Family 1-32-13-a

A hyperplane and quadric section of $\operatorname{LG}(3,6)$, Küchle’s fourfold of type (c2).

Gr(3, 6) · 53.271 (preferred model)

Defining data

Ambient: Gr(3, 6)
Bundle: $E^{ω_{2}} \oplus E^{ω_{3}} \oplus E^{2 ω_{3}}$ E^(w2) + E^(w3) + E^(2w3)
Bundle: $⋀^{2} U^{\lor} \oplus O (1) \oplus O (2)$ wedge^2 U^v + O(1) + O(2)
ZL62 label: 53.271

Line bundles on the ambient $Y$

$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(6) \\ \det\mathcal{E} &\cong \mathcal{O}(5) \end{aligned} \end{equation}$$

Numerical invariants

Fano index	=	1
$\mathrm{h}^0(X, \omega_X^\vee)$	=	13
$\operatorname{vol}(X) = \mathrm{c}_1^4$	=	32
$\chi(X, \mathrm{T}_X)$	=	-61

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

Chern numbers of $\mathrm{T}_X$

$\mathrm{c}_1^4$	=	32
$\mathrm{c}_1^2\mathrm{c}_2$	=	80
$\mathrm{c}_2^2$	=	200
$\mathrm{c}_1\mathrm{c}_3$	=	-36
$\mathrm{c}_4$	=	132

Hodge diamond

				1
			0		0
		0		1		0
	0		0		0		0
0		14		100		14		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 + 61	0	x₀ + 61	x₂
$\mathrm{HH}^{ 3 }$ = x₁	0	0	x₁	0
$\mathrm{HH}^{ 4 }$ = x₁ - x₂ + 69	0	0	x₁ - x₂ + 56	0	13
$\mathrm{HH}^{ 5 }$ = 0		0	0	0	0
$\mathrm{HH}^{ 6 }$ = 1			0	1	0
$\mathrm{HH}^{ 7 }$ = 0				0	0
$\mathrm{HH}^{ 8 }$ = 0					0

Hilbert series

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

$\displaystyle\frac{ 1 + 8t + 14t^{2} + 8t^{3} + t^{4} }{(1-t)^{ 5 }}$

$=$

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

$$\begin{equation} L_X=\left( 8t + 1 \right) ^{3} \left( 448t^{2} + 80t - 1 \right)D^{4} + 16 t \left( 40t + 7 \right) \left( 56t + 1 \right) \left( 8t + 1 \right) ^{2}D^{3} + 8 t \left( 8t + 1 \right) \left( 125440t^{3} + 29952t^{2} + 1640t + 9 \right)D^{2} + 16 t \left( 716800t^{4} + 222208t^{3} + 21408t^{2} + 624t + 1 \right)D + 256 t ^{2} \left( 21504t^{3} + 5952t^{2} + 492t + 11 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Gr(3, 6)

Show multiplicities

Sector dimension: 5
Spectral radius: $85.254834$
Diagonalizable at 1: No
Simple spectrum at 1: No

Eigenvalue approximations and Jordan block sizes

eigenvalue	multiplicity	Jordan blocks
$85.25483$	1	1
$-8$	3	3
$-5.254834$	1	1

Characteristic polynomial

$$\begin{equation}\left(\lambda + 8\right)^{3} \left(\lambda^{2} - 80 \lambda - 448\right)\notag\end{equation}$$

Minimal polynomial