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Flat complex torus: canonical triviality and non-full SU(n) holonomy for n ≥ 2

Example

Assume full AC. Let $n \geq 1$ , let $Λ = \sum_{a = 1}^{2 n} Z λ_{a} \subset C^{n}$ be a full lattice generated by a real basis of $C^{n} ≅ R^{2 n}$ , and put $X = C^{n} / Λ$ . The standard Euclidean Hermitian metric $g_{0} = \sum_{j} (d x_{j}^{2} + d y_{j}^{2})$ descends, with real fundamental form $ω_{0} = \sum_{j} d x_{j} \land d y_{j}$ . Then $X$ is a compact connected Kähler manifold and its canonical bundle is trivialized by the descended nowhere-zero holomorphic form $Ω_{0} = d z^{1} \land \dots \land d z^{n}$ . Let $D$ be the Levi-Civita connection of the descended metric and define its curvature by $R^{D} (X, Y) Z : = D_{X} D_{Y} Z - D_{Y} D_{X} Z - D_{[X, Y]} Z$ . The metric is flat when $R^{D} = 0$ ; in this example it is flat and has trivial holonomy. For $n \geq 2$ , trivial holonomy is a proper subgroup of $S U (n)$ , so the broad Calabi–Yau convention does not force exact $S U (n)$ holonomy. For $n = 1$ , $S U (1)$ is itself trivial and this example does not separate the groups.

Example references

Compact Calabi–Yau manifold conventions

The Axiom of Choice

Facts & Assumptions

Given: Full AC; a real basis $λ_{1}, \dots, λ_{2 n}$ of $C^{n}$ ; the standard complex structure and Euclidean metric; and the broad convention from the preceding item.

[A1]

Full AC is the choice-function principle of The Axiom of Choice and is inherited from the complete convention supplier.

[F1]

The broad convention requires a connected compact Kähler $n$ -fold with a specified holomorphic trivialization of its canonical line bundle; it adds no holonomy equality assumption (Compact Calabi–Yau manifold conventions).

[F2]

For a supplied smooth Riemannian metric the Levi-Civita connection is the unique torsion-free, metric-compatible connection; parallel transport is defined by parallel sections, and holonomy is generated by transports around based loops (Compact Calabi–Yau manifold conventions).

Verification

technique · direct

1.1A1givenconstructalgebra

The real-linear isomorphism $L : R^{2 n} \to C^{n}$ sending the standard basis to $(λ_{a})$ maps $Z^{2 n}$ onto $Λ$ , so $Λ$ is discrete; bounded sets meet $Λ$ finitely because their inverse images under $L$ are bounded and bounded sets meet $Z^{2 n}$ finitely. For every compact $K \subset C^{n}$ , if $(K + λ) \cap K \neq \emptyset$ then $λ \in K - K$ , a bounded set meeting the lattice finitely; this proves proper discontinuity, and freeness follows because a translation fixing a point has vector $0$ . The shortest nonzero lattice length is positive. Distinct orbits have positive distance: for fixed representatives $z, w$ , only finitely many $λ$ can satisfy $∣ z - (w + λ) ∣ \leq R$ for any fixed $R$ , so if no $w + λ = z$ the minimum distance is positive. Small disjoint balls therefore separate distinct orbits and give Hausdorff quotient charts; choosing their radii below half the shortest lattice length makes their nonidentity translates disjoint. The chart transitions are translations and therefore holomorphic. The closed fundamental parallelepiped ${\sum_{a} t_{a} λ_{a} : 0 \leq t_{a} \leq 1}$ is compact and surjects onto $X$ , proving compactness; the quotient map is continuous and surjective, proving connectedness.

2.1F1givenstep 1.1algebra

Translations preserve $g_{0}$ , $J$ , and $ω_{0}$ , so these descend. The descended $ω_{0}$ is real, positive, and closed, hence Kähler. Translations also preserve $Ω_{0}$ ; it descends to a nowhere-zero holomorphic section of $K_{X}$ , which trivializes $K_{X}$ by the preceding convention [F1].

2.2F2givenstep 1.1algebra

On $C^{n}$ , define $D_{X} (\sum_{a} f_{a} E_{a}) = \sum_{a} X (f_{a}) E_{a}$ for the constant real coordinate frame $(E_{a}) = (\partial_{x_{1}}, \partial_{y_{1}}, \dots, \partial_{x_{n}}, \partial_{y_{n}})$ . Deck translations have identity differential, so this connection descends and the frame remains global and parallel. The fields $E_{a}$ commute, giving $T^{D} (E_{a}, E_{b}) = 0$ ; torsion is tensorial, so $T^{D} = 0$ . Since $g_{0} (E_{a}, E_{b}) = δ_{a b}$ is constant and $D E_{a} = 0$ , the connection is metric-compatible, hence $D$ is Levi-Civita by [F2]. Its curvature annihilates each parallel $E_{a}$ , and since these form a frame, $R^{D} = 0$ . Along any piecewise smooth loop, a parallel field has constant coefficients in this global frame, so its endpoint equals its initial value; the holonomy group is trivial.

3.1step 2.2algebra∎

For $n \geq 2$ , choose $t \notin 2 π Z$ . The matrix $diag (e^{i t}, e^{- i t}, 1, \dots, 1)$ is unitary, has determinant $1$ , and is not the identity; hence $S U (n)$ is nontrivial and cannot equal the trivial holonomy group. For $n = 1$ , $S U (1) = {1}$ , as stated.

Flat complex torus: canonical triviality and non-full SU(n) holonomy for n ≥ 2

Example

Example references

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels