Torsors on semistable curves and degenerations

Abstract

In this paper, we answer two long-standing questions on the classification of G-torsors on curves for an almost simple, simply connected algebraic group G over the field of complex numbers. The first is the construction of a flat degeneration of the moduli of G-torsors on smooth projective curves when the smooth curve degenerates to an irreducible nodal curve and the second one is to give an intrinsic definition of (semi)stability for a G-torsor on an irreducible projective nodal curve. A generalization of the classical Bruhat–Tits group schemes to two-dimensional regular local rings and an application of the geometric formulation of the McKay correspondence provide the key tools.

Appendices

Appendices

Appendix A. Appendix to Part I

1.1 Quasi-Gieseker bundles

In the first part of the appendix, we will outline a small variant of the theme developed in [24, 27, 38].

We recall the notion of an admissible vector bundle V on a curve $C^{{(d)}}$ [27, Definition 3.11], [8, Definition 3.6] and add a variant, namely the notion of a quasi-admissible bundle. In fact, Kiem and Li in [32, Lemma 1.2(a)] just call these admissible bundles. In [38, Definition 1, page 167], we have the notion of a standard vector bundle on $C^{{(d)}}$ as a preliminary notion.

DEFINITION A1.1

Let V be a vector bundle of rank n on a chain $E^{{(d)}}$. Let $V\mid _{{R_i}} = \oplus _{j=1}^{n} {{\mathscr {O}}}(a_{ij})$, where the $R_{j}$ are the ${{\mathbb {P}}}^{1}$’s on the chain $E^{{(d)}}$. Say that V is standard if the $a_{ij}$ are 0 or 1. The bundle V is called strictly standard if moreover, for every i there is an index j such that $a_{ij} = 1$.

A vector bundle V on $C^{{(d)}}$ of rank n is called admissible (resp. quasi-admissible), if, for $d \ge 1$, the restriction $V|{{{E^{{(d)}}}}}$ is strictly standard (resp. standard) and the direct image $(p_{d})_{*}(V)$ is a torsion-free ${{\mathscr {O}}}_{C}$-module, where $p_{d}:C^{{(d)}} \rightarrow C$ is the canonical morphism which contracts the chain to the node.

The notion of admissibility (resp. quasi-admissibility) extends obviously to vector bundles on any modification $\mathtt M$ over $T \in \text {Sch} \big /{A}.$ Let V be a standard vector bundle on $C^{{(d)}}$ of $\text {rank}(V) = n$. Then, by the discussions in [38, page 168-171], after twisting the vector bundles sufficiently to ensure the vanishing of the first cohomology and ensure generation by sections, we get a canonical morphism $\phi _{V}:C^{{(d)}} \rightarrow \text {Grass}(\text {H}^{0}(V), n)$. This morphism contracts the $R_{j} = {{\mathbb {P}}}^{1}$’s on the chain $E^{{(d)}} \subset C^{{(d)}}$ such that the restriction $V {{\mid _{{R_{j}}}}}$ is trivial. The condition that V is strictly standard is shown to be equivalent to the morphism $\phi _{V}$ being a closed immersion.

Let $N := \mathtt{dim}(\text {H}^{0}(V))$. Let W[j] be the j-th standard model with $C^{{(j)}} \subset W[j]$ as the central fibre (4.1). Recall that this a smooth quasi-projective scheme with a tautological morphism $W[j] \rightarrow C \times _{{A}} B[j]$. For each $j \le n$, fix the coordinate plane embedding ${{\mathbb {A}}}^{{j+1}} \subset {{\mathbb {A}}}^{{n+1}}$ by the first coordinates. This gives an identification $W[j] \simeq W[n]\times _{{B[n]}} B[j]$ compatible with the tautological morphism [33, page 526]. Define

$$\begin{aligned} W(N,n) := W[n] \times _{{A}} \text {Grass}(N, n). \end{aligned}$$

(A1.1)

If V is a standard bundle on $C^{{(j)}}$ for some j, we get a closed immersion

$$\begin{aligned} C^{{(j)}} \subset W(N,n), \end{aligned}$$

(A1.2)

via the inclusions $\text {Graph}(\phi _{V}) \subset C^{{(j)}} \times \text {Grass}(N, n) \subset W(N,n)$.

Following [24, page 179] and [38, Definition 7, page 185], we have the definition.

DEFINITION A1.2

Let ${{{\mathscr {G}}}}^{{\mathtt q}}_{{N}}:\mathrm{Sch}_{{A}} \rightarrow \mathrm{Sets}$, be the functor defined as

(A1.3)

where

(A1.4)

is a closed embedding in the product and such that, (a) the projection $j:\mathtt M \rightarrow T\times _{k} W(N,n)$ is a closed immersion, (b) the projection $\pi :\mathtt M \rightarrow C \times _{{A}} T$ is a modification as in Definition 6.1, and (c) the projection $q_{T}:\mathtt M \rightarrow T$ is a flat family of curves $\mathtt{M}$, $t \in T$ as in Definition 6.1. (d) Moreover, the chain lengths d occuring in $\mathtt{M}$ is bounded above by n.

Further, if V is the tautological quotient bundle of rank n on $\text {Grass}(N,n)$ and $V_{T}$ its pull-back to $T\times \ W(N,n)$, then the pull-back ${{\mathscr {V}}}_{T}:= j^*(V_{T})$ is such that, ${{\mathscr {V}}}_{T}$ is a quasi-admissible vector bundle of rank n(A1.1) for the modification $\mathtt M \rightarrow C \times _{{A}} T$.

By the definition of ${{\mathscr {V}}}_{T}$, for each $t \in T$ we get a quotient morphism ${{\mathscr {O}}}^{N}_{\mathtt{M}} \twoheadrightarrow {{\mathscr {V}}}_{t}$, and we assume that this map induces an isomorphism: $H^{0}({{\mathscr {O}}}^{N}_{\mathtt{M}}) \simeq H^{0}({{\mathscr {V}}}_{t})$. In particular, we have $\mathtt{dim}(H^{0}({{\mathscr {V}}}_{t})) = N$ and it follows that

$$\begin{aligned} H^{1}({{\mathscr {V}}}_{t}) = 0. \end{aligned}$$

(A1.5)

As in [24] and [39, Proposition 8], it is easily seen that this new functor ${{{\mathscr {G}}}}^{{{\mathtt q}}}_{{{N}}}$ is also represented by a $\text {PGL}(N)$-invariant open subscheme $\mathtt Y$ of the Hilbert scheme $\text {Hilb}(C_{{A}} \times _{{A}} W(N,n))$ for the natural polarization on W(N, n).

Let ${\mathbb {M}} \subset C_{{A}} \times _{{A}} \mathtt Y\times _{k} \ W(N,n)$ be the universal object defining the functor ${{{\mathscr {G}}}}^{{\mathtt q}}_{{{N}}}$. This defines a universal modification ${\mathbb {M}} \rightarrow \mathtt Y$ together with a universal quasi-admissible vector bundle ${\mathbb {V}}$ on ${\mathbb {M}}$. The representability of the functor ${{{\mathscr {G}}}}^{{\mathtt q}}_{{{N}}}$ implies that for any quasi-admissible vector bundle V on a modification $\mathtt M_{{T}}$ there exists a unique morphism $\psi :T \rightarrow \mathtt Y$ and $\phi :\mathtt M_{{T}} \rightarrow {\mathbb {M}}$ so that $\phi ^{{*}}({\mathbb {V}})$ is V.

The stack $\text {GVB}^{{\mathtt q}}_{n}(C_{{A}})$ (cf. [27, Definition 3.11]): $(\mathtt M, {\mathscr {V}}) \in \text {GVB}^{{\mathtt q}}_{n}(C_{{A}})(T)$ is such that (1) ${\mathscr {V}}$ is a quasi-admissible vector bundle on the modification $\mathtt M$ and (2) $d \le n$ for chains $E^{{(d)}}$ in $\mathtt M$ . We may call $(\mathtt M, {\mathscr {V}})$ a quasi-Gieseker bundle. Modifications with bounded chain lengths is easily seen to be a stack and $\text {GVB}^{{\mathtt q}}_{n}(C_{{A}})$ is easily checked to be an Artin stack.

As in [27, Definition 3.22], if we fix a very ample sheaf on C. Then for a quasi-Gieseker vector bundle $(\mathtt M, {{\mathscr {V}}})$ for T a A-scheme and for an integer $N'$ we have the quasi-admissible bundle ${{\mathscr {V}}}(N')$ and for every pair of integers $N \ge n, N' \ge 0$, we have a canonical morphism of A-groupoids

$$\begin{aligned} {{{\mathscr {G}}}}^{{\mathtt q}}_{{N}} \rightarrow \text {GVB}^{{\mathtt q}}_{n}(C_{{A}}). \end{aligned}$$

(A1.6)

Analogous to [27, Lemma 3.23], given a quasi-Gieseker bundle $(\mathtt{M} \rightarrow C \times _{{A}} T, {{\mathscr {V}}})$, we again have an open subschemes $T_{{N,N'}} \subset T$ which has properties (1) and (2) in [27, Lemma 3.23], with the added observation that the scheme W(N, n), which replaces the Grassmannian in loc cit, ensures that $\forall t \in T_{{N,N'}}$, the induced morphism $\mathtt{M_{t}} \rightarrow W(N,n)$ is a closed immersion.

For the analogue of [27, Proposition 3.24], we need to do a bit more.

PROPOSITION A1.3

The morphism of A-groupoids

$$\begin{aligned} \coprod _{{N \ge n, N'\ge 0}} {{{\mathscr {G}}}}^{{\mathtt q}}_{{N}} \rightarrow \mathrm{GVB}^{{\mathtt q}}_{n}(C_{{A}}) \end{aligned}$$

(A1.7)

is smooth and surjective.

Proof

Let T be a A-scheme and let $T \rightarrow \text {GVB}^{{\mathtt q}}_{n}(C_{{A}})$ a T-point on $\text {GVB}^{{\mathtt q}}_{n}(C_{{A}})$ given by a quasi-Gieseker bundle $(\mathtt{M} \rightarrow C \times _{{A}} T, {{\mathscr {V}}})$. Let Z be the A-groupoid defined by the cartesian square

(A1.8)

Let $\{T_{\alpha }\}$ be en étale cover of T so that we have a morphism $T_{\alpha } \rightarrow B[d_{\alpha }]$ and modification $\mathtt M_{\alpha }$ comes as a pull-back. For each quasi-Gieseker bundle $(\mathtt M_{\alpha }, {{\mathscr {V}}}_{\alpha })$, we again have open subschemes $T_{{N,N',\alpha }} \subset T_{\alpha }$ with properties as stated above. We in fact have a morphism $\mathtt M \mid _{{T_{{N,N',\alpha }}}} \longrightarrow T_{{N,N',\alpha }} \times W[d_{\alpha }] \times \text {Grass}(N,n)$ and hence a morphism $\mathtt M \mid _{{T_{{N,N',\alpha }}}} \longrightarrow T_{{N,N',\alpha }} \times W(N,n)$. This morphism is proper and for each $\forall t \in T_{{N,N',\alpha }}$, the induced morphism $\mathtt M_{t} \rightarrow W(N,n)$ is a closed immersion. Hence by [27, Lemma 3.13], we get a closed immersion $\mathtt M \mid _{{T_{{N,N',\alpha }}}} \hookrightarrow T_{{N,N',\alpha }} \times W(N,n)$.

Let $Z_{\alpha } = T_{\alpha } \times _{T} Z$.Then, following the arguments in [27, page 4913], we again have the identification $Z_{\alpha } = \text {Isom}({{\mathscr {O}}}^{N}_{{T_{{N,N',\alpha }}}}, \pi _{*}({{\mathscr {V}}}_{\alpha })(N') \mid _{{T_{{N,N',\alpha }}}})$, where $\pi :\mathtt M_{\alpha } \rightarrow T_{\alpha }$. Thus, $Z_{\alpha }$ is smooth and surjective over $T_{{N,N',\alpha }}$ and since the $T_{{N,N',\alpha }}$ cover $T_{\alpha }$ for each $\alpha$ we are done. $\square$

Remark A1.4

The analogues of [27, Theorem 3.21] hold without any serious difficulty. In particular, the deformation theory works to show that $\mathtt Y$ is regular, its generic fibre over A is smooth while its special fibre $\mathtt Y_{\mathtt{o}}$ is a divisor with normal crossings. The proof of (7.7) gets easily adapted to this case.

1.2 Kawamata coverings

Let X be a smooth quasi-projective variety and let $D = \sum _{{i=1}}^{r} D_i$ be the decomposition of the simple or reduced normal crossing divisor D into its smooth components (intersecting transversally). The ‘covering lemma’ of Kawamata [30, Theorem 17] (see also [54, Lemma 2.5, page 56]) says that, given positive integers $N_{1}, \ldots , N_{r}$, there is a connected smooth quasi-projective variety Z over ${{\mathbb {C}}}$ and a Galois covering morphism

$$\begin{aligned} \kappa :Z \rightarrow X \end{aligned}$$

(A1.9)

such that the reduced divisor $\kappa ^{*}{D}:= \,({\kappa }^{*}D)_{{\text {red}}}$ is a normal crossing divisor on Z and furthermore, ${\kappa }^{*}D_{i}= N_{i}.({\kappa }^{*}D_{i})_{{\text {red}}}$. Let $\Gamma$ denote the Galois group for the covering map $\kappa$.

The isotropy group of any point $z \in Z$, for the action of $\Gamma$ on Z, will be denoted by ${\Gamma }_{z}$. It is easy to see that the stabilizer at generic points of the irreducible components of $(\kappa ^{*}D_i)_{{\text {red}}}$ are cyclic of order $N_{i}$. By an equivariant principal G-torsor P on Z of local type $\tau = \{\tau _{i}\}_{{i=1}}^{r}$, we mean

• (1) the restriction of the G-torsor $P_{{U_{z}}}$ to an étale neighbourhood at a generic point z of an irreducible component of $(\kappa ^{*}D_i)_{{\text {red}}}$ is given by a representation $\rho _{i}:\Gamma _{z} \rightarrow G$;

• (2) for a general point y of an irreducible component of a ramification divisor for $\kappa$ not contained in $(\kappa ^{*}D)_{{\text {red}}}$, the action of ${\Gamma }_{y}$ on P is the trivial action.

Such a P will always exists as an algebraic space with a G-action and can be obtained by gluing trivial $(\Gamma _{z},G)$-torsors given by $\rho _{i}$, in $U_{z}$ for the generic point z of $(\kappa ^{*}D_i)_{{\text {red}}}$ with pull-backs of G-torsors on $X \setminus D$ to Z. By a Hartogs type argument, it is easily checked that equivariant G-torsors are uniquely defined on Z once given on a subscheme of codimension bigger than 1.

Appendix B. Appendix to Part II

1.1 Laced vector bundles

In this subsection we analyse the special case of laced torsors when G is the linear group. Much of the early material in this subsection is adapted from [48].

Notation B1.1

Let ${\text {Vect}}_{{(\tilde{{{\mathscr {C}}}_{d}},\mathbf{z})}}^{\mathbf{d}}$ denote the category of vector bundles W on $(\tilde{{{\mathscr {C}}}_{d}}, \mathbf{z})$, i.e., balanced vector bundles on $(\tilde{{{\mathscr {C}}}_{d}}, \mathbf{z})$ with descent datum (12.0.4) which translates as an isomorphism $V_{{z_{1}}} \simeq V^{*}_{{z_{2}}}$.

DEFINITION B1.2

A balanced parabolic structure on a vector bundle V of rank n on a doubly marked curve $({\tilde{C}}, \mathbf{c})$ is given by the following datum:

• (1) For $1 \le s \le n$, weights, $(\alpha _{1}, \ldots , \alpha _{s})$, which are rational numbers such that

  $$\begin{aligned} 0 \le \alpha _{1}< \alpha _{2}<\cdots< \alpha _{s} < 1. \end{aligned}$$

  (B1.1)

  and ‘dual weights’:

  $$\begin{aligned} (\beta _{1}, \ldots , \beta _{s}) = \left\{ \begin{array}{l} (1-\alpha _{s}, 1-\alpha _{{s-1}}, \ldots , 1-\alpha _{1})~~\mathrm{if}~ \alpha _{1} \ne 0 \\ (0, ~~~~ 1-\alpha _{s}, \ldots ,1-\alpha _{2}) ~~~\mathrm{if} ~\alpha _{1}=0. \end{array} \right. \end{aligned}$$

  (B1.2)

• (2) A balanced parabolic structure on V at $c_{j}$, $j = 1,2$, i.e., strictly decreasing flags

  $$\begin{aligned} V_{c_{j}}\, =\, {{\mathscr {F}}}_{c_{j}}^{1}\, \supset \, {{\mathscr {F}}}_{c_{j}}^{2}\, \supset \, \cdots \,\supset \, {{\mathscr {F}}}_{c_{j}}^{2}\,\supset \, {{\mathscr {F}}}_{c_{j}}^{{s+1}}\,=\, 0, ~~j=1,2 \end{aligned}$$

  (B1.3)

  together with weights are given as follows:

  • The weight of ${{\mathscr {F}}}_{c_{1}}^{m}$ is $\alpha _{m}$, where $\alpha _{1},\ldots ,\alpha _{s}$ as in (B1.1).

  • The weight of ${{\mathscr {F}}}_{c_{2}}^{m}$ is $\beta _{m}$, where $\beta _{1}, \ldots , \beta _{s}$ are as in (B1.2).

Let $\mathrm{PVect}^{\mathrm{bal}}_{{({\tilde{C}}, \mathbf{c})}}$ denote the category of vector bundes on $({\tilde{C}}, \mathbf{c})$ with balanced parabolic structure.

Let V be an object in $\text {PVect}^{\mathrm{bal}}_{{({\tilde{C}}, \mathbf{c})}}$.

• (1) The flag ${{\mathscr {F}}}_{c_{2}}$ and $V_{c_{2}}$ induces on the dual $V_{c_{2}}^*$ of $V_{c_{2}}$, the natural dual flag ${{\mathscr {F}}}_{c_{2}}^*$ and the weights of ${{\mathscr {F}}}_{c_{2}}^*$ are ‘dual’ to those of ${{\mathscr {F}}}_{c_{2}}$, i.e., they coincide with $\alpha _{1},\ldots , \alpha _{s}$, the weights associated to ${{\mathscr {F}}}_{y_{1}}$.

• (2) For $i = 1,2$, define

  $$\begin{aligned} \mathtt{gr}(V_{c_{i}}):= & {} \bigoplus _{m} \mathtt{gr}^{m} {{\mathscr {F}}}_{c_{i}},~\mathrm{with} \end{aligned}$$

  (B1.4)
  $$\begin{aligned} \mathtt{gr}^{m} {{\mathscr {F}}}_{c_{i}}:= & {} {{\mathscr {F}}}_{c_{i}}^{m}/{{\mathscr {F}}}_{c_{i}}^{{m+1}}. \end{aligned}$$

  (B1.5)

  The graded pieces, $\mathtt{gr}(V_{c_{2}}^*)$ gets identified with $\mathtt{gr}(V_{c_{2}})$ by a shifting of degrees as follows:

  $$\begin{aligned} \left\{ \begin{array}{l} \mathtt{gr}^{m} {{\mathscr {F}}}^*_{c_{2}} = \mathtt{gr}^{{s+1-m}} {{\mathscr {F}}}_{c_{2}}~for~ ~1 \le m \le s,~\mathrm{if} ~ \alpha _{1} \ne 0 \\ \mathtt{gr}^{1} {{\mathscr {F}}}^*_{c_{2}} = \mathtt{gr}^{1}{{\mathscr {F}}}_{c_{2}}~\mathrm{and}~ \mathtt{gr}^{m} {{\mathscr {F}}}_{c_{2}}^* = \mathtt{gr}^{{s+2-m}} {{\mathscr {F}}}_{c_{2}} ~\mathrm{if} ~ \alpha _{1}=0. \end{array} \right. \end{aligned}$$

  (B1.6)

DEFINITION B1.3

Let V be an object in $\mathrm{PVect}^{\mathrm{bal}}_{{({\tilde{C}}, \mathbf{c})}}$. A lacing on V (or more precisely a s-lacing) is a s-tuple

$$\begin{aligned} \wp : = \{\wp _{m}: \mathtt{gr}^{m} {{\mathscr {F}}}_{c_{1}} \rightarrow \mathtt{gr}^{m} {{\mathscr {F}}}^*_{c_{2}}\}^{{s}}_{{m = 1}} \end{aligned}$$

(B1.7)

of linear isomorphisms.

DEFINITION B1.4

A balanced parabolic vector bundle endowed with a lacing $\wp$ will be called a laced vector bundle, i.e., given by the datum

$$\begin{aligned} V_{\wp } := (V_{\star }, \wp ), \end{aligned}$$

(B1.8)

where $V_{\star }$ is a balanced parabolic bundle on $({\tilde{C}}, \mathbf{c}).$

DEFINITION B1.5

The parabolic degree of a laced bunde $V_{\wp }$ is defined as

$$\begin{aligned} \mathtt{par.deg}_{{{\tilde{C}}}}(V_{\wp }):= \mathtt{par.deg}_{{{\tilde{C}}}}(V_{\star }). \end{aligned}$$

(B1.9)

Lemma B1.6

Let $V_{\wp }$ be a laced bundle on $({\tilde{C}}, \mathbf{c})$ and let $k = k_1$ denote the multiplicity of the weight $\alpha _1$. Let $l = (n-k)$. Then

$$\begin{aligned} \mathtt{par.deg}_{{{\tilde{C}}}}~V_{\wp } = \deg V + (n-k) = \deg V + l. \end{aligned}$$

(B1.10)

As a consequence, the parabolic degree of a laced bundle on $({\tilde{C}}, \mathbf{c})$ does not depend on the choice of the parabolic weights.

Proof

[48]. By the definition of parabolic degree, we see that

$$\begin{aligned} \mathtt{par.deg}_{{{\tilde{C}}}}~V_{\wp } = \left\{ \begin{array}{rcl} \deg V + \sum _{{m=1}}^{s} k_{m} \alpha _{m} + \sum _{{m=1}}^{s} k_{m}(1-\alpha _{m}) ~\mathrm{if}~\alpha _{1} \ne 0 \\ \deg V + \sum _{{m=2}}^{s} k_{m} \alpha _{m} + \sum _{{m=2}}^{s} k_{m}(1-\alpha _{m}) ~\mathrm{if}~\alpha _{1} = 0.\\ \end{array}\right. \nonumber \\ \end{aligned}$$

(B1.11)

Hence

$$\begin{aligned} \mathtt{par.deg}_{{{\tilde{C}}}}~V_{\wp } = \left\{ \begin{array}{lcl} \deg V+n \mathrm{~if~} \alpha _1 \ne 0 \\ \deg V + (n-k_1) \mathrm{~if~} \alpha _1 = 0.\\ \end{array}\right. \end{aligned}$$

(B1.12)

which give the equation (B1.10). $\square$

We summarize the following from [48].

PROPOSITION B1.7

Let $V_{\wp }$ be a laced bundle on ${\tilde{C}}$. Then the direct image $\nu _{*}(V_{\wp }) = {{\mathscr {F}}}$ is a torsion-free sheaf on C and conversely, $\nu ^{*}({{\mathscr {F}}})/{tors}$ recovers the underlying vector bundle of $V_{\wp }$. Moreover,

$$\begin{aligned} \mathtt{par.deg}_{{{\tilde{C}}}}(V_{{\wp }}) = \mathtt{deg}_{{C}}({{\mathscr {F}}}). \end{aligned}$$

(B1.13)

1.2 Some remarks on parabolic subgroups

Remark B2.1

Let $\lambda :{{\mathbb {G}}}_{m} \rightarrow G$ be a one-parameter subgroup and $P(\lambda )$ be the associated parabolic subgroup and $H(\lambda )$ the Levi quotient which canonically defines a Levi subgroup $L(\lambda )$ as the centralizer of $\lambda$. Let $\eta :G \hookrightarrow \text {GL}(W)$ be a faithful representation. Then the one-parameter subgroup given by the composition $\eta \circ \lambda :{{\mathbb {G}}}_{m} \rightarrow \text {GL}(W)$ defines a parabolic and Levi subgroups $P(\eta \circ \lambda )$ and $L(\eta \circ \lambda )$ of $\text {GL}(W)$.

We can view the parabolic subgroup $P(\eta \circ \lambda )$ as the stabilizer of the flag

$$\begin{aligned} (W_{\bullet }(\lambda ), \epsilon _{\bullet }) : 0 \subsetneq W_{1} \subsetneq W_{2} \cdots W_{s} \subsetneq W_{{s+1}} = W\subsetneq W \end{aligned}$$

(B2.1)

where $W_{i}:= \oplus _{j = 1}^{{i}} W^{j}$, with $W^{j}$ being the eigenspace of the ${{\mathbb {G}}}_{m}$-action via $\lambda$ for the character $z \mapsto z^{{\gamma _j}}$, and $\gamma _{1}< \cdot < \gamma _{{s+1}}$ are the distinct weights which occur. Set $\epsilon _{i} := (\gamma _{{i+1}} - \gamma _{i})/ \mathtt{dim}(V)$, $i = 1, \ldots , s$. The pair $(W_{\bullet }(\lambda ), \alpha _{\bullet })$ is called the associated weighted filtration of $\lambda$. The weighted filtration $(W_{\bullet }(\lambda ), \alpha _{\bullet })$ has an associated graded:

$$\begin{aligned} \mathtt{gr}_{\lambda }(W) := \bigoplus _{{j = 1}}^{{s}} W_{{j+1}}/W_{j} = \bigoplus _{{j = 1}}^{{s}} W^{{j+1}} \end{aligned}$$

(B2.2)

and it is easy to see that as $H = H(\lambda )$-modules, $W \simeq \mathtt{gr}_{\lambda }(W)$. Further, H fixes the $\lambda$-eigenspaces $W^{{j+1}}$, i.e., the above decomposition is a decomposition of H-modules. We also have an obvious weighted filtration $(\mathtt{gr}(W)^{{\bullet }}, \underline{\epsilon })$ with the same weights $\underline{\epsilon }$:

The 1-PS $\eta \circ \lambda$ also defines a canonical anti-dominant character $\chi _{{\eta \circ \lambda }}:P(\eta \circ \lambda ) \rightarrow H(\eta \circ \lambda ) \rightarrow {{\mathbb {G}}}_{m}$ dual to $\eta \circ \lambda$ [47, 2.4.9]. For instance, if $\underline{m} := (m_{1}, \ldots , m_{{s+1}})$ is a point of the Levi $H(\eta \circ \lambda )$ as block matrices, then $\chi _{\lambda }(\underline{(}m)) := \otimes _{j} \text {det}(m_{j})^{{\gamma _{j}}}$. This which restricts to an anti-dominant character $\chi _{\lambda }$ of $P(\lambda )$. We recall the following result.

Lemma B2.2

([47, Proposition 2.4.9.1]). Let $\chi :P(\lambda ) \rightarrow {{\mathbb {G}}}_{m}$ be any anti-dominant character. Then there is a positive rational number r such that $\chi = r \cdot \chi _{\lambda }.$

Remark B2.3

Let E be a G-torsor and suppose that we are given a reduction of structure group $E_{P} \subset E$ to the parabolic P. There is a canonical anti-dominant character $\chi _{\lambda }:P \rightarrow {{\mathbb {G}}}_{m}$ () which defines a line bundle $E_{P}(\chi _{\lambda })$ on Y.

Again, the representation $\eta$ gives a weighted filtration (B2.1) stabilized by $P(\eta \circ \lambda )$. We can take the associated vector bundle $E_{P}(W)$ which comes with its weighted filtration

$$\begin{aligned} (E_{P}(W))_{\bullet }, \underline{\epsilon }) : 0 \subsetneq E_{P}(W_{1}) \subsetneq \cdots \subsetneq E_{P}(W_{s}) \subsetneq E_{P}(W_{{s+1}}) = E_{P}(W)\nonumber \\ \end{aligned}$$

(B2.3)

and the weighted slope defined by Schmitt [43]:

$$\begin{aligned}&L((E_{P}(W)_{\bullet }, \epsilon _{\bullet })) \nonumber \\&:= \sum _{i = 1}^{s} \epsilon _{i}\{\mathtt{deg}_{{C}}(E_{P}(W))\cdot rk E_{P}(W_{{i}}) - \mathtt{deg}E_{P}(W_{{i}}) \cdot rk(E_{P}(W))\}.\nonumber \\ \end{aligned}$$

(B2.4)

Claim.

$$\begin{aligned} \mathtt{deg}(E_{P}(\chi _{\lambda }))= & {} L((E_{P}(W)_{\bullet }, \epsilon _{\bullet })), \end{aligned}$$

(B2.5)

$$\begin{aligned} \mathtt{deg}(E_{H}(\chi _{\lambda }))= & {} L((E_{H}(\mathtt{gr}(W))_{\bullet }, \epsilon _{\bullet })). \end{aligned}$$

(B2.6)

To see this, note that the line bundle $E_{H}(\chi _{\lambda }) \simeq \bigotimes \text {det}(E_{H}(W_{i}))^{{-\epsilon _{i}}}$ with $\epsilon _{i} := (\gamma _{{i+1}} - \gamma _{i})/ \mathtt{dim}(V)$ as above (see [47, Exercise 2.4.9.2, page 209]).

Remark B2.4

Let Y be a smooth projective curve and let $\mathsf P_{{\underline{v}}}$ be a parahoric group scheme generically split with fibre G, with parahoric structures $\underline{v} := (v_{j})$ at points $y_{j} \in Y$ given by a tuple of points in the affine apartment ${{\mathscr {A}}}_{T}$ [10]. Given a faithful representation $\eta :G \hookrightarrow \text {GL}(W)$, we get a corresponding group parahoric group scheme $\mathsf{P}_{{\text {GL}(W)}}$ with generic fibre $\text {GL}(W)$. If E is a $\mathsf P_{{\underline{v}}}$-torsor then we get an associated parabolic vector bundle $E(W)_{*}$ with parabolic structures at $y_{j}$. If $\lambda :{{\mathbb {G}}}_{m} \rightarrow G$ is a 1-PS, and the setting be as in the previous paragraph, then we have a parahoric Levi-type torsor $E_{H}$ for a parahoric group scheme with generic fibre isomorphic to H and associated parabolic line bundles $E_{H}(\chi _{\lambda })_{*}$. The standard properties of degrees of direct sum of vector bundles in terms of the determinants obviously go through in the parabolic setting by replacing degrees with parabolic degrees and tensor products with parabolic tensor products. This follows by expressing parabolic bundles in terms of orbifold bundles and push-forwards. Thus the entire formalism goes through and we get a relation $\mathtt{par.deg}(E_{H}(\chi _{\lambda })_{*}) = L\big ((E_{H}(\mathtt{gr}(W))_{\bullet }, \epsilon _{\bullet })\big )$ with parabolic degrees everywhere.

We apply it in the main paper for the laced bundle $E_{\wp }$ on ${\tilde{C}}$ which has an underlying parahoric structure at the two points $c_{i}$.

1.3 A counter example to a simplistic generalization of Ramanathan’s definition in the nodal case

Let E be a principal G-bundle on the nodal curve C. A naïve generalization of the usual definition along the lines of A. Ramanathan’s definition turns out to be false even when $G = \text {GL}(2).$

For every maximal parabolic subgroup $P \subset G$ and for every reduction of structure group $E_{P}$ of E over $C -c$, consider the Lie algebra sub-bundle $E_{P}({\mathfrak {p}}) \subset E({\mathfrak {g}})|_{{C - c}}.$ Let $\overline{E_{P}({\mathfrak {p}})}$ be the torsion-free sheaf which is the saturation of the sub-bundle $E_{P}({\mathfrak {p}})$ in $E({\mathfrak {g}})$ over C. The bundle E is ‘conjecturally’ (semi)stable if

$$\begin{aligned} \text {deg}(\overline{E_{P}({\mathfrak {p}})}) < 0 (\le 0). \end{aligned}$$

(B3.1)

For the failure of this ‘conjectural definition’ of (semi)stability of G-torsors on nodal curves even when $G = \text {GL}(2)$, we give the following counter-example which essentially comes from a remark due to Seshadri.

Let L, M be torsion-free sheaves on C of rank 1 and degree 0 which are not locally free. In particular, they are of local type ${\mathfrak {m}}$. Consider the group $\mathrm{Ext}^{1}(L,M)$ of extensions of M by L. We claim that there is a locally free sheaf V such that

$$\begin{aligned} 0 \rightarrow L \rightarrow V \rightarrow M \rightarrow 0 \end{aligned}$$

(B3.2)

and hence automatically V is semistable of degree 0. To see the existence of such a V, we consider the local-global spectral sequence for $\text {Ext}$ [26, Section 4.2] which gives (since $\text {dim}(C) = 1$)

$$\begin{aligned} \text {H}^{1}(C, {{\mathscr {H}}}\mathrm{om}(L,M)) \rightarrow \text {Ext}^{1}(L,M) \rightarrow \text {H}^{0}(C, {{\mathscr {E}}}\mathrm{xt}^{1}(L, M)) \rightarrow 0. \end{aligned}$$

(B3.3)

Note that $\text {H}^{0}(C, {{\mathscr {E}}}xt^{1}(L, M)) = \text {Ext}^{1}_{{A}}(L_{c}, M_{c})$, where $A = {{{\mathscr {O}}}}_{{C,c}} \simeq {{\mathbb {C}}}[x,y]/(xy)$. Locally we have ${{\mathfrak {m}}} = (x,y)$. Using these as generators, we have an embedding ${\mathfrak {m}} \hookrightarrow {{\mathscr {O}}}_{C} \oplus {{\mathscr {O}}}_{C}$ and hence an extension

$$\begin{aligned} 0 \rightarrow {\mathfrak {m}} \rightarrow {{\mathscr {O}}}_{C} \oplus {{\mathscr {O}}}_{C} \rightarrow {\mathfrak {m}} \rightarrow 0. \end{aligned}$$

(B3.4)

This gives an element in $\text {Ext}^{1}_{A}(L_{c}, M_{c})$ which lifts to give an element in $\text {Ext}^{1}(L,M)$. Clearly this extension is locally free since it is so at the node and we get the required V. This V is semistable of degree 0.

Giving a reduction of structure group of the principal $\text {GL}(2)$-bundle underlying V is expressing it in an exact sequence of vector bundles (B3.2) and the conjectural definition of semistability is equivalent to saying that for the sub-bundle $L \otimes M^*\subset V \otimes V^*$, we have

$$\begin{aligned} \text {deg}(\overline{L \otimes M^*}) \le 0 , \end{aligned}$$

(B3.5)

where $\overline{L \otimes M^*}$ denotes the saturation in $V \otimes V^*$.

Claim.

$$\begin{aligned} \text {deg}(\overline{L \otimes M^*}) = 1. \end{aligned}$$

(B3.6)

In particular, $V \otimes V^*$ is not semistable. Let $L'$ (resp. $M'$) denote $p^{*}(L)/tors$ (resp. $p^{*}(M)/tors$). Then the line sub-bundle of $p^{*}(V)$ (resp. $p^{*}(V^{*})$) generated by $L'$ (resp. $M'$) is of the form $L'(y_{1}+y_{2})$ (resp. $M'(y_{1}+y_{2}))$. We have $\text {deg}~L' = \text {deg}~M' = -1$, so that $\text {deg}~ L' (y_{1}+y_{2}) = \text {deg}~ M'(y_{1}+y_{2})=1$. Then we see that the line bundle

$$\begin{aligned} N = (L'(y_{1}+y_{2}) \otimes M' (y_{1}+y_{2}) (-y_{1}- y_{2}) \end{aligned}$$

(B3.7)

descends to a torsion free subsheaf of $V \otimes V^*$, which is the saturation $\overline{L \otimes M^*}$. Since $\deg N =0$, we see that $\text {deg}(\overline{L \otimes M^*}) = 1$.

Remark B3.1

The lesson is to avoid taking the saturation after taking tensor products. The degree exceeds the bound. Instead, one has to take some sort of a ‘parabolic tensor product’ and then take a saturation, both of these operations need to be carried out on the normalization Y. This can be made precise. We proceed differently in Section 13 to achieve this.

AbhyavasthāH prajāyante pra vavrer vavriś ciketa, upasthe mātur vi cashte

States upon states are born, covering over covering awakens to knowledge, in the lap of the universal mother he wholly sees.