*} is an isomorphism and $h^{-1}(X_{\sigma}) = \sum_{i \in I} m_i D_i$ has only simple normal crossings as a subvariety of $X'$. For $\varnothing \neq J \subset I$, we define $D_J, D_J^{\circ}$ as in the previous post. We denote by $\rho_J: \widetilde{D}^{\circ}_J \longrightarrow D_J^{\circ}$ the étale cover of $D^{\circ}_J$, locally defined as follows. For any $x \in D_J^{\circ}$, there exists an affine open neighborhood $U$ of $x$ in $X$ (for the Zariski topology), a regular sequence of elements $(t_j)_{j \in J}$ of the ring $\Gamma(U, \mathcal{O}_X)$, and a unit $u \in \Gamma(U,\mathcal{O}_X^{\times})$ such that
\begin{equation*}
f = u \prod_{j \in J} t_j^{N_j}
\end{equation*} and such that the component $D_j \cap U$ of $D \cap U$, for any $j \in J$, can be identified with the affine closed subscheme $V(t_j)$ of $U$. The base change of $\rho_J$ along the opent immersion $U \cap D_J^{\circ} \hookrightarrow D_J^{\circ}$ is identified with the finite étale morphism of $k$-schemes
\begin{equation*}
\mathrm{Spec}\left(\mathcal{O}_{U \cap D_J^{\circ}}[T]/(T^{N_J}- u) \right) \longrightarrow U \cap D^{\circ}_J,
\end{equation*} where the strictly positive ineger $N_J$ is the greatest common divisor of the $N_j$ for all $j \in J$.

Theorem 38. Let $h: X' \longrightarrow X$ be an embedded resolution of the singularities of $(X,X_{\sigma})$. Let us denote by $D = h^{-1}(X_{\sigma}) = \sum_{i\in I} m_i D_i$ the exceptional divisor of $h$, that is supposed to be a simple normal crossings divisor, with $D_i$, $i \in I$, as (reduced) irreducible components. There exist strictly positive integers $n_i$, $i \in I$, such that we have the following formulars
\begin{equation*}
Z_f(T) = \sum_{\varnothing \neq J \subset I}(\mathbb{L}-1)^{\left|J \right|-1}[\widetilde{D}^{\circ}_J]\prod_{j \in J}\frac{1}{T^{-m_j}\mathbb{L}^{n_j}-1} \in \mathscr{M}_{X_{\sigma}}[[T]],
\end{equation*} and
\begin{equation*}
Z_{f,x}(T)  = \sum_{\varnothing \neq J \subset I}(\mathbb{L}-1)^{\left|J \right|-1}[\widetilde{D}^{\circ}_J \cap h^{-1}(x)]\prod_{j \in J}\frac{1}{T^{-m_j}\mathbb{L}^{n_j}-1} \in \mathscr{M}_{k}[[T]],
\end{equation*} for any $x \in X_{\sigma}(k)$.

Definition 39. The motivic nearby cycle $\psi_f$ is defined as, thanks to the rationality of the motivic zeta function,
\begin{equation*}
\psi_f = -\left(\underset{T \longrightarrow +\infty}{\lim} Z_f(T) \right)= \sum_{\varnothing \neq J \subset I} [\widetilde{D}^{\circ}_J](1 - \mathbb{L})^{\left|J \right|-1} \in \mathscr{M}_{X_{\sigma}}.
\end{equation*} If $x \in X_{\sigma}(k)$, we define the motivic Milnor fiber as follows.
\begin{equation*}
\psi_{f,x} = -\left(\underset{T \longrightarrow +\infty}{\lim} Z_{f,x}(T) \right)= \sum_{\varnothing \neq J \subset I} [\widetilde{D}^{\circ}_J \cap h^{-1}(x)](1 - \mathbb{L})^{\left|J \right|-1} \in \mathscr{M}_{k}.
\end{equation*}

Bài viết đã được chỉnh sửa nội dung bởi bangbang1412: 13-04-2022 - 21:20

$$[\Psi_f(\mathbb{1}_{X_{\eta}}) ] = \sum_{\varnothing \neq J} (-1)^{\left|J \right|-1} [\mathrm{M}_{X_{\sigma},c}^{\vee}(\widetilde{D}_J^{\circ} \times_k \mathbf{G}_{m,k}^{\left|J \right|-1})] \in K_0(\mathbf{SH}_{\mathfrak{M},ct}(X_{\sigma})).$$

### #5 Zaraki Đã gửi 16-04-2022 - 07:11

Zaraki

Hồi đầu năm cũng có học khoán về motivic integration, lúc đó có lập được một cái bảng so sánh với p-adic integration như thế này:

• Mihnea Popa https://people.math..../571/index.html
• Francois Loeser Arizona winter school notes
• Devlin Mallory notes Motivic integration
• Willem Veys Arc spaces, motivic integration and stringy invariants.

Trong motivic integration, Bằng có biết có công thức change of variables tổng quát cho bất kì $\alpha: C\to \mathbb{Z}\cup \{\infty\}$ thay vì chỉ $\alpha: J_{\infty}(X)\to \mathbb{N}$ không nhỉ?

Bài viết đã được chỉnh sửa nội dung bởi Zaraki: 16-04-2022 - 07:12

Discovery is a child’s privilege. I mean the small child, the child who is not afraid to be wrong, to look silly, to not be serious, and to act differently from everyone else. He is also not afraid that the things he is interested in are in bad taste or turn out to be different from his expectations, from what they should be, or rather he is not afraid of what they actually are. He ignores the silent and flawless consensus that is part of the air we breathe – the consensus of all the people who are, or are reputed to be, reasonable.

Grothendieck, Récoltes et Semailles (“Crops and Seeds”).

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