Family 1-5-6-a

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

ℙ⁵P5 · 50.K (preferred model)

Defining data

Ambient: ℙ⁵P5
Bundle: $E^{5 ω_{1}}$ E^(5w1)
Bundle: $O (5)$ O(5)
ZL62 label: 50.K

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
$\mathrm{h}^0(X, \omega_X^\vee)$	=	6
$\operatorname{vol}(X) = \mathrm{c}_1^4$	=	5
$\chi(X, \mathrm{T}_X)$	=	-216

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

Chern numbers of $\mathrm{T}_X$

$\mathrm{c}_1^4$	=	5
$\mathrm{c}_1^2\mathrm{c}_2$	=	50
$\mathrm{c}_2^2$	=	500
$\mathrm{c}_1\mathrm{c}_3$	=	-150
$\mathrm{c}_4$	=	825

Hodge diamond

				1
			0		0
		0		1		0
	0		0		0		0
0		120		581		120		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
$\mathrm{HH}^{ 2 }$ = 216	0	216	0
$\mathrm{HH}^{ 3 }$ = 0	0	0	0	0
$\mathrm{HH}^{ 4 }$ = 552	0	0	546	0	6
$\mathrm{HH}^{ 5 }$ = 0		0	0	0	0
$\mathrm{HH}^{ 6 }$ = 56			0	56	0
$\mathrm{HH}^{ 7 }$ = 0				0	0
$\mathrm{HH}^{ 8 }$ = 0					0

Hilbert series

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

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

$=$

Regularized quantum period

Quantum period in the literature

catalogue	matching entries
Kalashnikov (2019)	CKK(749)
Coates et al. (2020)	No match
Coates–Kasprzyk–Prince (2015)	CKP(738)

Quantum differential equation

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

$$\begin{equation} L_X=\left( 3005t - 1 \right) \left( 120t + 1 \right) ^{4}D^{4} + 50 t \left( 72120t + 101 \right) \left( 120t + 1 \right) ^{3}D^{3} + 25 t \left( 120t + 1 \right) ^{2} \left( 60580800t^{2} + 304080t + 127 \right)D^{2} + 50 t \left( 120t + 1 \right) \left( 5192640000t^{3} + 52689600t^{2} + 116040t + 13 \right)D + 144000 t ^{2} \left( 103852800t^{3} + 1713600t^{2} + 8400t + 11 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ⁵P5

Show multiplicities

Sector dimension: 5
Spectral radius: $3005$
Diagonalizable at 1: No
Simple spectrum at 1: No

Eigenvalue approximations and Jordan block sizes

eigenvalue	multiplicity	Jordan blocks
$3005$	1	1
$-120$	4	4

Characteristic polynomial

$$\begin{equation}\left(\lambda + 120\right)^{4} \left(\lambda - 3005\right)\notag\end{equation}$$

Minimal polynomial