Flat compact Hermite–Lorentz manifolds in dimension 4

Abstract

We give a classification, up to finite cover, of flat compact complete Hermite–Lorentz manifolds up to complex dimension 4.

Appendix A

For $\mathbb {K}=\mathbb {R}$ the following is a non redundant list, up to isomorphism, of the 8-dimensional nilpotent Lie algebras that appear as Lie algebras of unipotent simply transitive subgroups of $\mathrm {U}(3,1)\ltimes \mathbb {C}^{3+1}$. They are found putting together Propositions 4.3, 4.4, 5.19 and 5.34. Furthermore taking $\mathbb {K}=\mathbb {Q}$ this is also the complete non redundant list of the $\mathbb {Q}$-isomorphism classes of $\mathbb {Q}$-forms in the aforementioned Lie algebras and hence of the abstract commensurability classes of nilpotent crystallographic subgroups of $\mathrm {U}(3,1)\ltimes \mathbb {C}^{3+1}$. They are found putting together Propositions 6.4, 6.6 and 6.12. We present these Lie algebras defined over the field $\mathbb {K}$ in a compact version that is valid for both $\mathbb {K}=\mathbb {R}$ or $\mathbb {K}=\mathbb {Q}$. The presentation is given in the basis $\{x_1,\ldots ,x_8\}$ and we will write only the non zero Lie brackets. For the Lie algebras that decompose as a direct sum of an abelian ideal and a smaller dimensional Lie algebra we have written in brackets the corresponding names in the lists of de Graaf [15] for dimension up to 6 and of Gong [14] for dimension 7.

For the case $\pi (\gamma _3(i\xi )-J\gamma _3(\xi ))=0$, see Proposition 4.3, we have

• $L_1$: abelian,

• $L_2$: $[x_1,x_2]=x_3$$\qquad (L_{3,2})$,

• $L_3$: $[x_1,x_2]=x_4,[x_1,x_3]=x_5$$\qquad (L_{5,8})$,

• $L_4$: $[x_1,x_2]=x_4, [x_1,x_3]=x_5, [x_1,x_4]=x_6, [x_1,x_5]=x_7$$\qquad (247A)$,

• $L_5$: $[x_1,x_2]=x_4, [x_1,x_3]=x_5, [x_1,x_4]=x_6, [x_1,x_5]=x_7, [x_2,x_3]=x_6$$\qquad (247L)$,

• $L_6^\mathbb {K}(\varepsilon )$: $[x_1,x_2]=x_4, [x_1,x_3]=x_5, [x_1,x_4]=x_7, [x_1,x_5]=x_8, [x_2,x_6]=\varepsilon x_8, [x_3,x_6]=-x_7$ with $\varepsilon \in \mathbb {K}_{>0}$ and $L_{6}(\varepsilon )\cong L_6(\varepsilon ')$ if and only if there exists $\alpha \in \mathbb {K}^*$ such that $\varepsilon '=\alpha ^2\varepsilon$.

For the case $\pi (\gamma _3(i\xi )-J\gamma _3(\xi ))\not =0$ and $\gamma _2=0$, see Proposition 4.4, we have

• $N_1$: $[x_1,x_2]=x_3,[x_1,x_3]=x_5,[x_2,x_4]=x_6$$\qquad (L_{6,19}(0))$,

• $N_2$: $[x_1,x_2]=x_3,[x_1,x_3]=x_5,[x_1,x_4]=x_6$$\qquad (L_{6,25})$,

• $N_3^\mathbb {K}(\varepsilon )$: $[x_1,x_2]=x_3,[x_1,x_3]=x_5,[x_1,x_4]=\varepsilon x_6, [x_2,x_3]=x_6,[x_2,x_4]=x_5$ with $\varepsilon \in \mathbb {K}$ and $N_3(\varepsilon )\cong N_3(\varepsilon ')$ if and only if there exists $\alpha \in \mathbb {K}^*$ such that $\varepsilon '=\alpha ^2\varepsilon$$\qquad (L_{6,24}(\varepsilon ))$,

• $N_4$: $[x_1,x_2]=x_3,[x_1,x_3]=x_5,[x_1,x_4]=x_6,[x_2,x_4]=x_5$$\qquad (L_{6,23})$,

• $N_5: [x_1,x_2]=x_3, [x_1,x_3]=x_6, [x_1,x_5]=x_7, [x_2,x_4]=x_6$$\qquad (257A)$,

• $N_6$: $[x_1,x_2]=x_3, [x_1,x_3]=x_6, [x_1,x_4]=x_7, [x_2,x_5]=x_7$$\qquad (257B)$,

• $N_7$: $[x_1,x_2]=x_3, [x_1,x_3]=x_6, [x_2,x_4]=x_6, [x_2,x_5]=x_7$$\qquad (257C)$,

• $N_8$: $[x_1,x_2]=x_3, [x_1,x_3]=x_6, [x_1,x_4]=x_7, [x_2,x_4]=x_6, [x_2,x_5]=x_7$$\qquad (257D)$,

• $N_9^\mathbb {K}(\varepsilon )$: $[x_1,x_2]=x_3, [x_1,x_3]=x_6, [x_1,x_4]=x_6, [x_1,x_5]=x_7, [x_2,x_3]=x_7, [x_2,x_5]=\varepsilon x_6$ with $\varepsilon \in \mathbb {K}$ and $N_{9}(\varepsilon )\cong N_{9}(\varepsilon ')$ if and only if there exists $\alpha \in \mathbb {K}^*$ such that $\varepsilon '=\alpha ^6\varepsilon$. Over $\mathbb {R}$ to classify equivalence classes the parameter $\varepsilon$ can then take three values $\varepsilon \in \{0,1,-1\}$ that correspond to $(257I),(257J_1)$ and (257J) respectively,

• $N_{10}: [x_1,x_2]=x_3, [x_1,x_3]=x_7, [x_1,x_4]=x_8, [x_2,x_5]=x_7, [x_2,x_6]=x_8$

• $N_{11}: [x_1,x_2]=x_3, [x_1,x_3]=x_7, [x_1,x_4]=x_8, [x_2,x_3]=x_8, [x_2,x_5]=x_7, [x_2,x_6]=x_8$

• $N_{12}: [x_1,x_2]=x_6, [x_1,x_3]=x_7, [x_1,x_4]=x_8, [x_2,x_5]=x_8$

• $N_{13}: [x_1,x_2]=x_6, [x_1,x_3]=x_7, [x_1,x_4]=x_8, [x_2,x_4]=x_7, [x_2,x_5]=x_8$.

For the case $\pi (\gamma _3(i\xi )-J\gamma _3(\xi ))\not =0$ and $\gamma _2\ne 0$, see Proposition  5.19, 5.34, 6.4 and 6.6, we have

• $\mathfrak {g}_\mathbb {K}(0,0,0,0)$: $[x_1,x_2]=x_4, [x_1,x_3]=x_5, [x_2,x_3]=x_6, [x_1,x_4]=x_7, [x_1,x_5]=x_8$

• $\mathfrak {g}_\mathbb {K}(0,0,\varepsilon ,1)$: $[x_1,x_2]=x_4, [x_1,x_3]=x_5, [x_2,x_3]=x_6, [x_1,x_4]=x_7, [x_1,x_5]=x_8, [x_2,x_6]=\varepsilon x_8,$$[x_3,x_6]=x_7$ with $\varepsilon \in \mathbb {K}$ such that $\mathfrak {g}_\mathbb {K}(0,0,\varepsilon ',1)\cong \mathfrak {g}_\mathbb {K}(0,0,\varepsilon ,1)$ if and only if there exists $\alpha \in \mathbb {K}^*$ such that $\varepsilon '=\alpha ^2\varepsilon$

• $\mathfrak {g}_\mathbb {R}(a,b,c):\ [x_1,x_2]=x_4, [x_1,x_3]=x_5, [x_2,x_3]=x_6, [x_1,x_4]=x_7, [x_1,x_5]=x_8, [x_2,x_4]=x_7,$$[x_2,x_5]=- x_8, [x_3,x_4]=-x_8, [x_3,x_5]=3x_7, [x_2,x_6]=ax_7+bx_8, [x_3,x_6]=cx_7-ax_8$ with $a,b,c\in \mathbb {R}, (a,b)\ne (0,-2) \text { and }\mathfrak {g}_\mathbb {R}(a,b,c)\cong \mathfrak {g}_\mathbb {R}(-a,b,c)$

• $\mathfrak {g}_\mathbb {Q}(e,f,g,h,j,k,l)$ with $e,f,g,h,j,k,l\in \mathbb {Q}$. This is a general family of Lie algebras of the form: $[x_1,x_2]=x_4, [x_1,x_3]=x_5, [x_2,x_3]=x_6, [x_1,x_4]=x_7, [x_1,x_5]=x_8, [x_2,x_4]=fx_7+hx_8, [x_2,x_5]=-gx_7-f x_8, [x_3,x_4]=-gx_7-fx_8, [x_3,x_5]=ex_7+gx_8,[x_2,x_6]=jx_7+kx_8, [x_3,x_6]=lx_7-jx_8$ . Each isomorphism class of this family is represented by one of the Lie algebras of the following lists. For all of them we will ask

  $$\begin{aligned}&3f^2g^2+6efgh-4f^3h-4g^3e\\&\quad -e^2h^2<0 \hbox { and } \begin{pmatrix} hl+2gj-fk \\ -eh+fg+j\\ 2g^2-2fh+k\end{pmatrix}\wedge \begin{pmatrix} -gl+ek-2fj\\ 2eg-2f^2+l\\ eh-fg-j \end{pmatrix}\ne 0. \end{aligned}$$

  • $\mathfrak {h}^1_\mathbb {Q}([e],[g],j,k,l):$ with $j,k,l\in \mathbb {Q}$ and $(e,g)\in \mathbb {Q}^2$ representatives of the equivalence class defined by $(e,g)\sim (e',g')$ if and only if there exists $\sigma _1,\sigma _2\in \mathbb {Q}^*$ such that $e'=\sigma _1^3 e$ and $g'=\sigma _1\sigma _2^2g$. This family is defined by $f=h=0$.

  • $\mathfrak {h}^2_\mathbb {Q}([e],m,j,k,l,[t]):$ with $j,k,l\in \mathbb {Q}$, $e\in \mathbb {Q}^*$ a representative of the equivalent class defined by $e\sim e'$ if and only if there exists $\mu \in \mathbb {Q}^*$ such that $e'=\mu ^9e$, $m\in \mathbb {Q}\cup \{\infty \}$ and $t\in \mathbb {Q}{\setminus }\mathbb {Q}^3$ a representative of the equivalence class defined by $t\sim t'$ if and only if there exists $\mu \in \mathbb {Q}$ such that $t'=\mu ^3t^j$ with $j=1,2$.

    • if $m\in \mathbb {Q}$: this family is defined by $f=0,g=-emt$ and $h=-et(m^3t+1)$,

    • if $m=\infty$: this family is defined by $f=0,g=0$ and $h=-et^2$.

  • $\mathfrak {h}^3_\mathbb {Q}([e],m,j,k,l,[t]):$ with $j,k,l\in \mathbb {Q}$, $e\in \mathbb {Q}^*$ a representative of the equivalent class defined by $e\sim e'$ if and only if there exists $\mu \in \mathbb {Q}^*$ such that $e'=\mu ^9e$, $m\in \mathbb {Q}\cup \{\infty \}$ and $t\in \mathbb {Q}$ such that $t^2-4>0$, $t^2-4\notin \mathbb {Q}^2$ and t not in the image of the function $f(x)=x^3-3x$ over the rational. Finally we choose t as a representative of the equivalence class defined by $t\sim t'$ if and only if there exists $\alpha ,\beta \in \mathbb {Q}$ with $\alpha ^2+t\alpha \beta +\beta ^2=1$ such that $t'=-3t\alpha ^2\beta +t\beta ^3+6\alpha +\alpha ^3t^2-8\alpha ^3$.

    • $m\in \mathbb {Q}$: this family is defined by $f=-2em,g=e(3m^2-tm-1)$ and $h=et(3m^2-m^3t-1)$,

    • $m=\infty$: this family is defined by $f=-2e,g=3e$ and $h=-et^2$.

  Let us remark that the presentation of the Lie algebras depend on the choice of representative of the class [t].

• $\mathfrak {g}_\mathbb {K}(0,-2,c): [x_1,x_2]=x_4, [x_1,x_3]=x_5, [x_2,x_3]=x_6, [x_1,x_4]=x_7, [x_1,x_5]=x_8, [x_2,x_4]=x_7,$$[x_2,x_5]=- x_8, [x_3,x_4]=-x_8, [x_3,x_5]=3x_7, [x_2,x_6]=-2x_8, [x_3,x_6]=cx_7$ with $c\in \mathbb {K}$.

Remark A.1

Since it is not written in Gong’s thesis we point out that the isomorphism between $N_9^\mathbb {R}(-1)$ and (37B) is given by $x_1'=x_2-x_4,x_2'=x_1-x_3,x_3'=x_1+x_3,x_4'=x_2+x_4x_5'=2(x_7-x_5), x_6'=2x_6, x_7'=2(x_5+x_7)$.