Conceptual explanation for $\chi(\mathcal{A_g})=\chi(\mathcal M_{1} \times … \times \mathcal M_{g})$?

Let $ \mathcal{A_g}$ be the moduli space of principally polarized abelian varieties over $ \mathbb C$ , $ \mathcal{M_g}$ be the moduli space of smooth projective curves of genus $ g$ over $ \mathbb C$ , and we regard both as Deligne-Mumford stacks or orbifolds. By computations we know their Euler characteristics :

$ \chi(\mathcal{A_g})=\chi(\mathcal M_{1})\chi(\mathcal M_{2})…\chi(\mathcal M_{g})=\prod_{k=1}^g \zeta(1-2k)$ .

Is there a conceptual explanation of this using Torelli map? Can we update it to an identity of Poincare polynomials?

If $\forall s\in\mathbb R^d, \forall F \in \mathcal F\ E[e^{i}I_F]=E[e^{i}]P[F]$ then $X$ is independen of $\mathcal F$

The claim in the title seems very plausible since the characteristic function “characterizes” or determines the distribution of $ X$ , but I don’t know how to derive it. There is a similar result for the characteristic functions of two Random variables, eg here, but I’m not sure if it could be deduced from that.

Any help would be appreciated!

Show that two processes have the same distribution knowing that their paths up to time $τ$ and their distributions conditioned on $\mathcal F_τ$ agree

Let

  • $ (\Omega,\mathcal A,\operatorname P)$ be a probability space
  • $ (E,\mathcal E)$ be a measurable space
  • $ (Y_n)_{n\in\mathbb N_0}$ and $ (\tilde Y_n)_{n\in\mathbb N_0}$ be time-homogeneous Markov chains on $ (\Omega,\mathcal A,\operatorname P)$
  • $ \kappa$ denote the transition kernel of $ Y$
  • $ (\mathcal F_n)_{n\in\mathbb N_0}$ be a filtration on $ (\Omega,\mathcal A)$ to which $ \tilde Y$ is adapted
  • $ \tau$ be an $ \mathcal F$ -stopping time with $ $ 1_{\left\{\:n\:<\:\tau\:\right\}}Y_n=1_{\left\{\:n\:<\:\tau\:\right\}}\tilde Y_n\;\;\;\text{for all }n\in\mathbb N_0$ $ and$ ^1$ $ ^2$ $ $ \operatorname P\left[\tau<\infty,\left(\tilde Y_{\tau+n_1},\ldots,\tilde Y_{\tau+n_k}\right)\mid\mathcal F_\tau\right]=1_{\left\{\:\tau\:<\:\infty\:\right\}}\bigotimes_{i=1}^k\kappa^{n_i-n_{i-1}}(Y_\tau,B)\tag1$ $ almost surely for all $ k\in\mathbb N_0$ , $ n_0,\ldots,n_k\in\mathbb N_0$ with $ 0=n_0<\cdots<n_k$ and $ B\in\mathcal E^{\otimes k}$

Are we able to conclude that $ Y$ and $ \tilde Y$ have the same distribution?

It’s clear to me that the distribution is uniquely determined by the finite-dimensional distributions. For simplicity and as a first step, I’ve tried to prove $ $ \operatorname P\left[Y_n\in B\right]=\operatorname P\left[\tilde Y_n\in B\right]\tag2$ $ for some fixed $ n\in\mathbb N_0$ and $ B\in\mathcal E$ . My idea is to write $ $ \operatorname P\left[\tilde Y\in B\right]=\operatorname P\left[n<\tau,\tilde Y\in B\right]+\operatorname P\left[n\ge\tau,\tilde Y\in B\right].\tag3$ $ By $ (1)$ , it’s sufficient to show the second term on the right-hand side of $ (3)$ is equal to $ \operatorname P\left[n\ge\tau,Y\in B\right]$ in order to conclude $ (2)$ . Noting that $ \left\{n\ge\tau\right\}=\biguplus_{m=0}^n\left\{\tau=m\right\}$ , we obtain \begin{equation}\begin{split}\operatorname P\left[n\ge\tau,\tilde Y\in B\right]&=\sum_{m=0}^n\operatorname P\left[\tau=m,\tilde Y_{\tau+(n-m)}\in B\right]\&=\sum_{m=0}^n\operatorname E\left[1_{\left\{\:\tau\:=\:m\:\right\}}\kappa^{n-m}(Y_\tau,B)\right]\&=\operatorname E\left[1_{\left\{\:n\:\ge\:\tau\:\right\}}\kappa^{n-\tau}(Y_\tau,B)\right]\end{split}\tag4\end{equation} by $ (1)$ .

However, I don’t know how to conclude $ (2)$ from $ (4)$ . And I don’t see how this approach can be generalized in order to show the actual claim.

(By the way, I would be interested in a solution which would still work if the index set $ \mathbb N_0$ is replaced by something uncountable like $ [0,\infty)$ – but that’s not mandatory; if you only know a solution for the countable case that would be fine for me).


$ ^1$ $ \mathcal F_\tau$ denotes the $ \sigma$ -algebra of $ \tau$ -past.

$ ^2$ $ \bigotimes$ denotes the product of transition kernels.

Monoidal functors $\mathcal C \to [\mathcal D,\mathcal V]$ are monoidal functors $\mathcal C \otimes \mathcal D \to \mathcal V$?

It is well known (e.g., Reference for "lax monoidal functors" = "monoids under Day convolution" ) that if $ \mathcal C$ is a monoidal $ \mathcal V$ -enriched category, then a monoid in $ [\mathcal C, \mathcal V]$ is the same thing as a lax monoidal functor $ \mathcal C \to \mathcal V$ , where $ [\mathcal C,\mathcal V]$ carries the monoidal structure given by the Day convolution.

Is the following also true?

If $ \mathcal C,\mathcal D$ are monoidal $ \mathcal V$ -enriched categories, where $ \mathcal V$ is cocomplete, then a lax monoidal functor $ \mathcal C \to [\mathcal D,\mathcal V]$ is the same thing as a lax monoidal functor $ \mathcal C \otimes \mathcal D \to \mathcal V$ .

(In particular, if $ \mathcal C$ is the unit category, then we get the original formulation.)

It certainly seems to be the case: if $ \mathcal F \colon \mathcal C \otimes \mathcal D \to \mathcal V$ is a lax monoidal functor, then we have natural coherences \begin{gather} \mathcal F(c,\_)\otimes_{\text{Day}}\mathcal F(c’,\_) \to \mathcal F(c\otimes c’,\_) \ I_{\text{Day}} \to F(I, \_)\,, \end{gather} where the multiplicative coherence is given by the composite \begin{align} & \int^{d,d’\colon\mathcal D}(\mathcal F(c,d)\otimes\mathcal F(c’,d’))\otimes \mathcal D(d\otimes d’,x)\ \to & \int^{d,d’\colon\mathcal D}\mathcal F(c\otimes c’,d\otimes d’)\otimes\mathcal D(d\otimes d’,x)\ \to & \mathcal F(c\otimes c’,x)\,. \end{align} and the unital coherence $ \mathcal D(I_{\mathcal D},x)\to\mathcal F(I_{\mathcal C},x)$ comes (via the enriched Yoneda lemma) from the monoidal unit $ I_{\mathcal V}\to\mathcal F(I_{\mathcal C},I_{\mathcal D})$ for $ \mathcal F$ .

I have not checked whether these satisfy the coherence conditions for a monoidal functor, but I would be surprised if they did not.

In the other direction, if $ \mathcal F\colon \mathcal C \to [\mathcal D,\mathcal V]$ is lax monoidal, then we have coherences given by \begin{align} &\mathcal G(c)(d) \otimes \mathcal G(c’)(d’)\ \cong & \int^{e,e’\colon\mathcal D} (\mathcal G(c)(e) \otimes \mathcal G(c’)(e’)) \otimes (\mathcal D(e,d) \otimes \mathcal D(e’,d’)) \ \to & \int^{e,e’\colon\mathcal D} (\mathcal G(c)(e) \otimes \mathcal G(c’)(e’)) \otimes \mathcal D(e\otimes e’,d\otimes d’)\ = & (\mathcal G(c) \otimes_{\text{Day}}\mathcal G(c’))(d \otimes d’)\ \to &\mathcal G(c\otimes c’)(d\otimes d’) \end{align} and monoidal unit $ I_{\mathcal V}\to \mathcal F(I_{\mathcal C},I_{\mathcal D})$ by the enriched Yoneda lemma as before.

I haven’t checked whether these coherences actually satisfy the appropriate diagrams, nor whether these two maps are indeed inverses.

Is this fact true? And if so, has it been proved in the literature somewhere?

Any subcollection $\mathcal A_1$ of a locally finite collection $\mathcal A$ of subsets of $X$ is locally finite.

Any subcollection $ \mathcal A_1$ of a locally finite collection $ \mathcal A$ of subsets of $ X$ is locally finite.

Proof

Let $ \mathcal A_1$ be any subcollection of $ \mathcal A$ i.e., $ \mathcal A_1\subset \mathcal A.$

Let $ x\in X$ and $ \mathcal A$ is locally finite, so there exists a nbhd $ U$ of $ x$ that intersects only finitely many elements $ A_1, A_2, A_3,…, A_n$ of $ \mathcal A$ .

More precisely,$ A_i\cap U\neq \phi$ ,where $ 1\le i \le n $ .

Since,$ \mathcal A_1\subset \mathcal A.$ ,so if any element of {$ A_i:1\le i \le n$ } is in $ \mathcal A_1,$ so $ U$ intersects only finitely many elements of $ \mathcal A_1$ ,making it locally finite collection of subsets of $ X$ .

Please check the proof critically up to here, especially the use of quantifiers.. If there is some scope of improvement in the proof or some elegant method other than this please let me know…

My question is “What if no element of {$ A_i:1\le i \le n$ } is in $ \mathcal A_1$ ?”