Let $X$ be an irreducible smooth projective curve defined over the field of complex numbers $\mathbb{C}$. Let $E$ be an algebraic vector bundle over $X$. Let $P\in X$.

Defn : A quasi-parabolic structure on $E$ over the point of $P$ is given by a strictly decreasing filtration of subspaces (such a filtration is called a flag) of the fibre $E_{P}$ over $P$ $$ E_{P}=F^1(E_{P})\supsetneq F^2(E_{P})\supsetneq \cdots F^{n_i}(E_{P})\supsetneq F^{n_i+1}(E_{P})=\{0\}$$

Defn : A parabolic structure on $E$ over a point $P$ is given by a quasi-parabolic structure with the following additional data: rational numbers $\{\alpha^j_{P}\}$ associated to each subspace $F^j(E_{P})$ satisfying the condition, $$0\leq \alpha^1_{P}\lneq \alpha^2_{P}\cdots \lneq \alpha^{n_i}_{P}\lneq 1$$

Defn : Let $\mathcal{E}:=\big(E,P, \{F^j(E_{P})\},\{\alpha^j_{P}\}\big)$ and $\mathcal{F}:=\big(F,P, \{F^j(F_{P})\},\{\beta^j_{P}\}\big)$ be two parabolic vector bundles over $X$ with quasi-parabolic structures at the same point $P\in X$. A parabolic morphism from $\mathcal{E}$ to $\mathcal{F}$ is given by a vector bundle morphism $\psi$ such that given a weight $\alpha^s_{P}$ if $\beta^t_{P}$ is the smallest weight such that $\alpha^s_{P}\leq\beta^t_{P}$ (i.e., $\beta ^{t-1}_{P}< \alpha^s_{P}\leq\beta ^{t}_{P}$) then we demand $$ \psi_{P}(F^{s}(E_{P})) \subset F^{t}(F_{P}) $$ If there is no such $\beta^t_{P}$ then we demand $$ \psi_{P}(F^{s}(E_{P})) =0 $$

Given two rank $n$ parabolic vector bundles $\mathcal E=\big(E,P,\{F^j(E_P)\}_{j=1}^{n+1},\{\alpha_P^j\}\big)$ and $\mathcal F=\big(F,P,\{F^j(F_P)\}_{j=1}^{n+1},\{\beta_P^j\}\big)$ over $X$ with parabolic structure at same point and full flags. We can define their direct sum and even give it a natural quasi parabolic structure at $P\in X$. (Just give the flag as $\{F^j((E\oplus F)_P):=F^j(E_P)\oplus F^j(F_P)\}$).

My question : How do you give $\mathcal{E\oplus F}$ parabolic structure? That is what weights do you give each member of the flag?

The weights need to be given so that the natural maps $\mathcal{E\to E\oplus F}$ and $\mathcal{F\to E\oplus F}$ are parabolic morphisms. So assigning the new weights $\gamma_P^j$ as $\max\text{ or }\min\{\alpha_P^j,\beta_P^j\}$ won't work. Is there some natural way to do it?

Thank you.


1 Answer 1


Maybe it's a little bit late to answer after three years but I'll try the same...

I think that the problem is that the definition of the flag for $E \oplus F$ over $P$ should not be the natural one. If I got it correctly it should depend also on the weights for $E,F$ in the following manner.

You start putting $F^1(E \oplus F)=E \oplus F$. We can then consider the following situations:

  • $\beta_1 > \alpha_1$. We then put $F^2(E\oplus F)=F^2E \oplus F$.
  • $\beta_1=\alpha_1$. We then put $F^2(E \oplus F)=E \oplus F$.
  • $\beta_1 < \alpha_1 $. This is symmetric to the first one.

One then proceeds inductively in the same way to complete the flag. One should then assign the following weights $\gamma_i$ to this quasiparabolic structure. I work out just the case $\gamma_1$, the others are defined in the same way.

So for each of the case above we put:

  • $\gamma_1=\alpha_1$
  • $\gamma_1=\alpha_1$
  • $\gamma_1=\beta_1$

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