Family 1-36-14-a

A smooth quadric section of the adjoint $G_2$ fivefold.

Gr(2, 7) · 601.27 (preferred model)

Defining data

Ambient: Gr(2, 7)
Bundle: $E^{ω_{3}} \oplus E^{2 ω_{2}}$ E^(w3) + E^(2w2)
Bundle: $Q^{\lor} (1) \oplus O (2)$ Q^v(1) + O(2)
ZL62 label: 601.27

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
$\mathrm{h}^0(X, \omega_X^\vee)$	=	14
$\operatorname{vol}(X) = \mathrm{c}_1^4$	=	36
$\chi(X, \mathrm{T}_X)$	=	-62

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

Chern numbers of $\mathrm{T}_X$

$\mathrm{c}_1^4$	=	36
$\mathrm{c}_1^2\mathrm{c}_2$	=	84
$\mathrm{c}_2^2$	=	196
$\mathrm{c}_1\mathrm{c}_3$	=	-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 + 62	0	x₀ + 62	x₂
$\mathrm{HH}^{ 3 }$ = x₁ + 1	0	0	x₁	1
$\mathrm{HH}^{ 4 }$ = x₁ - x₂ + 69	0	0	x₁ - x₂ + 55	0	14
$\mathrm{HH}^{ 5 }$ = 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

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

$\displaystyle\frac{ 1 + 9t + 16t^{2} + 9t^{3} + t^{4} }{(1-t)^{ 5 }}$

$=$

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

$$\begin{equation} L_X=\left( 6t + 1 \right) ^{3} \left( 828t^{2} + 60t - 1 \right)D^{4} + 12 t \left( 6t + 1 \right) ^{2} \left( 4140t^{2} + 516t + 7 \right)D^{3} + 54 t \left( 6t + 1 \right) \left( 19320t^{3} + 4048t^{2} + 198t + 1 \right)D^{2} + 12 t \left( 745200t^{4} + 238896t^{3} + 23058t^{2} + 636t + 1 \right)D + 144 t ^{2} \left( 29808t^{3} + 8532t^{2} + 693t + 14 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Gr(2, 7)

Show multiplicities

Sector dimension: 5
Spectral radius: $71.5692194$
Diagonalizable at 1: No
Simple spectrum at 1: No

Eigenvalue approximations and Jordan block sizes

eigenvalue	multiplicity	Jordan blocks
$71.56922$	1	1
$-11.56922$	1	1
$-6$	3	3

Characteristic polynomial

$$\begin{equation}\left(\lambda + 6\right)^{3} \left(\lambda^{2} - 60 \lambda - 828\right)\notag\end{equation}$$

Minimal polynomial