On the geometric P=W conjecture

Abstract

We formulate the geometric P=W conjecture for singular character varieties. We establish it for compact Riemann surfaces of genus one, and obtain partial results in arbitrary genus. To this end, we employ non-Archimedean, birational and degeneration techniques to study the topology of the dual boundary complex of character varieties. We also clarify the relation between the geometric and the cohomological P=W conjectures.

Appendices

Appendix A: Independence of the geometric P=W conjecture of auxiliary choices

In this appendix we check the independence of the geometric P=W conjecture of all the auxiliary choices and claims made in Sect. 4.2.

Lemma A.0.1

The inclusion $\Psi (N^*_{\mathrm {Dol}})\hookrightarrow N^*_{\mathrm {B}}$ is a homotopy equivalence.

Proof

Choose rug functions $\beta _{\mathrm {B}}:{\overline{M}}_{\mathrm {B}} \rightarrow {\mathbb {R}}$ and $\beta _{\mathrm {Dol}}:{\overline{M}}_{\mathrm {Dol}} \rightarrow {\mathbb {R}}$, and real numbers $\delta _{{\mathrm {B}}}$ and $\delta _{\mathrm {Dol}}$ such that $N_{\mathrm {B}} = \beta _{\mathrm {B}}^{-1}[0, \delta _{\mathrm {B}})$ and $N_{\mathrm {Dol}} = \beta _\mathrm {Dol}^{-1}[0, \delta _\mathrm {Dol})$. Observe that the homotopy types of $N_{\mathrm {B}}$ and $N_{\mathrm {Dol}}$ is independent of these choices by the uniqueness of semialgebraic neighbourhoods; see Proposition 4.1.2.

In order to show that $\Psi (N^*_{\mathrm {Dol}}) \hookrightarrow N^*_{\mathrm {B}}$ is a homotopy equivalence, we argue as in the proof of [20, Proposition 1.6]. Consider the system of neighbourhoods of $\partial M_{\mathrm {Dol}}$ and $\partial M_{\mathrm {B}}$ respectively

$$\begin{aligned} N_{\mathrm {Dol}, k}= \beta _{\mathrm {Dol}}^{-1}\bigg [0, \frac{\delta _{\mathrm {Dol}}}{k}\bigg ) \subseteq {\overline{M}}_{\mathrm {Dol}} \qquad N_{B,k} = \beta _{\mathrm {B}}^{-1}\bigg [0, \frac{\delta _{\mathrm {B}}}{k} \bigg ) \subseteq {\overline{M}}_{\mathrm {B}} \quad \mathrm { with }\, k \in {\mathbb {N}}^*. \end{aligned}$$

Choose $k_1, k_2\in {\mathbb {N}}$ such that $\Psi (N^*_{\mathrm {Dol}, k_1}) \subseteq N^*_{B, k_2} \subseteq \Psi (N^*_{\mathrm {Dol}}) \subseteq N^*_{B}$. The inclusion $N^*_{B, k_2} \subseteq N^*_{B}$ is a homotopy equivalence by Proposition 4.1.2. The same holds for $\Psi (N^*_{\mathrm {Dol}, k_1}) \subseteq \Psi (N^*_{\mathrm {Dol}})$, since $N^*_{\mathrm {Dol}, k_1}$ and $N^*_{\mathrm {Dol}}$ are stratified isotopic by Proposition 4.1.2, and $\Psi$ is a homeomorphism. Therefore, we have the following sequence in homotopy

$$\begin{aligned} \pi _i(\Psi (N^*_{\mathrm {Dol}, k_1})) \rightarrow \pi _i( N^*_{\mathrm {B}, k_2}) \rightarrow \pi _i(\Psi (N^*_{\mathrm {Dol}})) \rightarrow \pi _i(N^*_{\mathrm {B}}), \end{aligned}$$

which has the property that the composition of two consecutive morphisms is an isomorphism. Hence, the inclusion $N^*_{B, k_2} \subseteq \Psi (N^*_{\mathrm {Dol}})$ induces isomorphisms of homotopy groups, and it is a homotopy equivalence. We conclude that $\Psi (N^*_{\mathrm {Dol}}) \rightarrow N^*_{\mathrm {B}}$ is a homotopy equivalence. $\square$

The geometric P=W conjecture is independent of the choice of the dlt compactification, the semialgebraic deleted neighbourhood $N_{\mathrm {B}}^*$, $N^*_i$, and of the partition of unity $\{\varphi _i\}$, as shown in Lemma A.0.5.

To this end, we first extend the construction of a retraction between dual complexes of snc pairs in [7, Sect. 4.2] to the ${\mathbb {Q}}$-factorial dlt setting. Let $(X, \Delta _X = \sum _{i \in I}\Delta _{X,i})$ be a dlt compactification of $U = X \setminus \Delta _X$ satisfying Assumption 4.2.4. We say that a morphism of reduced dlt pairs $g :(Z, \Delta _Z = \sum _{r \in R} \Delta _{Z,r}) \rightarrow (Y, \Delta _Y = \sum _{k \in K}\Delta _{Y,k})$ over $(X, \Delta _X)$ is a birational morphism of dlt compactifications over $(X, \Delta _X)$ if

• The structural morphisms $f_Z :Z \rightarrow X$ and $f_Y :Y \rightarrow X$ are birational, and $f_Z = f_Y \circ g$;

• $U \simeq Y \setminus \Delta _Y \simeq X \setminus \Delta _X$;

• $(Z,\Delta _Z)$ and $(Y,\Delta _Y)$ satisfy Assumption 4.2.4.

Suppose that X, Y and Z are ${\mathbb {Q}}$-factorial. Write $f_Y^*{\Delta _X}= \sum _{k \in K}n_k \Delta _{Y,k}$ for some $n_k \in {\mathbb {Q}}$, and

$$\begin{aligned} f_Z^*{\Delta _X} = g^*(f_Y^*{\Delta _X}) = \sum _{k \in K}n_k g^*(\Delta _{Y,k}) = \sum _{k \in K}n_k \bigg (\sum _{r \in R} a_{kr} \Delta _{Z,r} \bigg ) {=:}\sum _{r \in R}m_r \Delta _{Z,r} \end{aligned}$$

for some $a_{kr} \in {\mathbb {Q}}$ and $m_r= \sum _{k \in K} n_k a_{kr}$.

A stratum $\Delta _{Z,S}$ of $\Delta _Z$, with $S \subseteq R$, corresponds to a cell $\sigma _S$ of ${\mathcal {D}}(\Delta _Z)$, which we identify with the standard simplex $\{\sum _{s \in S} w_s =1\}\subset {\mathbb {R}}^S_{\ge 0} \subseteq {\mathbb {R}}^R$. Let $\Delta _{Y,L}$ be the minimum stratum of $\Delta _Y$ such that $g(\Delta _{Z,S}) \subseteq \Delta _{Y,L}$. Denote by $\sigma _L$ the corresponding cell in ${\mathcal {D}}(\Delta _Y)$, identified with the standard simplex $\{\sum _{l \in L} u_l =1\}\subset {\mathbb {R}}^L_{\ge 0} \subseteq {\mathbb {R}}^K$.

We define the map $r_{ZY} :{\mathcal {D}}(\Delta _Z) \rightarrow {\mathcal {D}}(\Delta _Y)$ on each cell $\sigma _S$ by

$$\begin{aligned} (w_s)_{s \in S} \in \sigma _S \mapsto \bigg (n_l \sum _{s \in S} a_{ls} \frac{w_s}{m_s}\bigg )_{l \in L} \in \sigma _L. \end{aligned}$$

(A.0.2)

Observe that $r_{ZX} = r_{YX} \circ r_{ZY}$, with $f_X= \text {Id}_X$. We show now that the map $r_{YX}$ is a retraction. To this end, we first recall the definition of simple blow-up.

Definition A.0.3

Let $t:(V_2, \Delta _{V_2}{:=}(t^{-1}\Delta _{V_1})_{red }) \rightarrow (V_1, \Delta _{V_1})$ is a morphism of snc pairs. We say that $\rho$ is a simple blow-up if it is a blowup along a smooth, connected subvariety W of $\Delta _{V_1}$ meeting transversely (or not at all) every irreducible component of $\Delta _{V_1}$ that does not contain W; see for instance [42, Definition 22].

Lemma A.0.4

The map $r_{YX}$ is a retraction.

Proof

The idea of the proof is to reduce to the snc case and apply [7, Proposition 4.3]. Let $X^{\mathrm {snc}}$ be the largest locus in X where the pair $(X, \Delta _X)$ is snc. There exists a birational modification $g:Z \rightarrow Y$ such that $h {:=}f \circ g :Z \rightarrow X$ is a composition of simple blow-up over $X^{\mathrm {snc}}$; see for instance [7, Lemma 4.1]. Up to passing to a dlt modification of Z which is an isomorphism over $h^{-1}(X^{\mathrm {snc}})$, we can suppose that $g:(Z, \Delta _Z) \rightarrow (Y, \Delta _Y)$ is a morphism of dlt compactifications. Note that $(X^{\mathrm {snc}}, \Delta ^{\mathrm {snc}}_X{:=}\Delta _X|_{X^{\mathrm {snc}}})$ and $(h^{-1}(X^{\mathrm {snc}}), h^{-1}(\Delta ^{\mathrm {snc}}_X))$ are snc pairs by definition of dlt pair and simple blow-up.

Taking dual complexes, we have that ${\mathcal {D}}(\Delta _X) = {\mathcal {D}}(\Delta ^{\mathrm {snc}}_X)$ again by definition of dlt pair, and ${\mathcal {D}}(h^{-1}(\Delta ^{\mathrm {snc}}_X))$ is a subcomplex of ${\mathcal {D}}(\Delta _Z)$. By construction, the restriction $r_{ZX}$ to ${\mathcal {D}}(h^{-1}(\Delta ^{\mathrm {snc}}_X))$ coincides with $r_{h^{-1}(X^{\mathrm {snc}}) X^{\mathrm {snc}}}$, which is a retraction by [7, Lemma 4.7]. The commutativity of the square

implies that $r_{ZX}$ is a retraction, and so $r_{YX}$ is a retraction too, since $r_{ZX}= r_{YX} \circ r_{ZY}$. $\square$

Lemma A.0.5

(Independence of auxiliary choices) Let $(X, \Delta _X = \sum _{i \in I}\Delta _{X,i})$ and $(Y, \Delta _Y = \sum _{k \in K}\Delta _{Y,k})$ be dlt compactifications of $M_{\mathrm {B}}$. Let $ev_X:N^*_X \rightarrow {\mathcal {D}}(\Delta _X)$ and $ev_Y:N^*_Y \rightarrow {\mathcal {D}}(\Delta _Y)$ be evaluation maps for suitable semialgebraic neighbourhoods $N_X$ of $\Delta _X$ and $N_Y$ of $\Delta _Y$ containing $\Psi (N^*_{\mathrm {Dol}})$. If ${\mathcal {D}}(\Delta _X)$ (equivalently ${\mathcal {D}}(\Delta _Y)$) has the homotopy type of a sphere, then there exists a homotopy equivalence ${\mathfrak {s}}:{\mathcal {D}}(\Delta _Y) \rightarrow {\mathcal {D}}(\Delta _X)$ which makes the following diagram homotopy commutative:

Proof

We proceed in several steps.

• Step 1 We can suppose that $(X, \Delta _X)$ and $(Y, \Delta _Y)$ are ${\mathbb {Q}}$-factorial. If $(X, \Delta _X)$ is not so (the same argument holds for $(Y, \Delta _Y)$ too), we take a small dlt ${\mathbb {Q}}$-factorial modification $\pi _1:(X_1, \Delta _1) \rightarrow (X, \Delta _X)$ such that ${{\,\mathrm{Supp}\,}}(\Delta _1)=\pi _1^{-1}({{\,\mathrm{Supp}\,}}(\Delta _X))$ as in [17, Definition 26] (mind that in our case $\Delta _1$ and $\Delta _X$ are reduced). Since $M_{\mathrm {B}}$ has ${\mathbb {Q}}$-factorial singularities, $\pi _1$ is an isomorphism over $M_{\mathrm {B}}$, and so $(X_1, \Delta _1)$ is a ${\mathbb {Q}}$-factorial dlt compactification of $M_{\mathrm {B}}$. Note that ${\mathcal {D}}(\Delta _1)={\mathcal {D}}(\Delta _X)$, and rug functions for $\Delta _{X, i}$ pull-back to rug functions for the divisor $\pi _1^{-1}(\Delta _{X, i})$. We conclude that the map $ev \circ \pi _1 :\pi _1^{-1}(N^*_X) \rightarrow {\mathcal {D}}(\Delta _X)$ is an evaluation map associated to the pair $(X_1, \Delta _1)$, and it can be identified with ev itself via the isomorphisms $\pi _1^{-1}(N^*_X) \simeq N^*_X$ and ${\mathcal {D}}(\Delta _1)={\mathcal {D}}(\Delta _X)$.

• Step 2 We can suppose that there exists a birational morphism $f:(Y, \Delta _Y) \rightarrow (X, \Delta _X)$. By construction, there exists a priori only a birational map $X \dashrightarrow Y$. Then choose $X_2$ a resolution of indeterminacy of the map which is an isomorphism over $M_{\mathrm {B}}$, i.e.

  

  Up to taking a dlt modification of $X_2$, we can suppose that $(X_2, \Delta _2 {:=}f^{-1}(\Delta _1)_{\mathrm {red}})$ is a ${\mathbb {Q}}$-factorial dlt compactification of $M_{\mathrm {B}}$. Therefore, Lemma A.0.5 for $(X, \Delta _X)$ and $(Y, \Delta _Y)$ holds if and only if it it holds simultaneously for the pairs $(X, \Delta _X)$ and $(X_2, \Delta _2)$, and for the pairs $(X_2, \Delta _2)$ and $(Y, \Delta _Y)$. Hence, we can suppose that Y dominates X, and we can now take ${\mathfrak {s}}{:=}r_{YX}$ as defined in (A.0.2).

• Step 3 For any $L \subseteq K$, let $\Delta _{X, J}$ be the minimum stratum of $\Delta _X$ such that $f(\Delta _{Y, L}) \subseteq \Delta _{X,J}$, with $J \subseteq I$. Assume first that $f(N_{Y, L})\subset N_{X,J}$. Consider the open cover of $N_Y$

  $$\begin{aligned} U_i {:=}\bigcup _{k :f(N_{Y,k})\subset N_{X,i}} N_{Y,k}, \text { with }i \in I. \end{aligned}$$

  By construction $\{{\mathfrak {s}} \circ ev_Y(i)\}_{i \in I}$ is a partition of unity subordinate to $\{U_i\}_{i \in I}$, while $\{ev_X (i) \circ f\}_{i \in I}$ is a partition of unity subordinate to $\{f^{-1}(N_i)\}_{i \in I}$. Note that for any $y \in U^{\circ }_J{:=}U_J \setminus \bigcup _{i \not \in J} U_i$, there exists $M \subseteq I$ containing J such that $y \in f^{-1}((N_{X,M})^{\circ })$. Denoting $\sigma _J$ the (closed) cell in ${\mathcal {D}}(\Delta _X)$ corresponding to $\Delta _{X,J}$, we have that ${\mathfrak {s}} \circ ev_Y(y) \in \sigma _J \subseteq \sigma _M$ and $ev_X \circ f(y) \in \sigma _M$. Hence, the segment $t \, ({\mathfrak {s}} \circ ev_Y(y)) +(1-t)\, (ev_X \circ f(y))\subset {\mathbb {R}}^M$, with $t \in [0,1]$, is entirely contained in $\sigma _M$, and so the convex convolution of ${\mathfrak {s}} \circ ev_Y \circ \Psi$ and $ev_X \circ \Psi$ gives the desired homotopy.

• Step 4 In Step 3 we assumed that $f(N_{Y,L})\subset N_{X,J}$ for any $L \subset K$. If this is not the case, we can shrink $N_{Y}= \beta ^{-1}[0, \delta )$ to $N'_{Y}{:=}\beta ^{-1}[0, \delta ')$, with $\delta '\ll \delta$, such that $f(N'_{Y, L})\subset N_{X,J}$. Choose an auxiliary evaluation map $ev'_Y :N'_{Y} \rightarrow {\mathcal {D}}(\Delta _Y)$. Applying Step 3 with $f=\mathrm {Id}_Y :Y \rightarrow Y$, we obtain that the maps $ev'_Y \circ \Psi$ and $ev_Y \circ \Psi$ are homotopic equivalent, and so again by Step 3 the following maps are homotopic to each other

  $$\begin{aligned} {\mathfrak {s}} \circ ev_Y \circ \Psi \sim {\mathfrak {s}} \circ ev'_Y \circ \Psi \sim ev_X \circ \Psi . \end{aligned}$$

• Step 5 By [17, Theorem 28] ${\mathcal {D}}(\Delta _X)$ and ${\mathcal {D}}(\Delta _Y)$ have the same homotopy type. If ${\mathcal {D}}(\Delta _X)$ has the homotopy of a sphere, then the retraction $r_{YX}$ has topological degree $\pm 1$, and so ${\mathfrak {s}}=r_{YX}$ is a homotopy equivalence by Hopf theorem.

$\square$

Appendix B: Local computations on the Tate curve

The goal of this appendix is to prove Corollary B.0.7, which is a technical ingredient needed in the proof of Lemma 5.3.4. To this end, we recall the construction of the Tate curve. Following [18, VII], it is a model over $R{:=}{\mathbb {C}}[[t]]$ of the multiplicative group ${\mathbb {G}}_{m}$ with special fibre given by an infinite chain of ${\mathbb {P}}^1$’s.

(B.0.1) Let $(x_i,y_{i+1})_{i \in {\mathbb {Z}}}$ be a collection of indeterminates. The Tate curve ${\overline{\mathscr {G}}}_m$ over $R$ is the union of the affine charts $(U_{i+1/2})_{i \in {\mathbb {Z}}}$ given by

$$\begin{aligned} U_{i+1/2}{:=}{{\,\mathrm{Spec}\,}}\left( \frac{R[x_i, y_{i+1}]}{(x_iy_{i+1}-t)} \right) . \end{aligned}$$

For each $i \in {\mathbb {Z}}$, the charts $U_{i-1/2}$ and $U_{i+1/2}$ are glued along the open subscheme

$$\begin{aligned} T_i {:=}U_{i-1/2} \cap U_{i+1/2}&= {{\,\mathrm{Spec}\,}}\left( {\mathcal {O}}(U_{i+1/2})[x_i^{-1}] \right) = {{\,\mathrm{Spec}\,}}\left( R[x_i, x^{-1}_i ]\right) \qquad \quad&(y_{i+1}= t/x_i)\\&= {{\,\mathrm{Spec}\,}}\left( {\mathcal {O}}(U_{i-1/2})[y_i^{-1}]\right) = {{\,\mathrm{Spec}\,}}\left( R[y_i, y^{-1}_i ] \right) \qquad \quad&(x_{i-1}= t/y_i) \end{aligned}$$

via the identification $x_iy_i =1.$

(B.0.2) The $R$-group scheme $\mathscr {G}_m {:=}\bigcup _{i \in {\mathbb {Z}}} T_i$, obtained from ${\overline{\mathscr {G}}}_m$ by removing the nodes in the special fibre, is the Néron model of the multiplicative group ${\mathbb {G}}_m$, as explained in [18, Example 1.2.c]. In particular, the n-th multiplication map ${\mathbb {G}}_m \times \ldots \times {\mathbb {G}}_m \rightarrow {\mathbb {G}}_m$ extends to a homomorphism

$$\begin{aligned} \mu _n :\mathscr {G}^n_m {:=}\mathscr {G}_m \times _R\ldots \times _R\mathscr {G}_m \longrightarrow \mathscr {G}_m \end{aligned}$$

of $R$-group schemes. When $n=2$, $\mu _n$ is given in local charts by

As $x_{i-1}y_i =t$ and $x_i y_i =1$, it follows that $x_i = t^{-i}x_0$. In particular, the identity section ${\mathcal {I}}d$ of $\mathscr {G}_m$ is cut out in the chart $T_i$ by the equation $x_i = t^{-i}$.

Let $\mathscr {V}_{n-1}{:=}\mu _n^{-1}({\mathcal {I}}d)$ be the fibre of the identity section ${\mathcal {I}}d$ via $\mu _n$, and let ${\overline{\mathscr {V}}}_{n-1}$ denote the closure of $\mathscr {V}_{n-1}$ in ${\overline{\mathscr {G}}}_m^n$. Proposition B.0.3 describes some properties of $({\overline{\mathscr {V}}}_{n-1}, {\overline{\mathscr {V}}}_{n-1,0})$, where ${\overline{\mathscr {V}}}_{n-1,0}$ is the special fibre of ${\overline{\mathscr {V}}}_{n-1}$;

Proposition B.0.3

The pair $({\overline{\mathscr {V}}}_{n-1}, {\overline{\mathscr {V}}}_{n-1, 0})$ is normal, reduced, and toric, i.e. ${\overline{\mathscr {V}}}_{n-1}$ is the formal completion of a normal toric scheme along its reduced toric boundary ${\overline{\mathscr {V}}}_{n-1, 0}$. Furthermore, the intersection ${\overline{\mathscr {V}}}_{n-1} \setminus \mathscr {V}_{n-1}$ has codimension two in ${\overline{\mathscr {V}}}_{n-1}$.

The Proposition B.0.3 is an immediate corollary of Lemma B.0.4 below. Indeed, the assertions in Proposition B.0.3 are local: we may work on the the open subsets

$$\begin{aligned} U_{\alpha }\,{:=}\, U_{\alpha _1 + 1/2} \times _{R} \ldots \times _{R} U_{\alpha _n + 1/2}, \end{aligned}$$

for any multi-index $\alpha =(\alpha _1, \ldots , \alpha _n) \in {\mathbb {Z}}^n$, since the $U_{\alpha }$’s cover ${\overline{\mathscr {G}}}_m^n$. Let $\mathscr {V}_{\alpha }$ be the restriction of $\mathscr {V}_{n-1}$ to $T_{\alpha }{:=}T_{\alpha _1} \times _{R} \ldots \times _{R} T_{\alpha _n}$, ${\overline{\mathscr {V}}}_{\alpha }$ be its closure in $U_{\alpha }$, and ${\overline{\mathscr {V}}}_{\alpha , 0}$ be the special fibre of ${\overline{\mathscr {V}}}_{\alpha }$. In local coordinates,

Lemma B.0.4

For any $\alpha \in {\mathbb {Z}}^n$, the pair $({\overline{\mathscr {V}}}_{\alpha }, {\overline{\mathscr {V}}}_{\alpha , 0})$ is normal, reduced, and toric. Furthermore, the intersection ${\overline{\mathscr {V}}}_{\alpha } \setminus \mathscr {V}_\alpha$ has codimension two in ${\overline{\mathscr {V}}}_{\alpha }$.

Proof

The proof is divided into cases depending on the sign of $|\alpha | {:=}\sum _{i=1}^n \alpha _i$.

• Case 1. Assume $|\alpha | >0$. The equation $t^{|\alpha |} \prod _{i=1}^n x_{\alpha _i} = 1$ vanishes on ${\overline{\mathscr {V}}}_\alpha$. This implies that t and all $x_{\alpha _i}$ are invertible on it, thus ${\overline{\mathscr {V}}}_{\alpha ,0} = \emptyset$ and ${\overline{\mathscr {V}}}_\alpha = \mathscr {V}_\alpha \simeq {\mathbb {G}}^{n-1}_{{\mathbb {C}}((t))}$.

• Case 2. Assume $|\alpha |=0$. The equation $\prod _{i=1}^n x_{\alpha _i} = 1$ implies that all $x_{\alpha _i}$ are invertible on ${\overline{\mathscr {V}}}_\alpha$. We obtain that

  $$\begin{aligned} {\overline{\mathscr {V}}}_{\alpha } \subseteq \left\{ \prod _{i=1}^n x_{\alpha _i} = 1, \, x_{\alpha _i}y_{\alpha _i +1} = t \right\} \simeq {\mathbb {G}}^{n-1}_R, \end{aligned}$$

  thus $({\overline{\mathscr {V}}}_{\alpha }, {\overline{\mathscr {V}}}_{\alpha , 0})=({\mathbb {G}}^{n-1}_R, {\mathbb {G}}^{n-1}_{\mathbb {C}})$ and ${\overline{\mathscr {V}}}_{\alpha } \setminus \mathscr {V}_\alpha = \emptyset$.

• Case 3. Assume $|\alpha |<0$. We will show that ${\overline{\mathscr {V}}}_{\alpha }$ is normal by proving the conditions $\mathrm {S}_2$ and $\mathrm {R}_1$, and in the process we deduce that $({\overline{\mathscr {V}}}_{\alpha }, {\overline{\mathscr {V}}}_{\alpha , 0})$ is toric, ${\overline{\mathscr {V}}}_{\alpha , 0}$ is reduced and ${\overline{\mathscr {V}}}_{\alpha }\setminus {\mathscr {V}}_{\alpha }$ has codimension two in ${\overline{\mathscr {V}}}_{\alpha }$.

  • Step 1. We prove that ${\overline{\mathscr {V}}}_{\alpha }$ is Cohen-Macaulay (hence $\mathrm {S}_2$) by applying [36, Lemma 7]: if a Gorenstein scheme of pure dimension d is a union of two closed subschemes of pure dimension d and one is Cohen-Macaulay, then the other is Cohen-Macaulay. Let $\mathscr {Z}_\alpha$ be the closed toric subscheme of $U_\alpha$ given by the equations

    $$\begin{aligned} \mathscr {Z}_\alpha {:=}{\left\{ \begin{array}{ll} t \big ( \prod ^n_{i=1} x_{\alpha _i}- t^{-|\alpha |} \big )=0,\\ x_{\alpha _1}y_{\alpha _1 +1}=\ldots =x_{\alpha _n}y_{\alpha _n +1}=t. \end{array}\right. } \end{aligned}$$

    We have

    $$\begin{aligned}&\mathscr {Z}_{\alpha , 0} = U_{\alpha ,0} = \{x_{\alpha _1}y_{\alpha _1 +1}=\ldots =x_{\alpha _n}y_{\alpha _n +1}=0 \}, \\&\mathscr {Z}_\alpha \cap \{t \ne 0 \} = \mathscr {V}_\alpha \cap \{t \ne 0 \} \simeq {\mathbb {G}}_{m, {\mathbb {C}}((t))}^{n-1}, \end{aligned}$$

    so $\mathscr {Z}_\alpha = U_{\alpha ,0} \cup {\overline{\mathscr {V}}}_\alpha$. We can then apply [36, Lemma 7] as $\mathscr {Z}_\alpha$ and $U_{\alpha ,0}$ are complete intersections. Hence, ${\overline{\mathscr {V}}}_\alpha$ is Cohen-Macaulay. In particular, the pair $({\overline{\mathscr {V}}}_{\alpha }, {\overline{\mathscr {V}}}_{\alpha , 0})$ is toric, as both ${\overline{\mathscr {V}}}_{\alpha }$ and ${\overline{\mathscr {V}}}_{\alpha , 0}$ are torus-invariant subschemes of $\mathscr {Z}_{\alpha }$ and ${\overline{\mathscr {V}}}_{\alpha } \setminus {\overline{\mathscr {V}}}_{\alpha ,0} \simeq {\mathbb {G}}_{m,{\mathbb {C}}((t))}^{n-1}$.

  • Step 2. To prove that ${\overline{\mathscr {V}}}_\alpha$ is $\mathrm {R}_1$, it suffices to check that ${\overline{\mathscr {V}}}_\alpha$ is smooth at the generic point of the irreducible components of ${\overline{\mathscr {V}}}_{\alpha ,0}$, since ${\overline{\mathscr {V}}}_\alpha \cap \{t \ne 0 \}=\mathscr {V}_\alpha \cap \{t \ne 0\}$ is smooth. Let $D \subseteq {\overline{\mathscr {V}}}_{\alpha ,0}$ be one of them. Up to relabelling, there exists an integer m such that the chart U of $U_{\alpha ,0}$

    $$\begin{aligned} U {:=}\left\{ \begin{aligned}&x_{\alpha _1}= \ldots = x_{\alpha _m} = y_{\alpha _{m+1} +1}= \ldots = y_{\alpha _{n} +1}=0 \\&y_{\alpha _{2}+1} \ne 0, \ldots , y_{\alpha _{m}+1} \ne 0, x_{\alpha _{m+1}}\ne 0, \ldots , x_{\alpha _{n}} \ne 0 \end{aligned} \right\} \simeq {\mathbb {G}}_{m, {\mathbb {C}}}^{n-1} \times {\mathbb {A}}^1_{\mathbb {C}}\end{aligned}$$

    contains the generic point of D. At the generic point of D we have

    $$\begin{aligned} {\left\{ \begin{array}{ll} x_{\alpha _i}= x_{\alpha _1} y_{\alpha _1 + 1} y_{\alpha _i+1}^{-1} &{} \text { for }2 \leqslant i \leqslant m \\ y_{\alpha _i+1}= x_{\alpha _1} y_{\alpha _1 + 1} x_{\alpha _i}^{-1} &{} \text { for }m+1 \leqslant i \leqslant n, \\ \end{array}\right. } \end{aligned}$$

    and ${\overline{\mathscr {V}}}_\alpha$ is an irreducible component of the locus given by the equation

    $$\begin{aligned} \prod _{i=1}^{n} x_{\alpha _i} = x_{\alpha _1}^{m} y_{\alpha _1 +1}^{m-1} \underbrace{\bigg ( \prod _{j=2}^m y_{\alpha _{j}+1}^{-1}\prod _{i=m+1}^n x_{\alpha _i}\bigg )}_{u} = x_{\alpha _1}^{-|\alpha |} y_{\alpha _1 +1}^{-|\alpha |}. \end{aligned}$$

    (B.0.5)

    If $m > - |\alpha |$, then (B.0.5) implies that $x_{\alpha _{1}}^{-|\alpha |} y_{\alpha _1 + 1}^{{-|\alpha |}}(u x_{\alpha _{1}}^{m+|\alpha |} y_{\alpha _1 + 1}^{{m-1+|\alpha |}}- 1)=0$. This is impossible: ${\overline{\mathscr {V}}}_\alpha$ would be cut by the equation $u x_{\alpha _{1}}^{m+|\alpha |} y_{\alpha _1 + 1}^{{m-1+|\alpha |}}= 1$ at the generic point of D, but then $D \not \subseteq {\overline{\mathscr {V}}}_\alpha$. The same argument proves that the case $m < - |\alpha |$ cannot occur. If $m = - |\alpha |$, then (B.0.5) implies that $x_{\alpha _{1}}^m y_{\alpha _1 + 1}^{m-1}(u- y_{\alpha _1 +1})=0$. It follows that

    $$\begin{aligned} {\overline{\mathscr {V}}}_\alpha {\mathop {=}\limits ^{\text {loc at { D}}}} \{u- y_{\alpha _1 +1} =0 \}, \end{aligned}$$

    (B.0.6)

    hence ${\overline{\mathscr {V}}}_\alpha$ is $\mathrm {R}_1$, and ${\overline{\mathscr {V}}}_{\alpha , 0} {\mathop {=}\limits ^{\text {loc at D}}} \{x_{\alpha _1} =0 \}$ is reduced.

  • Step 3 A point in ${\overline{\mathscr {V}}}_{\alpha } \setminus \mathscr {V}_\alpha \subseteq U_{\alpha } \setminus T_\alpha$ is characterised by the property that a pair of coordinates $(x_{\alpha _i}, y_{\alpha _i +1})$ vanishes simultaneously. Hence, (B.0.6) yields that $\mathscr {V}_\alpha$ coincides with ${\overline{\mathscr {V}}}_{\alpha }$ at the generic point of D. Therefore, ${\overline{\mathscr {V}}}_{\alpha } \setminus \mathscr {V}_\alpha$ has codimension two in ${\overline{\mathscr {V}}}_{\alpha }$.

$\square$

Let $\mathscr {X}_{n-1}$ and ${\overline{\mathscr {X}}}_{n-1}$ be as in Proposition 5.3.2 and consider the diagram (5.3.3). Given the uniformisation $p:{\overline{\mathscr {G}}}_m \rightarrow {\mathscr {E}}$, we have

Corollary B.0.7

The pair $({\overline{\mathscr {X}}}_{n-1}, {\overline{\mathscr {X}}}_{n-1,0})$ is reduced lc and ${\overline{\mathscr {X}}}_{n-1} \setminus \mathscr {X}_{n-1}$ has codimension two in ${\overline{\mathscr {X}}}_{n-1}$.

Proof

The restriction of P to ${\overline{\mathscr {V}}}_{n-1} \times _R {\overline{\mathscr {V}}}_{n-1}$ is étale. Hence it suffices to prove the statement for the pair $\big ({\overline{\mathscr {V}}}_{n-1} \times _R {\overline{\mathscr {V}}}_{n-1} , ({\overline{\mathscr {V}}}_{n-1} \times _R {\overline{\mathscr {V}}}_{n-1})\,_0 \big )$. This follows directly from Proposition B.0.3, taking the fibre product. $\square$