Family 1-512-105-a

A smooth quadric fourfold in $\mathbb P^5$.

Defining data

Ambient: ℙ⁵P5
Bundle: $E^{2 ω_{1}}$ E^(2w1)
Bundle: $O (2)$ O(2)
ZL62 label: 50.5

Line bundles on the ambient $Y$

$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(6) \\ \det\mathcal{E} &\cong \mathcal{O}(2) \end{aligned} \end{equation}$$

Numerical invariants

Fano index	=	4
$\mathrm{h}^0(X, \omega_X^\vee)$	=	105
$\operatorname{vol}(X) = \mathrm{c}_1^4$	=	512
$\chi(X, \mathrm{T}_X)$	=	15

$\omega_X^\vee \cong \mathcal{O}_X(4)$

Chern numbers of $\mathrm{T}_X$

$\mathrm{c}_1^4$	=	512
$\mathrm{c}_1^2\mathrm{c}_2$	=	224
$\mathrm{c}_2^2$	=	98
$\mathrm{c}_1\mathrm{c}_3$	=	48
$\mathrm{c}_4$	=	6

Hodge diamond

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

Polyvector parallelogram & Hochschild cohomology

$\mathrm{HH}^{ 0 }$ = 1	1
$\mathrm{HH}^{ 1 }$ = 15	0	15
$\mathrm{HH}^{ 2 }$ = 90	0	0	90
$\mathrm{HH}^{ 3 }$ = 175	0	0	0	175
$\mathrm{HH}^{ 4 }$ = 105	0	0	0	0	105
$\mathrm{HH}^{ 5 }$ = 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

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

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

$=$

Regularized quantum period

Quantum period in the literature

catalogue	matching entries
Kalashnikov (2019)	CKK(3)
Coates et al. (2020)	Q4
Coates–Kasprzyk–Prince (2015)	CKP(3)

Quantum differential equation

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

$$\begin{equation} L_X=\left( 32t^{2} - 1 \right) \left( 32t^{2} + 1 \right)D^{4} + 8192 t ^{4}D^{3} + 23552 t ^{4}D^{2} + 28672 t ^{4}D + 12288 t ^{4} \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ⁵P5

Sector dimension: 5
Spectral radius: $5.65685425$
Diagonalizable at 1: Yes
Simple spectrum at 1: Yes

Eigenvalue approximations and Jordan block sizes

eigenvalue	multiplicity	Jordan blocks
$-5.656854$	1	1
$0 - 5.656854\mathrm{i}$	1	1
$0 + 5.656854\mathrm{i}$	1	1
$5.656854$	1	1
$0$	1	1

Characteristic polynomial

$$\begin{equation}\lambda \left(\lambda^{2} + 32\right) \left(\lambda^{2} - 32\right)\notag\end{equation}$$

Minimal polynomial