Family 1-625-126-a

Projective four-space $\mathbb P^4$.

ℙ⁴P4 · 40 (preferred model)

Defining data

Ambient: ℙ⁴P4
Bundle: The bundle is zero: this presentation is the ambient variety itself.
ZL62 label: 40

Line bundles on the ambient $Y$

$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(5) \\ \det\mathcal{E} &\cong \mathcal{O}(0) \end{aligned} \end{equation}$$

Numerical invariants

Fano index	=	5
$\mathrm{h}^0(X, \omega_X^\vee)$	=	126
$\operatorname{vol}(X) = \mathrm{c}_1^4$	=	625
$\chi(X, \mathrm{T}_X)$	=	24

$\omega_X^\vee \cong \mathcal{O}_X(5)$

Chern numbers of $\mathrm{T}_X$

$\mathrm{c}_1^4$	=	625
$\mathrm{c}_1^2\mathrm{c}_2$	=	250
$\mathrm{c}_2^2$	=	100
$\mathrm{c}_1\mathrm{c}_3$	=	50
$\mathrm{c}_4$	=	5

Hodge diamond

				1
			0		0
		0		1		0
	0		0		0		0
0		0		1		0		0
	0		0		0		0
		0		1		0
			0		0
				1

Polyvector parallelogram & Hochschild cohomology

$\mathrm{HH}^{ 0 }$ = 1	1
$\mathrm{HH}^{ 1 }$ = 24	0	24
$\mathrm{HH}^{ 2 }$ = 126	0	0	126
$\mathrm{HH}^{ 3 }$ = 224	0	0	0	224
$\mathrm{HH}^{ 4 }$ = 126	0	0	0	0	126
$\mathrm{HH}^{ 5 }$ = 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

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

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

$=$

Regularized quantum period

Quantum period in the literature

catalogue	matching entries
Kalashnikov (2019)	CKK(1)
Coates et al. (2020)	P4
Coates–Kasprzyk–Prince (2015)	CKP(1)

Quantum differential equation

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

$$\begin{equation} L_X=\left( 5t - 1 \right) \left( 625t^{4} + 125t^{3} + 25t^{2} + 5t + 1 \right)D^{4} + 31250 t ^{5}D^{3} + 109375 t ^{5}D^{2} + 156250 t ^{5}D + 75000 t ^{5} \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ⁴P4

Sector dimension: 5
Spectral radius: $5$
Diagonalizable at 1: Yes
Simple spectrum at 1: Yes

Eigenvalue approximations and Jordan block sizes

eigenvalue	multiplicity	Jordan blocks
$-4.045085 - 2.938926\mathrm{i}$	1	1
$-4.045085 + 2.938926\mathrm{i}$	1	1
$1.545085 - 4.755283\mathrm{i}$	1	1
$1.545085 + 4.755283\mathrm{i}$	1	1
$5$	1	1

Characteristic polynomial

$$\begin{equation}\left(\lambda - 5\right) \left(\lambda^{4} + 5 \lambda^{3} + 25 \lambda^{2} + 125 \lambda + 625\right)\notag\end{equation}$$

Minimal polynomial