Family 1-64-21-a

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

ℙ⁵P5 · 50.F (preferred model)

Defining data

Ambient: ℙ⁵P5
Bundle: $E^{4 ω_{1}}$ E^(4w1)
Bundle: $O (4)$ O(4)
ZL62 label: 50.F

Line bundles on the ambient $Y$

$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(6) \\ \det\mathcal{E} &\cong \mathcal{O}(4) \end{aligned} \end{equation}$$

Numerical invariants

Fano index	=	2
$\mathrm{h}^0(X, \omega_X^\vee)$	=	21
$\operatorname{vol}(X) = \mathrm{c}_1^4$	=	64
$\chi(X, \mathrm{T}_X)$	=	-90

$\omega_X^\vee \cong \mathcal{O}_X(2)$

Chern numbers of $\mathrm{T}_X$

$\mathrm{c}_1^4$	=	64
$\mathrm{c}_1^2\mathrm{c}_2$	=	112
$\mathrm{c}_2^2$	=	196
$\mathrm{c}_1\mathrm{c}_3$	=	-64
$\mathrm{c}_4$	=	188

Hodge diamond

				1
			0		0
		0		1		0
	0		0		0		0
0		21		142		21		0
	0		0		0		0
		0		1		0
			0		0
				1

Polyvector parallelogram & Hochschild cohomology

$\mathrm{HH}^{ 0 }$ = 1	1
$\mathrm{HH}^{ 1 }$ = 0	0	0
$\mathrm{HH}^{ 2 }$ = 90	0	90	0
$\mathrm{HH}^{ 3 }$ = 15	0	0	0	15
$\mathrm{HH}^{ 4 }$ = 111	0	0	90	0	21
$\mathrm{HH}^{ 5 }$ = 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

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

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

$=$

Regularized quantum period

Quantum period in the literature

catalogue	matching entries
Kalashnikov (2019)	CKK(683)
Coates et al. (2020)	V(4,4)
Coates–Kasprzyk–Prince (2015)	CKP(682)

Quantum differential equation

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

$$\begin{equation} L_X=\left( 32t - 1 \right) \left( 32t + 1 \right)D^{4} + 4096 t ^{2}D^{3} + 5888 t ^{2}D^{2} + 3584 t ^{2}D + 768 t ^{2} \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ⁵P5

Show multiplicities

Sector dimension: 5
Spectral radius: $32$
Diagonalizable at 1: No
Simple spectrum at 1: No

Eigenvalue approximations and Jordan block sizes

eigenvalue	multiplicity	Jordan blocks
$-32$	1	1
$32$	1	1
$0$	3	3

Characteristic polynomial

$$\begin{equation}\lambda^{3} \left(\lambda + 32\right) \left(\lambda - 32\right)\notag\end{equation}$$

Minimal polynomial