Family 1-72-21-a

Küchle’s fourfold of type (c3).

Gr(3, 7) · 603.33 (preferred model)

Defining data

Ambient: Gr(3, 7)
Bundle: $(E^{ω_{4}})^{\oplus 2}$ 2 copies of E^(w4)
Bundle: $(Q^{\lor} (1))^{\oplus 2}$ 2 copies of Q^v(1)
ZL62 label: 603.33

Line bundles on the ambient $Y$

$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(7) \\ \det\mathcal{E} &\cong \mathcal{O}(6) \end{aligned} \end{equation}$$

Numerical invariants

Fano index	=	1
$\mathrm{h}^0(X, \omega_X^\vee)$	=	21
$\operatorname{vol}(X) = \mathrm{c}_1^4$	=	72
$\chi(X, \mathrm{T}_X)$	=	-18

$\omega_X^\vee \cong \mathcal{O}_X(1)$

Chern numbers of $\mathrm{T}_X$

$\mathrm{c}_1^4$	=	72
$\mathrm{c}_1^2\mathrm{c}_2$	=	96
$\mathrm{c}_2^2$	=	135
$\mathrm{c}_1\mathrm{c}_3$	=	22
$\mathrm{c}_4$	=	19

Hodge diamond

				1
			0		0
		0		1		0
	0		0		0		0
0		0		15		0		0
	0		0		0		0
		0		1		0
			0		0
				1

Polyvector parallelogram & Hochschild cohomology

$\mathrm{HH}^{ 0 }$ = 1	1
$\mathrm{HH}^{ 1 }$ = x₀	0	x₀
$\mathrm{HH}^{ 2 }$ = x₀ + x₂ + 18	0	x₀ + 18	x₂
$\mathrm{HH}^{ 3 }$ = x₁	0	0	x₁	0
$\mathrm{HH}^{ 4 }$ = x₁ - x₂	0	0	x₁ - x₂ - 21	0	21
$\mathrm{HH}^{ 5 }$ = 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

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

$\displaystyle\frac{ 1 + 16t + 38t^{2} + 16t^{3} + t^{4} }{(1-t)^{ 5 }}$

$=$

Regularized quantum period

Quantum period in the literature

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

Quantum differential equation

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

$$\begin{equation} L_X=\left( 3t + 1 \right) \left( 4t + 1 \right) \left( 8t + 9 \right) ^{2} \left( 1408t^{4} + 1024t^{3} + 256t^{2} + 20t - 1 \right)D^{4} + 2 t \left( 8t + 9 \right) \left( 675840t^{6} + 1620992t^{5} + 1276032t^{4} + 468224t^{3} + 85584t^{2} + 6912t + 125 \right)D^{3} + t \left( 37847040t^{7} + 132349952t^{6} + 165777920t^{5} + 91994496t^{4} + 25327744t^{3} + 3434832t^{2} + 190448t + 1449 \right)D^{2} + 4 t \left( 13516800t^{7} + 47024128t^{6} + 57428736t^{5} + 29291456t^{4} + 7077184t^{3} + 798552t^{2} + 33300t + 81 \right)D + 64 t ^{2} \left( 405504t^{6} + 1406208t^{5} + 1691072t^{4} + 811952t^{3} + 178024t^{2} + 17388t + 567 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Gr(3, 7)

Sector dimension: 6
Spectral radius: $29.7977466$
Diagonalizable at 1: Yes
Simple spectrum at 1: Yes

Eigenvalue approximations and Jordan block sizes

eigenvalue	multiplicity	Jordan blocks
$29.79775$	1	1
$-4$	1	1
$-3.327723 - 1.990883\mathrm{i}$	1	1
$-3.327723 + 1.990883\mathrm{i}$	1	1
$-3.142301$	1	1
$-3$	1	1

Characteristic polynomial

$$\begin{equation}\left(\lambda + 3\right) \left(\lambda + 4\right) \left(\lambda^{4} - 20 \lambda^{3} - 256 \lambda^{2} - 1024 \lambda - 1408\right)\notag\end{equation}$$

Minimal polynomial