Family 1-57-18-a

Küchle’s fourfold of type (b11).

Gr(2, 8) · 701.211 (preferred model)

Defining data

Ambient: Gr(2, 8)
Bundle: $(E^{ω_{2}})^{\oplus 2} \oplus E^{ω_{3}}$ 2 copies of E^(w2) + E^(w3)
Bundle: $O (1)^{\oplus 2} \oplus Q^{\lor} (1)$ 2 copies of O(1) + Q^v(1)
ZL62 label: 701.211

Line bundles on the ambient $Y$

$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(8) \\ \det\mathcal{E} &\cong \mathcal{O}(7) \end{aligned} \end{equation}$$

Numerical invariants

Fano index	=	1
$\mathrm{h}^0(X, \omega_X^\vee)$	=	18
$\operatorname{vol}(X) = \mathrm{c}_1^4$	=	57
$\chi(X, \mathrm{T}_X)$	=	-28

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

Chern numbers of $\mathrm{T}_X$

$\mathrm{c}_1^4$	=	57
$\mathrm{c}_1^2\mathrm{c}_2$	=	90
$\mathrm{c}_2^2$	=	150
$\mathrm{c}_1\mathrm{c}_3$	=	6
$\mathrm{c}_4$	=	39

Hodge diamond

				1
			0		0
		0		1		0
	0		0		0		0
0		2		31		2		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 + 28	0	x₀ + 28	x₂
$\mathrm{HH}^{ 3 }$ = x₁	0	0	x₁	0
$\mathrm{HH}^{ 4 }$ = x₁ - x₂ + 10	0	0	x₁ - x₂ - 8	0	18
$\mathrm{HH}^{ 5 }$ = 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

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

$\displaystyle\frac{ 1 + 13t + 29t^{2} + 13t^{3} + t^{4} }{(1-t)^{ 5 }}$

$=$

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 $L_X\widehat{\mathrm{G}}_X=0$.

$$\begin{equation} L_X=\left( 4t + 1 \right) \left( 30t + 19 \right) ^{2} \left( 3t + 1 \right) ^{3} \left( 339t^{2} + 29t - 1 \right)D^{4} + 2 t \left( 30t + 19 \right) \left( 3t + 1 \right) ^{2} \left( 610200t^{4} + 708912t^{3} + 227367t^{2} + 22810t + 334 \right)D^{3} + t \left( 3t + 1 \right) \left( 384426000t^{6} + 779662260t^{5} + 573842016t^{4} + 186568173t^{3} + 26773377t^{2} + 1383136t + 8151 \right)D^{2} + t \left( 1647540000t^{7} + 3778202880t^{6} + 3304467576t^{5} + 1379326050t^{4} + 287055900t^{3} + 27736542t^{2} + 940804t + 1805 \right)D + 12 t ^{2} \left( 65901600t^{6} + 148351500t^{5} + 125653680t^{4} + 49436244t^{3} + 9359118t^{2} + 778791t + 20216 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Gr(2, 8)

Show multiplicities

Sector dimension: 6
Spectral radius: $37.9360833$
Diagonalizable at 1: No
Simple spectrum at 1: No

Eigenvalue approximations and Jordan block sizes

eigenvalue	multiplicity	Jordan blocks
$37.93608$	1	1
$-8.936083$	1	1
$-4$	1	1
$-3$	3	3

Characteristic polynomial

$$\begin{equation}\left(\lambda + 3\right)^{3} \left(\lambda + 4\right) \left(\lambda^{2} - 29 \lambda - 339\right)\notag\end{equation}$$

Minimal polynomial