Full exceptional collections of vector bundles on rank-two linear GIT quotients

TL;DR

This work extends Beilinson–Kapranov-style exceptional collections to linear GIT quotients by rank-two groups, proving that full strong collections of vector bundles exist on $X^{\rm ss}(\ell)/G$ when the quotient is Fano or nef–Fano and stabilizers are finite. The authors introduce barrel windows in weight space, coupled with a Van den Bergh–type local cohomology vanishing criterion, to generate collections from tautological bundles $\mathcal{O}_{X^{\rm ss}}\otimes U$ with weights in $\theta+\nabla_\ell$. They develop a decorated-quiver framework to construct many new examples via iterated fiber bundles and demonstrate fullness in the rank-two case (e.g., for $G=\mathrm{GL}_2$ and associated Grassmannians), including a GL${}_2$ instance and toric Deligne–Mumford stacks of Picard rank two that align with the Borisov– Hua window. The results connect to known toric and quiver-flag constructions, showing barrel windows reproduce Borisov– Hua-type collections in toric settings and offering a practical, modular route to producing full strong exceptional collections on a broad class of GIT quotients.

Abstract

We produce full strong exceptional collections consisting of vector bundles on the geometric invariant theory quotient of certain linear actions of a split reductive group $G$ of rank two. The vector bundles correspond to irreducible $G$-representations whose weights lie in an explicit bounded region in the weight space of $G$. We also describe a method for constructing more examples of linear GIT quotients with full strong exceptional collections of this kind as "decorated" quiver varieties.

Figures (7)

• Figure 1: The diagram to the left is the cylinder window $\overline{\nabla}$ for \ref{['ex:grassmannian_N_6']}. The $T$-weights indicated by blue crosses are those in the cylinder window that are excluded from the barrel window, thus the $T$-weights indicated by red dots are precisely those in the barrel window $\nabla$. We have marked the special weights $P_i := \zeta_{\lambda_i}/2$ for $i=1,2$ that become relevant in the passage from the cylinder window to the barrel window as is depicted in the diagram to the right. This diagram indicates a perturbation by some $\theta \in M_\bR^W$ of the regions $\overline{\nabla}_{\ell,t}$ defined in \ref{['R:t_perturbed_regions']} for selected values of $0 < t \ll 1$. In this particular example, $\theta= s \omega^\ast$, where $-s$ is a small positive number. Here the weight indicated by a blue cross is excluded in the $\theta$-perturbed barrel window $\theta + \nabla$.
• Figure 2: This diagram illustrates the "allowable" region for \ref{['Ex:allowable_region']}. The red circles indicate the $T$-weights that satisfy the inequality in \ref{['P:vanishing']}.
• Figure 3: This illustrates the reduction to the $\lambda_0$-strip for the $\mathop{\mathrm{GL}}\nolimits_2$ representation $X = \mathop{\mathrm{Sym}}\nolimits^3 \bC^2$ with $\lambda_0 = (-1,-1)$ and $\theta = 0$. The roots of $G$ are $(1,-1),(-1,1)$, where we take $(1,-1)$ to be the positive root. The weights of $X$ are $(0,3),(1,2),(2,1),(3,0)$, so $\eta_{\lambda_0} =12$. Here we are using the standard identification $M_\bR \cong N_\bR \cong \bR^2$. The $T$-weights indicated by blue crosses satisfy $\langle \lambda_0, \chi \rangle = \pm \alpha_\chi \eta_{\lambda_0}/2$, where $\alpha_\chi = 7/3$. In all three cases, the red crosses indicate the weights $\mu^+$ appearing in the respective complexes as described in \ref{['L:reduction_to_lambda_0']}. For $\chi_1$, the complex is $C_{\lambda_0, \chi_1}$ and all $\mu$ are dominant, so $\mu^+ = \mu$. For $\chi_3$, the complex is $C_{\lambda_0, \chi_3 + \det X}$ and some of the $\mu$ are not dominant. These weights are indicated by the black crosses when $\mu^+$ exists and by the black circles otherwise. The $\rho$-shifted Weyl group action $(-)^+$ is reflection along the dotted blue line as indicated by the arrows. One sees that all the resulting $\mu^+$ lie in a strictly smaller $\lambda_0$-strip.
• Figure 4: This diagram illustrates the reduction to the cylinder window $\overline{\nabla}$ for the $\mathop{\mathrm{GL}}\nolimits_2$ representation $X = \mathop{\mathrm{Sym}}\nolimits^3 \bC^2$ with $\lambda_0 = (-1,-1)$, $\lambda' = (-1, 1)$, and $\theta = 0$. As in \ref{['fig:lambda_0_reduction']}, $\eta_{\lambda_0} = 12$ and $\eta_{\lambda'} = 2$. In this example, $Q_{\lambda'} = 0$. The character $\chi_1$ falls into Case 2a and one selects the complex $C_{\lambda', \chi_1}$. On the other hand, $\chi_2$ falls into Case 2b and one selects the complex $D^\vee_{\lambda', \chi_2}$. In both cases, the resulting weights $\mu^+$ are indicated by the red crosses.
• Figure 5: The diagram illustrates the reduction from the cylinder window to the barrel window for our running $\mathop{\mathrm{GL}}\nolimits_2$ representation $X = \mathop{\mathrm{Sym}}\nolimits^3 \bC^2$ from \ref{['fig:lambda_0_reduction']} and \ref{['fig:cylinder_reduction']}. Here $\lambda' = (-1,1)$ and $Q_{\lambda'} = 0$. We have indicated by the blue diamonds the special weights $\pm \zeta_{\lambda'}/2 = \pm (-2,-1)$ and $\pm \zeta_{-\lambda'}/2 = \pm (-1,-2)$ that are relevant in the passage from the cylinder to the barrel window. The character $\chi$ lies in $M^+ \cap (\overline{\nabla} \setminus \nabla)$ and it falls into Case 3a. Thus, one takes the complex $C_{\lambda', \chi}$. The weights $\mu$ appearing as in \ref{['L:complex_1']} are not all dominant and we have indicated these by the black crosses. As in \ref{['fig:lambda_0_reduction']}, the $\rho$-shifted action $(-)^+$ for these weight is the reflection along the dotted line as shown by the blue arrows. The resulting $\mu^+$ are indicated by the red crosses.

Theorems & Definitions (69)

• Theorem 1.3: \ref{['P:exceptional_collections']}, \ref{['T:main_fano']}, \ref{['T:main_nef_fano']}
• Remark 1.4
• Definition 2.1
• Definition 2.2
• Example 2.3
• Definition 2.4
• Definition 2.5: $\lambda$-strip
• Definition 2.6: Barrel window
• Remark 2.7
• Example 2.8