Following the construction of [2], consider the full subcategory $\mathcal{D}_0 \subset D^\flat(\operatorname{Coh} \omega_{\mathbb{P}^1})$ consisting of complexes whose cohomology objects are supported on the zero divisor $\mathbb{P}^1 \hookrightarrow \omega_{\mathbb{P}^1}$. The space $\omega_{\mathbb{P}^1}$ is the minimal resolution of the Kleinian singularity $\mathbb{C}^2/\mathbb{Z}^2$ and the category $\mathcal{D}_0$ acts as a ‘local model’ for the derived category of a K3 surface near a ($-1$)-curve (with respect to a fixed $A_2$-configuration of spherical objects).

In [1], the authors consider one dimension higher: $\omega_{\mathbb{P}^2}$ is a quasi-projective Calabi–Yau threefold fibered over a projective plane, and one can analogously define a full subcategory of $D^\flat(\operatorname{Coh} \omega_{\mathbb{P}^2})$ giving rise to quivery hearts related by a braid group action. My question is more or less how this is considered as a local model for projective Calabi–Yau threefolds: while CY threefolds contain rational curves, it’s not clear to me when they contain rational divisors.

——

[1] Arend Bayer and Emanuele Macrì, The Space of Stability Conditions on the Local Projective Plane, Duke Math. J. 160 (2011), no. 2, 263–322, DOI 10.1215/00127094-1444249

[2] Tom Bridgeland, T-structures on some local Calabi–Yau varieties, Journal of Algebra 289 (2005), no. 2, 453-483, DOI https://doi.org/10.1016/j.jalgebra.2005.03.016.

Following the construction of [2], consider the full subcategory $\mathcal{D}_0 \subset D^\flat(\operatorname{Coh} \omega_{\mathbb{P}^1})$ consisting of complexes whose cohomology objects are supported on the zero divisor $\mathbb{P}^1 \hookrightarrow \omega_{\mathbb{P}^1}$. The space $\omega_{\mathbb{P}^1}$ is the minimal resolution of the Kleinian singularity $\mathbb{C}^2/\mathbb{Z}_2$ and the category $\mathcal{D}_0$ acts as a ‘local model’ for the derived category of a K3 surface near a ($-2$)-curve (with respect to a fixed $A_2$-configuration of spherical objects).

In [1], the authors consider one dimension higher: $\omega_{\mathbb{P}^2}$ is a quasi-projective Calabi–Yau threefold fibered over a projective plane, and one can analogously define a full subcategory of $D^\flat(\operatorname{Coh} \omega_{\mathbb{P}^2})$ giving rise to quivery hearts related by a braid group action. My question is more or less how this is considered as a local model for projective Calabi–Yau threefolds: while CY threefolds contain rational curves, it’s not clear to me when they contain rational divisors.

——

[1] Arend Bayer and Emanuele Macrì, The Space of Stability Conditions on the Local Projective Plane, Duke Math. J. 160 (2011), no. 2, 263–322, DOI 10.1215/00127094-1444249

[2] Tom Bridgeland, T-structures on some local Calabi–Yau varieties, Journal of Algebra 289 (2005), no. 2, 453-483, DOI https://doi.org/10.1016/j.jalgebra.2005.03.016.

Following the construction of [2], consider the full subcategory $\mathcal{D}_0 \subset D^\flat(\operatorname{Coh} \omega_{\mathbb{P}^1})$ consisting of complexes whose cohomology objects are supported on the zero divisor $\mathbb{P}^1 \hookrightarrow \omega_{\mathbb{P}^1}$. The space $\omega_{\mathbb{P}^1}$ is the minimal resolution of the Kleinian singularity $\mathbb{C}^2/\mathbb{Z}^2$ and the category $\mathcal{D}_0$ acts as a ‘local model’ for the derived category of a K3 surface near a (-1)-curve (with respect to a fixed $A_2$-configuration of spherical objects).

In [1], the authors consider one dimension higher: $\omega_{\mathbb{P}^2}$ is a quasi-projective Calabi-Yau threefold fibered over a projective plane, and one can analogously define a full subcategory of $D^\flat(\operatorname{Coh} \omega_{\mathbb{P}^2})$ giving rise to quivery hearts related by a braid group action. My question is more or less how this is considered as a local model for projective Calabi-Yau threefolds: while CY threefolds contain rational curves, it’s not clear to me when they contain rational divisors.

——

[1] Arend Bayer and Emanuele Macrì, The Space of Stability Conditions on the Local Projective Plane, Duke Math. J. 160 (2011), no. 2, 263–322, DOI 10.1215/00127094-1444249

[2] Tom Bridgeland, T-structures on some local Calabi–Yau varieties, Journal of Algebra 289 (2005), no. 2, 453-483, DOI https://doi.org/10.1016/j.jalgebra.2005.03.016.

Following the construction of [2], consider the full subcategory $\mathcal{D}_0 \subset D^\flat(\operatorname{Coh} \omega_{\mathbb{P}^1})$ consisting of complexes whose cohomology objects are supported on the zero divisor $\mathbb{P}^1 \hookrightarrow \omega_{\mathbb{P}^1}$. The space $\omega_{\mathbb{P}^1}$ is the minimal resolution of the Kleinian singularity $\mathbb{C}^2/\mathbb{Z}^2$ and the category $\mathcal{D}_0$ acts as a ‘local model’ for the derived category of a K3 surface near a ($-1$)-curve (with respect to a fixed $A_2$-configuration of spherical objects).

In [1], the authors consider one dimension higher: $\omega_{\mathbb{P}^2}$ is a quasi-projective Calabi–Yau threefold fibered over a projective plane, and one can analogously define a full subcategory of $D^\flat(\operatorname{Coh} \omega_{\mathbb{P}^2})$ giving rise to quivery hearts related by a braid group action. My question is more or less how this is considered as a local model for projective Calabi–Yau threefolds: while CY threefolds contain rational curves, it’s not clear to me when they contain rational divisors.

——

[1] Arend Bayer and Emanuele Macrì, The Space of Stability Conditions on the Local Projective Plane, Duke Math. J. 160 (2011), no. 2, 263–322, DOI 10.1215/00127094-1444249

[2] Tom Bridgeland, T-structures on some local Calabi–Yau varieties, Journal of Algebra 289 (2005), no. 2, 453-483, DOI https://doi.org/10.1016/j.jalgebra.2005.03.016.