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Dimension of the finite holonomy incidence manifold

Statement

Let $N \geq 2$ , $r \geq 1$ , and $1 \leq k < N$ . Regard $M_{N} (C)$ as $R^{2 N^{2}}$ by real and imaginary entries, write $Q^{*} = {\overline{Q}}^{T}$ , and set $U (N) = {Q : Q^{*} Q = I_{N}}$ and $S U (N) = {Q \in U (N) : \det Q = 1}$ . Equip $U (N)$ with the embedded regular-level smooth structure of $Q \mapsto Q^{*} Q$ on $G L_{N} (C)$ , and $S U (N)$ with the embedded regular-level smooth structure of $\det : U (N) \to S^{1}$ , where $S^{1} = {z \in C : ∣ z ∣ = 1}$ has its embedded circle structure. For each orthogonal projection $P$ onto a complex $k$ -plane in $C^{N}$ , write $V_{P} = ran P$ . Define

I_{k, r} = {(g_{1}, \dots, g_{r}, P) \in S U (N)^{r} \times {Gr}_{C} (k, N) : g_{i} V_{P} = V_{P} for every i},

Here ${Gr}_{C} (k, N) = {P : P^{*} = P, P^{2} = P, tr P = k}$ has the orthogonal-projection topology and the smooth atlas whose chart at a plane $V_{0}$ sends the graph of $T \in {Hom}_{C} (V_{0}, V_{0}^{⊥})$ to $T$ ; the chart domain consists of planes transverse to $V_{0}^{⊥}$ . Then $I_{k, r}$ is a closed embedded smooth submanifold,

\dim I_{k, r} = 2 k (N - k) + r (k^{2} + (N - k)^{2} - 1) .

Its codimension in $S U (N)^{r} \times {Gr}_{C} (k, N)$ is $2 r k (N - k)$ , and

\dim S U (N)^{r} - \dim I_{k, r} = 2 (r - 1) k (N - k) .

Facts & Assumptions

Given: Integers $N \geq 2$ , $r \geq 1$ , and $1 \leq k < N$ .

[F1]

${Gr}_{C} (k, N)$ is represented by Hermitian rank- $k$ orthogonal projections with the subspace topology inherited from the real and imaginary matrix entries (Compactness of the finite complex Grassmannian).

[F2]

If a smooth map between finite-dimensional smooth manifolds has a regular value with nonempty fiber, that fiber is an embedded submanifold of codimension equal to the target dimension (A regular level set is an embedded submanifold).

Proof

technique · direct

1.1F2givenalgebra

Regard $M_{m} (C)$ as $R^{2 m^{2}}$ and the Hermitian matrices as a real vector space of dimension $m^{2}$ . On the open set of invertible matrices, the smooth map $Θ (Q) = Q^{*} Q$ has derivative $d Θ_{Q} (X) = Q^{*} X + X^{*} Q$ . At a unitary $Q$ , every Hermitian $H$ is the derivative value for $X = \frac{1}{2} Q H$ . Thus $I_{m}$ is a regular value and [F2] makes $U (m) = Θ^{- 1} (I_{m})$ a manifold of dimension $m^{2}$ . The determinant map $U (m) \to S^{1}$ is a submersion: at the identity its derivative on a skew-Hermitian tangent matrix $X$ is $tr X$ , which reaches the tangent line $i R$ by $X = i t I_{m} / m$ , and translation gives surjectivity at every group element. Consequently $S U (m) = \det^{- 1} (1)$ has dimension $m^{2} - 1$ . The same determinant argument for $(U (k) \times U (N - k)) \to S^{1}$ , $(U, V) \mapsto \det U \det V$ , shows that $H_{k} = {diag (U, V) : U \in U (k), V \in U (N - k), \det U \det V = 1}$ is a closed Lie subgroup of dimension $k^{2} + (N - k)^{2} - 1$ .

1.2F1algebraconstruct

Fix a $k$ -plane $V_{0}$ and its orthogonal complement. A plane transverse to $V_{0}^{⊥}$ is the graph of a unique complex-linear map $T : V_{0} \to V_{0}^{⊥}$ . The graph projection is

P_{T} = (\begin{array}{c} I \\ T \end{array}) (I + T^{*} T)^{- 1} (\begin{array}{cc} I & T^{*} \end{array}),

and its inverse chart is $T = (P_{T})_{21} (P_{T})_{11}^{- 1}$ . These formulas are smooth, and such planes form an open neighborhood because invertibility of $(P)_{11}$ is an open matrix condition. Thus the Grassmannian has local charts modeled on ${Hom}_{C} (V_{0}, V_{0}^{⊥})$ and real dimension $2 k (N - k)$ .

2.1F1step 1.1step 1.2constructalgebra

On a sufficiently small graph chart, Gram–Schmidt applied to the columns of $(\binom{I}{T})$ and $(\binom{- T^{*}}{I})$ gives a smoothly varying unitary matrix $Q (T)$ carrying $V_{0}$ to the graph of $T$ ; all normalization denominators are positive because each list is linearly independent. A unitary matrix preserving a plane also preserves its orthogonal complement. Therefore $g_{i} V_{P} = V_{P}$ exactly when $Q (T)^{*} g_{i} Q (T)$ is block diagonal and belongs to $H_{k}$ . The map $(T, h_{1}, \dots, h_{r}) ⟼ (Q (T) h_{1} Q (T)^{*}, \dots, Q (T) h_{r} Q (T)^{*}, P_{T})$ and its inverse are smooth local coordinates for $I_{k, r}$ . Hence $I_{k, r}$ is an embedded submanifold with dimension $2 k (N - k) + r \dim H_{k}$ . It is closed because it is the zero set of the continuous matrix equations $g_{i} P - P g_{i} = 0$ .

3.1step 1.1step 1.2step 2.1algebra∎

Substituting $\dim H_{k} = k^{2} + (N - k)^{2} - 1$ gives the displayed dimension. Since $\dim S U (N) = N^{2} - 1$ , subtraction from the ambient product dimension gives codimension $r ((N^{2} - 1) - (k^{2} + (N - k)^{2} - 1)) = 2 r k (N - k)$ . Subtracting $\dim I_{k, r}$ from $\dim S U (N)^{r} = r (N^{2} - 1)$ instead gives $2 (r - 1) k (N - k)$ , the dimension deficit used when the plane variable is included in the source of a holonomy family.

Dimension of the finite holonomy incidence manifold

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources