Abstract
A geometric characterization of the structure of the group of automorphisms of an arbitrary Birkhoff–Grothendieck bundle splitting \(\oplus _{i=1}^{r} \mathcal {O}(m_{i})\) over \(\mathbb {C}\mathbb {P}^{1}\) is provided, in terms of its action on a suitable space of generalized flags in the fibers over a finite subset \(S\subset \mathbb {C}\mathbb {P}^{1}\). The relevance of such characterization derives from the possibility of constructing geometric models for diverse moduli spaces of stable objects in genus 0, such as parabolic bundles, parabolic Higgs bundles, and logarithmic connections, as collections of orbit spaces of parabolic structures and compatible geometric data satisfying a given stability criterion, under the actions of the different splitting types’ automorphism groups, that are glued in a concrete fashion. We illustrate an instance of such idea, on the existence of several natural representatives for the induced actions on the corresponding vector spaces of (orbits of) logarithmic connections with residues adapted to a parabolic structure.
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Notes
Grothendieck’s result holds for holomorphic principal bundles whose structure group is an arbitrary reductive group. For simplicity, we will only consider the case of \(\mathrm {GL}(r,\mathbb {C})\) (that is, the vector bundle case).
The splitting coefficients of each underlying bundle are only holomorphic (i.e., not topological) invariants of E. The different admissible splitting types for a given choice of parabolic weights determine moduli spaces’ stratifications.
It should also be remarked that Loray and Saito [13] have studied the natural holomorphic symplectic structure of the moduli spaces \(\mathscr {L}\) of logarithmic connections in the case of rank 2 and odd degree using algebro-geometric techniques, in such a way that the moduli spaces \(\mathscr {N}\) arise as the bases of natural Lagrangian fibrations. In particular, they specialize their construction to a choice of parabolic weights for which \(\mathscr {N}\) contains a single Harder–Narasimhan stratum, allowing them to introduce charts and manifold structure in \(\mathscr {L}\), and to study its birrational geometry in an explicit way.
As an illustration, we describe a minimal example of such behavior. Consider the case \(r = 2\), \(\deg (E) = -\,4\), and \(n = 4\), with arbitrary parabolic weights in the open chamber determined by the eight inequalities
$$\begin{aligned} \alpha _{12} +\alpha _{i2} +\sum _{j \ne 1,i}\alpha _{j1}> & {} 2,\quad i =2,3,4 \end{aligned}$$(2.1)$$\begin{aligned} \alpha _{i1} + \sum _{j \ne i} \alpha _{j2}< & {} 3, \quad i = 1,2,3,4, \end{aligned}$$(2.2)$$\begin{aligned} \sum _{i = 1}^{4}\alpha _{i1}< & {} 1 \end{aligned}$$(2.3)which is non-empty, since for any \(0< \delta< \alpha < 1/4\) we can choose
$$\begin{aligned} \alpha _{11} = \delta ,\quad \alpha _{21} = \alpha _{31} = \alpha _{41} = \alpha ,\qquad \alpha _{i2} = 1 - \alpha _{i1}. \end{aligned}$$In particular, a semi-stable parabolic structure is always strictly stable. Over the splitting \(E = \mathcal {O}(-\,2)^{2}\), the flag manifolds \(\mathscr {F}(E_{i})\) get identified with the aid of the subbundles \(\mathcal {O}(-\,2)\hookrightarrow E\). It can be verified that for any weights satisfying (2.1)–(2.2), the set \(\mathscr {P}^{s}\) of stable parabolic structures on such splitting is biholomorphic to \(\mathscr {C}_{4}\), a configuration space of 4 points in \(\mathscr {F}_{2} \cong \mathbb {C}\mathbb {P}^{1}\). Hence, the stratum \(\mathscr {N}_{N}\) of stable parabolic bundles with splitting type \(\mathcal {O}(-2)^{2}\) is obtained as the quotient
$$\begin{aligned} \mathscr {N}_{N} \cong P\left( {{\mathrm{Aut}}}\left( \mathcal {O}(-2)^{2}\right) \right) \setminus \mathscr {P}^{s} \cong \mathbb {C}\mathbb {P}^{1} \setminus \{0,1,\infty \}. \end{aligned}$$In turn, the splitting \(E = \mathcal {O}(-3)\oplus \mathcal {O}(-1)\) admits exactly one stable orbit. The unique subbundle \(\mathcal {O}(-1)\hookrightarrow E\) stratifies every \(\mathscr {F}(E|_{z})\)
$$\begin{aligned} \mathscr {F}(E|_{z}) \cong \mathbb {P}(E|_{z}) = \mathbb {C}\sqcup \{\infty \}, \end{aligned}$$with the lines \(\{\infty \}\in \mathbb {P}(E|_{z})\) being stabilized by \({{\mathrm{Aut}}}(E)|_{z}\). It readily follows from (2.2) that any \({{\mathrm{Aut}}}(E)\)-orbit containing a flag at \(\infty \) is automatically unstable. Moreover, the affine subspace
$$\begin{aligned} \mathbb {C}^{4}\subset \mathbb {P}(E|_{z_{1}})\times \mathbb {P}(E|_{z_{2}})\times \mathbb {P}(E|_{z_{3}})\times \mathbb {P}(E|_{z_{4}}), \end{aligned}$$is decomposed into exactly two \({{\mathrm{Aut}}}(E)\)-orbits, namely, \({{\mathrm{Aut}}}(E)\cdot (0,0,0,0)\) and its complement. It can be verified that such orbits correspond to the orbits of the standard \(\mathbb {C}^{*}\)-action \(t\cdot w = tw\) on \(\mathbb {C}\), and as a consequence of (2.2)–(2.3) only the latter (that is, the one corresponding to \(\mathbb {C}\setminus \{0\}\) in \(\mathbb {C}\)) is stable.
The picture that emerges from such considerations is the following. The moduli space \(\mathscr {N}\) can be reconstructed from the open stratum \(\mathscr {N}_{N}\subset \mathscr {N}\) as a 3-point compactification. A neighborhood \(\mathscr {U}_{i}\) of each of these points \(w_{1}, w_{2}, w_{3}\) corresponds to a local holomorphic family of stable parabolic bundles, whose underlying bundle \(E_{w}\) is isomorphic to \(\mathcal {O}(-2)^{2}\) if \(w \ne w_{i}\), and to \(\mathcal {O}(-\,3)\oplus \mathcal {O}(-\,1)\) otherwise.
The normalization \(z_{0}=0\) is not strictly necessary, but implementing it accounts for a simpler proof of Theorem 2.
An analogous normalization is used in [13] to introduce open charts in moduli spaces of rank 2 logarithmic connections for a special symmetric choice of parabolic weights that forces the condition \(\mathscr {N}= \mathscr {N}_{0}\) in our terminology. Thus, as another application of the construction of strong Bruhat gauges, we have a general mechanism to introduce affine charts for any stratum fibration \(\mathscr {L}_{N} \rightarrow \mathscr {N}_{N}\), which in fact works for arbitrary rank and admissible parabolic weights.
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I would like to kindly thank CIMAT (Mexico) for its generosity during the development of the present work.
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Meneses, C. Remarks on groups of bundle automorphisms over the Riemann sphere. Geom Dedicata 196, 63–90 (2018). https://doi.org/10.1007/s10711-017-0309-y
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DOI: https://doi.org/10.1007/s10711-017-0309-y