Family 2-384-81-a

The complete isotropic flag variety $\operatorname{Sp}_4/B$.

$\pi_1\colon X\to\mathbb P^3$ is a $\mathbb P^1$-bundle. Here $X$ is the complete isotropic flag variety of a symplectic form on a four-dimensional space.
$\pi_2\colon X\to\mathbb P^3$ is surjective with disconnected fibers (pairs of lines); its Stein factorization is a $\mathbb P^1$-bundle onto a smooth quadric threefold, followed by a double cover of $\mathbb P^3$ branched along a smooth quadric surface.

More details (pdf)

Defining data

Ambient: Fl(1, 3, 5)
Bundle: $E^{ω_{1}} \oplus E^{2 ω_{4}}$ E^(w1) + E^(2w4)
Bundle: $O (1, 0) \oplus Sym^{2} Q_{1}^{\lor} \otimes Σ_{(2, 2)} U_{2}^{\lor} \otimes Sym^{2} U_{3}^{\lor}$ O(1,0) + Sym^2 Q_1^v ot Sigma_(2,2) U_2^v ot Sym^2 U_3^v
ZL62 label: 44.70

Line bundles on the ambient $Y$

$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(3,4) \\ \det\mathcal{E} &\cong \mathcal{O}(1,3) \end{aligned} \end{equation}$$

Numerical invariants

Fano index	=	2
$\mathrm{h}^0(X, \omega_X^\vee)$	=	81
$\operatorname{vol}(X) = \mathrm{c}_1^4$	=	384
$\chi(X, \mathrm{T}_X)$	=	10

$\omega_X^\vee \cong \mathcal{O}_X(2)$

Chern numbers of $\mathrm{T}_X$

$\mathrm{c}_1^4$	=	384
$\mathrm{c}_1^2\mathrm{c}_2$	=	192
$\mathrm{c}_2^2$	=	96
$\mathrm{c}_1\mathrm{c}_3$	=	56
$\mathrm{c}_4$	=	8

Hodge diamond

				1
			0		0
		0		2		0
	0		0		0		0
0		0		2		0		0
	0		0		0		0
		0		2		0
			0		0
				1

Polyvector parallelogram & Hochschild cohomology

$\mathrm{HH}^{ 0 }$ = 1	1
$\mathrm{HH}^{ 1 }$ = x₀	0	x₀
$\mathrm{HH}^{ 2 }$ = x₀ + x₁ - 10	0	x₀ - 10	x₁
$\mathrm{HH}^{ 3 }$ = x₁ + 64	0	0	x₁ - 50	114
$\mathrm{HH}^{ 4 }$ = 81	0	0	0	0	81
$\mathrm{HH}^{ 5 }$ = 0		0	0	0	0
$\mathrm{HH}^{ 6 }$ = 0			0	0	0
$\mathrm{HH}^{ 7 }$ = 0				0	0
$\mathrm{HH}^{ 8 }$ = 0					0

Hilbert series

$\operatorname{Hilb}_X(t)$ $=$

$\displaystyle\frac{ 1 + 11t + 11t^{2} + t^{3} }{(1-t)^{ 5 }}$

$=$

Regularized quantum period

Quantum period in the literature

catalogue	matching entries
Kalashnikov (2019)	CKK(136)
Coates et al. (2020)	MW(11)
Coates–Kasprzyk–Prince (2015)	No match

Quantum differential equation

With $D=t\frac{\mathrm{d}}{\mathrm{d}t}$, the regularized period satisfies $L_X\widehat{\mathrm{G}}_X=0$.

$$\begin{equation} L_X=\left( 176t^{4} - 96t^{3} + 8t^{2} - 4t - 1 \right) \left( 176t^{4} + 96t^{3} + 8t^{2} + 4t - 1 \right) \left( 46963840t^{6} - 6320080t^{4} + 10817t^{2} - 60 \right)D^{6} + 2 \left( 16002270986240t^{14} - 4453229608960t^{12} + 179162526976t^{10} + 27549781760t^{8} - 288092960t^{6} + 25314336t^{4} - 28611t^{2} + 120 \right)D^{5} + 4 \left( 69100715622400t^{14} - 16714357862400t^{12} + 879319798400t^{10} + 38172435520t^{8} - 54159304t^{6} - 23633284t^{4} + 6819t^{2} - 60 \right)D^{4} + 8 t ^{2} \left( 149112070553600t^{12} - 33265162726400t^{10} + 1526941953280t^{8} + 17140032640t^{6} - 188273072t^{4} + 3160464t^{2} + 1615 \right)D^{3} + 128 t ^{2} \left( 21014345918720t^{12} - 4511525020640t^{10} + 157602014826t^{8} - 362882280t^{6} + 22427697t^{4} + 282960t^{2} + 30 \right)D^{2} + 512 t ^{4} \left( 5790594508160t^{10} - 1225795140240t^{8} + 31283552519t^{6} - 265518840t^{4} + 10109387t^{2} + 49308 \right)D + 15360 t ^{4} \left( 79556744960t^{10} - 16784238240t^{8} + 319629734t^{6} - 3585072t^{4} + 146945t^{2} + 456 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Fl(1, 3, 5)

Sector dimension: 8
Spectral radius: $7.32727615$
Diagonalizable at 1: Yes
Simple spectrum at 1: Yes

Eigenvalue approximations and Jordan block sizes

eigenvalue	multiplicity	Jordan blocks
$-7.327276$	1	1
$7.327276$	1	1
$-0.7819892 - 3.607022\mathrm{i}$	1	1
$-0.7819892 + 3.607022\mathrm{i}$	1	1
$0.7819892 - 3.607022\mathrm{i}$	1	1
$0.7819892 + 3.607022\mathrm{i}$	1	1
$-1.763298$	1	1
$1.763298$	1	1

Characteristic polynomial

$$\begin{equation}\left(\lambda^{4} + 4 \lambda^{3} - 8 \lambda^{2} + 96 \lambda - 176\right) \left(\lambda^{4} - 4 \lambda^{3} - 8 \lambda^{2} - 96 \lambda - 176\right)\notag\end{equation}$$

Minimal polynomial