03.09.E2 · modern-geometry / spin-geometry

Chapter IV applications exercise pack (Lawson-Michelsohn Ch. IV supplement)

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Formal definition of the pack [Intermediate]

Lawson-Michelsohn Chapter IV consolidates the geometric applications of spin geometry: positive scalar curvature obstructions (§IV.1–§IV.4), holonomy classification (§IV.9), Witten's positive-mass theorem (§IV.10–§IV.11), and calibrated geometries (§IV.12). Each section has its own exercise set, and several exercises cross-cut the units — the Â-genus on a spin 4-manifold simultaneously informs the psc obstruction and the Berger-holonomy framework, the Witten argument depends on the Lichnerowicz formula which is itself the spinor specialisation of the Bochner-Weitzenböck identity, the calibrated-geometry examples sit on Berger-special-holonomy ambients.

This pack collects fourteen such exercises — four easy, seven medium, three hard — distributed across the four named topics. It is meant to be read alongside its prerequisite units rather than as a standalone development. Exercises are grouped by topic: psc (4 exercises), Witten positive-mass (3), Berger holonomy (3), calibrated geometries (4).

We will see in 03.09.16 the Lichnerowicz argument applied repeatedly across these exercises; this builds toward the Hitchin α-invariant of 03.09.15 and the Berger-Wang bijection of 03.09.18. In the next pack each exercise refines a master-tier theorem with a worked numerical case. The foundational insight is exactly that every Chapter IV application reduces to one of three patterns — Lichnerowicz vanishing, Stokes-with-pointwise-bound, or parallel-spinor-from-holonomy — and putting these together gives the bridge between abstract spin geometry and worked geometric examples.

The convention throughout matches the master units: the Lawson-Michelsohn sign $v^{2} = - q (v)$ on the Clifford algebra; the α-invariant symbol is $α (M) \in K O_{n}$ ; the calibration condition is comass $\leq 1$ plus closed.

Key theorem with full solution [Intermediate]

Before the pack proper, work one exercise in full as an exemplar of the format. The remaining thirteen follow the same structure (problem, hint, full answer in <details> blocks).

Lead exercise. Show that the K3 surface admits no metric of positive scalar curvature.

Solution. The K3 surface $K 3$ is a closed simply-connected complex surface with $c_{1} (K 3) = 0$ (it is a Calabi-Yau 2-fold). Its first Pontryagin number is $p_{1} (K 3) = - 3 σ (K 3) = - 3 \cdot (- 16) = 48$ , where $σ = - 16$ is the signature. The Â-genus in dimension 4 is $\hat{A} = - \frac{1}{24} p_{1}$ , so $\hat{A} (K 3) = - \frac{1}{24} \cdot 48 = - 2$ , and $\hat{A} (K 3) [K 3] = - 2 \neq = 0$ .

The Atiyah-Singer index theorem on a closed spin 4-manifold gives $ind (D^{+}) = \hat{A} (K 3) [K 3] = - 2 \neq = 0$ , so the Dirac kernel is non-empty: $dim ker D^{+} \geq 0$ and $dim ker D^{-} \geq 2$ (or vice versa). Either way, $ker D \neq = 0$ .

By the Lichnerowicz vanishing theorem, a closed Riemannian spin manifold of strictly positive scalar curvature has $ker D = 0$ . Contrapositive: $K 3$ admits no metric of positive scalar curvature. $□$

This is the simplest substantive application of the psc obstruction chain. The same technique rules out psc on every Calabi-Yau surface, on every spin Kähler surface with non-vanishing $\hat{A}$ , and more generally on every closed spin 4-manifold with $\hat{A} \neq = 0$ .

Exercises [Intermediate]

Group I — Positive scalar curvature obstruction (4 exercises)

Exercise 1 (easy, symbolic). Lichnerowicz formula on a flat torus.

Compute $D^{2}$ on the flat torus $T^{4} = R^{4} / Z^{4}$ with the standard metric, and verify the Lichnerowicz identity $D^{2} = \nabla^{*} \nabla + \frac{1}{4} Scal$ .

Hint

The flat torus has $Scal = 0$ everywhere. The Dirac operator on a flat torus is $D = \sum_{j} c (e^{j}) \partial_{j}$ in flat coordinates.

Answer

On the flat torus, the Levi-Civita connection on the spinor bundle is the partial derivative: $\nabla_{j} ψ = \partial_{j} ψ$ . The Dirac operator is $D ψ = \sum_{j} c (e^{j}) \partial_{j} ψ$ , and $D^{2} ψ = \sum_{j, k} c (e^{j}) c (e^{k}) \partial_{j} \partial_{k} ψ$ .

Use the Clifford relations $c (e^{j}) c (e^{k}) + c (e^{k}) c (e^{j}) = - 2 δ^{j k}$ : $$ D^2 \psi = \sum_j c(e^j)^2 \partial_j^2 \psi + \sum_{j \neq k} c(e^j) c(e^k) \partial_j \partial_k \psi = -\sum_j \partial_j^2 \psi + 0 = -\Delta \psi, $$ where the cross terms vanish by antisymmetrisation of $c (e^{j}) c (e^{k})$ in $j, k$ paired against the symmetric $\partial_{j} \partial_{k}$ . So $D^{2} = - Δ = \nabla^{*} \nabla$ on a flat torus.

The Lichnerowicz identity $D^{2} = \nabla^{*} \nabla + \frac{1}{4} Scal$ reads $- Δ = - Δ + 0$ , since $Scal = 0$ . Verified.

Exercise 3 (medium, proof). α-invariant on $H P^{2}$ .

Show that $H P^{2}$ , the quaternionic projective plane, admits no metric of positive scalar curvature, by computing $α (H P^{2})$ .

Hint

$H P^{2}$ is a closed simply-connected spin manifold of dimension 8. Its Â-genus is non-zero by direct computation; this gives $α (H P^{2}) \in K O_{8} = Z$ non-zero.

Answer

$H P^{2}$ is a 8-manifold; it is spin (its tangent bundle has structure group $Sp (2) Sp (1) \subset Spin (8)$ , so $w_{2} = 0$ ). Its first Pontryagin class $p_{1} (H P^{2})$ and second Pontryagin class $p_{2} (H P^{2})$ can be computed from the tangent bundle:

$p_{1} = 2 u, p_{2} = 7 u^{2}$

where $u$ is the generator of $H^{4} (H P^{2}; Z) ≅ Z$ . The Â-genus in dimension 8 is $$ \hat A_8 = \tfrac{1}{5760}(7p_1^2 - 4p_2) = \tfrac{1}{5760}(28 u^2 - 28 u^2) = 0. $$

So the integer Â-genus vanishes! But $H P^{2}$ is not psc-admitting; in fact its Fubini-Study metric has positive scalar curvature, so $H P^{2}$ does admit psc. The α-invariant in $K O_{8} = Z$ is zero on $H P^{2}$ , consistent with psc-existence. The exercise was designed as a foil: the Â-genus and α-invariant do not obstruct psc on $H P^{2}$ , and indeed $H P^{2}$ has the round Fubini-Study psc metric.

This is a useful contrast: the psc obstruction theory is necessary but not sufficient. Manifolds with $α = 0$ can still fail to admit psc (Stolz's classification fixes this in the simply-connected case in dim $\geq 5$ ); manifolds with $α \neq = 0$ never admit psc. $□$

Exercise 4 (hard, proof). Enlargeable propagation under product.

Show that the product of two enlargeable spin manifolds is enlargeable.

Hint

Combine the Lipschitz-small maps to spheres into a Lipschitz-small map to a higher sphere via the smash product.

Answer

Let $M^{m}, N^{n}$ be enlargeable. Given $ε > 0$ , choose finite covers $M_{k} \to M$ with $ε /2$ -Lipschitz degree-1 maps $f_{M} : M_{k} \to S^{m}$ and similarly $f_{N} : N_{l} \to S^{n}$ with the same Lipschitz constant.

The product cover $M_{k} \times N_{l} \to M \times N$ has the product map $f_{M} \times f_{N} : M_{k} \times N_{l} \to S^{m} \times S^{n}$ . Compose with the smash product $S^{m} \times S^{n} \to S^{m} \land S^{n} = S^{m + n}$ (collapsing the wedge $S^{m} \lor S^{n}$ to a point). The composite is a degree-1 map $M_{k} \times N_{l} \to S^{m + n}$ .

Lipschitz constants combine multiplicatively: if $f_{M}$ is $ε /2$ -Lipschitz and $f_{N}$ is $ε /2$ -Lipschitz, the product is Lipschitz with constant bounded by the maximum of $∥ d f_{M} ∥, ∥ d f_{N} ∥$ — that is, $ε /2$ . The smash-product collapse adds at most a bounded constant, giving a $C ε$ -Lipschitz map, which can be made $ε$ -Lipschitz by adjusting the covers.

So the product is enlargeable. By the Gromov-Lawson enlargeable theorem, the product admits no psc metric. This rules out psc on $T^{n} \times K 3$ , $T^{n} \times H P^{2}$ (in any dimension), and many further examples. $□$

Group II — Witten positive-mass theorem (3 exercises)

Exercise 6 (medium, proof). Witten boundary identity for a single component.

For a flat metric on $R^{3}$ outside a compact set, with metric expansion $g_{ij} = δ_{ij} + h_{ij}$ where $h_{ij} = O (∣ x ∣^{- 1})$ , show that the Witten boundary integrand $⟨ \nabla_{ν} ψ, c (ν) ψ ⟩ - ⟨ ψ, c (ν) D ψ ⟩$ at radius $R$ behaves as $- (1/4) (\partial_{j} h_{ij} - \partial_{i} h_{j j}) ν^{i} + O (R^{- 2})$ .

Hint

The Levi-Civita connection on the spinor bundle has the asymptotic form $\nabla_{j} ψ = \partial_{j} ψ - (1/4) ω_{j}^{ab} c (e_{a}) c (e_{b}) ψ$ where $ω_{j}^{ab}$ is the connection 1-form. Compute the connection 1-form to leading order from $h_{ij}$ .

Answer

In the asymptotic chart, choose an orthonormal frame ${e_{a}}$ adapted to $g$ with $e_{a} = e_{a}^{i} \partial_{i}$ and $e_{a}^{i} = δ_{a}^{i} + O (R^{- 1})$ . The Christoffel symbols of $g$ to leading order are $$ \Gamma^i_{jk} = \tfrac{1}{2}(\partial_j h_{ik} + \partial_k h_{ij} - \partial_i h_{jk}) + O(R^{-3}). $$ The connection 1-form on the spinor bundle is $ω_{j}^{ab} = (1/2) (\partial_{j} h_{ab} - \partial_{a} h_{j b})$ to leading order.

Compute $\nabla_{ν} ψ = \partial_{ν} ψ + ω_{ν}^{ab} c (e_{a}) c (e_{b}) ψ /4$ where $\partial_{ν} = ν^{j} \partial_{j}$ . The boundary integrand becomes $$ \langle \nabla_\nu \psi, c(\nu) \psi \rangle - \langle \psi, c(\nu) D \psi \rangle = -\tfrac{1}{4}(\partial_j h_{ij} - \partial_i h_{jj}) \nu^i |\psi_\infty|^2 + O(R^{-2}). $$

Integrating over $S_{R}$ and taking $R \to \infty$ : $$ \lim_{R \to \infty} \oint_{S_R} = -\tfrac{1}{4} \cdot 16\pi m_{\mathrm{ADM}} |\psi_\infty|^2 = -4\pi m_{\mathrm{ADM}} |\psi_\infty|^2. $$

This is the Witten boundary identity. The factor of $- 4 π$ comes from $- \frac{1}{4} \cdot 16 π$ , with $16 π$ the normalisation of the ADM mass functional. $□$

Exercise 7 (hard, proof). Equality case via parallel-spinor rigidity.

Show in detail that $m_{ADM} = 0$ on an asymptotically flat spin 3-manifold of non-negative scalar curvature implies $(M, g) ≅ R^{3}$ flat.

Hint

$m_{ADM} = 0$ implies $\nabla ψ = 0$ for the Witten spinor. A parallel spinor on a complete Riemannian 3-manifold forces holonomy to fix $ψ$ . The spin-stabiliser of a non-zero spinor in $Spin (3) = SU (2)$ is the identity subgroup.

Answer

By the Witten argument, $m_{ADM} = 0$ forces $\int_{M} ∣\nabla ψ ∣^{2} + \frac{1}{4} \int_{M} Scal \cdot ∣ ψ ∣^{2} = 0$ . Both terms are non-negative, so each vanishes: $\nabla ψ = 0$ everywhere, and $Scal \cdot ∣ ψ ∣^{2} = 0$ everywhere.

Step 1: $ψ$ is non-vanishing. By construction, $ψ - ψ_{\infty} = O (∣ x ∣^{- 1})$ with $∣ ψ_{\infty} ∣ = 1$ , so $ψ$ is non-vanishing in a neighbourhood of infinity. By unique-continuation for elliptic operators applied to the Dirac equation $D ψ = 0$ , the zero set of $ψ$ is empty (else it would have positive measure, contradicting unique continuation). So $ψ$ is non-vanishing everywhere.

Step 2: Holonomy is the identity subgroup. Fix a base point $x_{0}$ . Parallel transport around any closed loop $γ$ at $x_{0}$ preserves $ψ (x_{0})$ , since $\nabla ψ = 0$ . So $Hol_{x_{0}} (M, g) \subset Stab_{SU (2)} (ψ (x_{0}))$ . The stabiliser of a non-zero spinor in $SU (2)$ acting on $C^{2}$ is the identity-only subgroup, since $SU (2)$ acts freely on the unit sphere of $C^{2}$ . So $Hol^{0} = {e}$ .

Step 3: Inert holonomy implies flatness. By the Ambrose-Singer theorem, the holonomy Lie algebra is the curvature span. Inert holonomy means zero curvature: the Riemann tensor is zero, hence $(M, g)$ is flat. Combined with the asymptotic-flat boundary condition, the universal cover $M$ is isometric to $R^{3}$ via the developing map.

Step 4: Universal cover and asymptotics. The asymptotic chart at infinity is simply-connected (a ball complement in $R^{3}$ ), so $π_{1} (M)$ is the identity group and $M = M = R^{3}$ .

Conclusion: $(M, g) ≅ (R^{3}, δ)$ , the equality case of the positive-mass theorem. $□$

Group III — Berger holonomy and parallel spinors (3 exercises)

Exercise 9 (medium, proof). Parallel-spinor count on K3 (Calabi-Yau 2-fold).

Compute the dimension of the space of parallel spinors on a K3 surface (Calabi-Yau 2-fold with Yau metric).

Hint

Wang's bijection: $SU (2)$ holonomy gives parallel-spinor count 2.

Answer

K3 has $SU (2)$ holonomy in the Yau metric. By Wang's parallel-spinor count for $SU (n)$ at $n = 2$ : 2 parallel spinors.

Geometrically: the two parallel spinors correspond to the constant function $1$ (the inert summand of the spinor representation under $SU (2)$ ) and the holomorphic 2-form $Ω$ (the top exterior power summand). Both are inert under $SU (2)$ but not under the larger $U (2)$ . Note: $Sp (1) = SU (2)$ , so K3 is also a hyperkähler manifold; the $Sp (1)$ count gives $1 + 1 = 2$ parallel spinors, same as the $SU (2)$ count.

The dimension of the Dirac kernel on K3 is therefore at least $2 \cdot 2 = 4$ (each parallel spinor splits into chiralities). This matches the Atiyah-Singer index $∣ \hat{A} (K 3) ∣ = 2$ for $D^{+}$ (so $dim ker D^{+} = 0, dim ker D^{-} = 2$ in K3 with this orientation), with the parallel spinors contributing.

Exercise 10 (hard, proof). Calabi-Yau manifolds are Ricci-flat.

Show in detail that a compact Kähler manifold with $c_{1} = 0$ admits a metric of zero Ricci tensor.

Hint

This is Yau's theorem (1977), the proof of the Calabi conjecture. Sketch the complex Monge-Ampère equation and the existence of a unique solution.

Answer

Calabi conjectured: on a compact Kähler manifold $(M, ω_{0})$ with $c_{1} (M) = 0$ in $H^{2} (M; R)$ , every Kähler class $[ω_{0}]$ contains a unique Kähler metric $ω$ with $ρ_{ω} = 0$ (vanishing Ricci form).

The Ricci form of a Kähler metric $ω$ is $ρ_{ω} = - i \partial \overset{ˉ}{\partial} lo g det g$ , and the vanishing $ρ_{ω} = 0$ is equivalent to $lo g (det g_{ω} / det g_{ω_{0}})$ being a constant. This reduces to the complex Monge-Ampère equation: find $φ$ with $ω = ω_{0} + i \partial \overset{ˉ}{\partial} φ$ and $$ (\omega_0 + i\partial\bar\partial\varphi)^n = e^F \omega_0^n $$ for a prescribed $F$ (the volume-form ratio fixing the cohomology class).

Yau's 1977 paper proved the existence and uniqueness of $φ \in C^{\infty} (M)$ solving this PDE, by a continuity method combined with $C^{0}, C^{2}, C^{3}$ estimates. The continuity method writes a one-parameter family of equations connecting the inert $F = 0$ case to the general $F$ , and shows the parameter set of solvable cases is open and closed, hence the full interval. The estimates use Calabi's third-order estimate (the Aubin-Yau inequality) and the maximum principle.

The resulting $ω$ is the Calabi-Yau metric: Ricci-flat, Kähler, in the prescribed Kähler class. By Wang 1989, this implies the holonomy reduces to $SU (n)$ , and 2 parallel spinors exist (the constant 1 and the holomorphic volume form $Ω$ ). $□$

Group IV — Calibrated geometries (4 exercises)

Exercise 12 (medium, proof). Wirtinger inequality on $C^{n}$ .

Prove that the Kähler form $ω = \sum_{j} d x^{j} \land d y^{j}$ on $C^{n}$ has comass $1$ , with equality on complex 1-planes.

Hint

For a unit decomposable 2-vector $ξ = v \land w$ with ${v, w}$ orthonormal, $ω (v, w) = g (J v, w)$ . By Cauchy-Schwarz, $∣ g (J v, w) ∣ \leq ∣ J v ∣∣ w ∣ = 1$ , with equality iff $w = \pm J v$ .

Answer

For an oriented orthonormal pair ${v, w}$ in $T_{x} C^{n} = R^{2 n}$ , $ω (v, w) = g (J v, w)$ where $J$ is the complex structure (the Kähler form is by definition $ω (ξ, η) = g (J ξ, η)$ on a Kähler manifold).

By Cauchy-Schwarz applied to the inner product $g$ , $∣ g (J v, w) ∣ \leq ∣ J v ∣ \cdot ∣ w ∣ = 1 \cdot 1 = 1$ . Equality iff $w$ is parallel to $J v$ , i.e., the 2-plane $span {v, w}$ is invariant under $J$ , equivalently a complex 1-plane.

So $ω$ has comass $1$ globally, and the contact set at $x$ is exactly the set of $J$ -invariant 2-planes (complex 1-planes) at $x$ . By the fundamental theorem, complex curves in $C^{n}$ are area-minimisers — the Federer-Fleming result, recovered from the calibration framework.

The same Wirtinger calculation generalises to $ω^{k} / k!$ on Hermitian inner product spaces: the comass is $1$ at each point, with contact set the complex $k$ -planes. Calibrated $2 k$ -submanifolds of a Kähler manifold are exactly the complex $k$ -dimensional submanifolds. $□$

Exercise 14 (hard, proof). G_2 calibration and the cross product.

Show that on $R^{7} = Im O$ with the $G_{2}$ three-form $ϕ$ , the calibrated 3-planes are exactly the imaginary parts of associative subalgebras of $O$ .

Hint

A 3-plane $V \subset Im O$ is associative iff for orthonormal $u, v \in V$ , the cross product $u \times v$ also lies in $V$ . The cross product is defined by $ϕ (u, v, w) = g (u \times v, w)$ .

Answer

The $G_{2}$ three-form $ϕ$ on $Im O = R^{7}$ is defined by $ϕ (u, v, w) = ⟨ u, v \cdot w ⟩$ where $\cdot$ is the octonion product. Equivalently $ϕ (u, v, w) = g (u \times v, w)$ with $u \times v ⊥ u, v$ a unit vector when $u, v$ are orthonormal (the cross-product property in $O$ ).

A 3-plane $V$ is calibrated by $ϕ$ iff for every oriented orthonormal basis ${u, v, w}$ of $V$ , $ϕ (u, v, w) = 1$ exactly. Equivalently, $g (u \times v, w) = 1$ for unit $u \times v$ , which forces $w = u \times v$ (up to sign). So $u \times v \in V$ .

By symmetry, the cross product of any two orthonormal vectors in $V$ lies in $V$ . So $V$ is closed under the cross product. The cross product on $Im O$ is the imaginary part of the octonion product: if $u, v \in Im O$ , then $u \cdot v = - g (u, v) + u \times v$ . Closure of $V$ under the cross product means closure under the octonion product (modulo the real part), which makes $V$ the imaginary part of a 4-dimensional associative subalgebra of $O$ .

The 4-dimensional associative subalgebras of $O$ are exactly the quaternionic subalgebras $H \subset O$ . So calibrated 3-planes in $R^{7}$ are exactly the imaginary parts of quaternionic subalgebras of $O$ , i.e., $Im H$ for various quaternionic embeddings $H ↪ O$ .

The moduli space of these 3-planes is $G_{2} / SO (4)$ , of real dimension $14 - 6 = 8$ , the space of associative 3-planes. Each such 3-plane minimises volume in its homology class by the fundamental theorem of calibrations. $□$

This makes precise the structural content of the name "associative": the calibrated submanifolds are the associative subalgebras (in their imaginary parts).

Group V — Vector-field problem and spinor cohomology (2 exercises)

Exercise 15 (hard, proof). Vector-field / immersion bound on $S^{n}$ via Clifford modules.

Let $ρ (n)$ denote the maximal number of pointwise-linearly-independent vector fields on $S^{n}$ (the Hurwitz-Radon number minus one). Using Clifford modules, derive Adams' formula

ρ (n) + 1 = 8 a + 2^{b}, n + 1 = 2^{4 a + b} \cdot (odd), 0 \leq b \leq 3,

and explain how the same Cl-module count obstructs immersions of $RP^{n}$ into low-dimensional Euclidean space.

Hint

The maximal number of pointwise-linearly-independent vector fields on $S^{n - 1}$ equals the number of orthonormal Clifford generators that can be packed onto $R^{n}$ — that is, the largest $k$ such that $R^{n}$ admits a $Cl_{k}$ -module structure compatible with the standard inner product. Use the real Clifford classification table 03.09.11 and count module dimensions. Adams' 1962 paper Vector fields on spheres (Annals 75) does the count explicitly. For the immersion application, recall that an immersion $RP^{n} ↪ R^{n + k}$ produces a normal bundle whose total Stiefel-Whitney class divides the inverse of $w (T RP^{n}) = (1 + a)^{n + 1}$ where $a \in H^{1} (RP^{n}; F_{2})$ generates; the resulting bound on $k$ is sharpened by spin / Cl-module obstructions.

Answer

Vector field count. The Hurwitz-Radon problem: how many pointwise-orthogonal vector fields can $S^{n - 1} \subset R^{n}$ admit? Equivalently, what is the largest $k = k (n)$ such that $R^{n}$ is a $Cl_{k}$ -module over the reals (with standard Euclidean form)? The answer is encoded in the real Clifford classification table $Cl_{k} ≅ M_{d} (F)$ 03.09.11: writing $n + 1 = 2^{4 a + b} \cdot m$ with $m$ odd and $0 \leq b \leq 3$ , the largest $k$ is

ρ (n) + 1 = 8 a + 2^{b} .

The proof has two halves. Lower bound (construction): explicit orthonormal vector fields are produced by Clifford multiplication. Take a $Cl_{ρ (n)}$ -module structure on $R^{n + 1}$ — guaranteed by the dimension count — and let $e_{1}, \dots, e_{ρ (n)}$ be the Cl-generators. For $x \in S^{n}$ , the vectors $e_{1} \cdot x, \dots, e_{ρ (n)} \cdot x$ are orthonormal to $x$ (by the Clifford relation $e_{i}^{2} = - 1$ and skew-symmetry of $e_{i} \cdot$ on the standard inner product) and pointwise linearly independent. Upper bound (Adams 1962): suppose $S^{n}$ admits $ρ (n) + 1$ orthonormal vector fields. Stiefel-Whitney class arguments combined with the secondary Steenrod operations from Adams' paper show this would force a non-existent K-theory class, contradicting Bott periodicity. The Adams calculation uses the J-homomorphism $J : K O^{- 1} (pt) \to π_{*}^{s}$ and explicit obstruction theory in the Cl-module category.

Immersion application. An immersion $i : RP^{n} ↪ R^{n + k}$ produces a normal bundle $ν$ of rank $k$ with $w (T RP^{n}) \cdot w (ν) = 1$ in $H^{*} (RP^{n}; F_{2}) = F_{2} [a] / (a^{n + 1})$ . Computing $w (T RP^{n}) = (1 + a)^{n + 1}$ pins down $w (ν)$ , and the smallest $k$ for which a rank- $k$ bundle with this Stiefel-Whitney class exists on $RP^{n}$ is bounded below by counting Cl-module dimensions in the geometric dimension of $ν$ . The sharpest bounds come from spin obstructions to the Cl-module structure: $RP^{n}$ immerses in $R^{2 n - α (n)}$ where $α (n)$ is the number of $1$ s in the binary expansion of $n + 1$ , a result of Cohen 1985 The immersion conjecture (Annals 122) that closes the immersion problem completely. The Lawson-Michelsohn perspective (§IV.2) is that the entire vector-field / immersion problem is one Cl-module bookkeeping calculation in two dressings: vector fields on $S^{n}$ count Cl-generators on $R^{n + 1}$ ; immersion codimensions count Cl-modules on the normal-bundle classifying space. $□$

This is the canonical example of Lawson-Michelsohn's framing principle: spin-geometric data — Cl-module classifications, Bott periodicity, J-homomorphism — controls classical geometric questions (vector fields, immersions) that look algebraic-topological at first glance.

Exercise 16 (hard, proof). Spinor cohomology on $c_{1} = 0$ Calabi-Yau manifolds.

Let $M$ be a closed Kähler manifold of complex dimension $n$ with $c_{1} (M) = 0$ (i.e., a Calabi-Yau). Show that the sheaf cohomology of the spinor bundle splits canonically as

H^{*} (M, S^{+}) \oplus H^{*} (M, S^{-}) ≅ p, q ⨁ H^{p, q} (M)

with $S^{+} = ⨁_{p even} Ω^{0, p}$ and $S^{-} = ⨁_{p odd} Ω^{0, p}$ , and conclude that the dimension of harmonic spinors on $M$ equals the alternating sum of Hodge numbers $\sum_{p} (- 1)^{p} h^{0, p} (M)$ , which is the holomorphic Euler characteristic $χ (M, O_{M})$ .

Hint

On a Kähler manifold the canonical $Spin^{c}$ structure has $S^{\pm} ≅ Ω^{0, even/odd}$ tensored with a square root of the canonical bundle 03.09.05. When $c_{1} = 0$ , the canonical bundle $K_{M}$ is topologically zero (and on a simply-connected CY, holomorphically zero), so the canonical $Spin^{c}$ structure refines to a genuine spin structure with $S^{\pm} = Ω^{0, even/odd}$ on the nose. Use $D = 2 (\overset{ˉ}{\partial} + \overset{ˉ}{\partial}^{*})$ to identify harmonic spinors with $\overset{ˉ}{\partial}$ -harmonic forms in $Ω^{0, *}$ , and apply Dolbeault's theorem.

Answer

Step 1: Spin structure from $c_{1} = 0$ . A Kähler manifold $M$ has canonical $Spin^{c}$ structure with determinant line bundle $K_{M}^{- 1}$ [03.09.05, Theorem D.2]. The condition $c_{1} (M) = 0$ means $c_{1} (K_{M}^{- 1}) = 0$ , so $K_{M}^{- 1}$ is topologically zero — it admits a square root, and the $Spin^{c}$ structure refines to a genuine spin structure on $M$ . Under this refinement,

S^{+} = Ω^{0, even} = p even ⨁ Λ^{0, p} T^{*} M, S^{-} = Ω^{0, odd} = p odd ⨁ Λ^{0, p} T^{*} M

as smooth vector bundles. The spin Dirac operator is $D = 2 (\overset{ˉ}{\partial} + \overset{ˉ}{\partial}^{*})$ , mapping $S^{+} \to S^{-}$ on the chiral piece (LM §IV.11).

Step 2: Identification of harmonic spinors with Dolbeault cohomology. A harmonic spinor $ψ \in ker D$ on a closed Kähler manifold satisfies both $\overset{ˉ}{\partial} ψ = 0$ and $\overset{ˉ}{\partial}^{*} ψ = 0$ (because $D^{2} = 2 (\overset{ˉ}{\partial} \overset{ˉ}{\partial}^{*} + \overset{ˉ}{\partial}^{*} \overset{ˉ}{\partial})$ is twice the $\overset{ˉ}{\partial}$ -Laplacian on $Ω^{0, *}$ when $c_{1} = 0$ , so $D ψ = 0$ iff both $\overset{ˉ}{\partial} ψ$ and $\overset{ˉ}{\partial}^{*} ψ$ vanish). The space of harmonic $(0, p)$ -forms is canonically isomorphic to the Dolbeault cohomology $H^{0, p} (M) = H^{p} (M, O_{M})$ by Hodge theory. Combining,

ker D ∣_{S^{+}} ≅ p even ⨁ H^{p} (M, O_{M}), ker D ∣_{S^{-}} ≅ p odd ⨁ H^{p} (M, O_{M}) .

Step 3: Index = holomorphic Euler characteristic.

ind (D^{+}) = dim ker D ∣_{S^{+}} - dim ker D ∣_{S^{-}} = p \sum (- 1)^{p} h^{0, p} (M) = χ (M, O_{M}) .

The Atiyah-Singer index theorem on the spin Dirac operator gives $ind (D^{+}) = A (M) [M]$ , while the Hirzebruch-Riemann-Roch formula on $M$ gives $χ (M, O_{M}) = Td (M) [M]$ . The identity $A (M) = Td (M)$ on a Calabi-Yau (where $c_{1} = 0$ ) is verified by direct expansion: $Td = e^{c_{1} /2} A$ and $c_{1} = 0$ collapses the exponential factor to $1$ . So both routes give the same index, confirming the cohomological identification above.

Specialisations. For $M$ a K3 surface ( $n = 2$ ), $h^{0, 0} = 1$ , $h^{0, 1} = 0$ , $h^{0, 2} = 1$ , so $χ (K 3, O) = 2$ and $K 3$ has two harmonic spinors (the parallel spinors of the Calabi-Yau holonomy 03.09.18). For $M$ a Calabi-Yau threefold ( $n = 3$ ), $h^{0, 0} = 1$ , $h^{0, 3} = 1$ , $h^{0, 1} = h^{0, 2} = 0$ (simply-connected case), so $χ (M, O) = 0$ — but each chiral piece has one harmonic spinor: the constant function on $S^{+}$ and the holomorphic volume form on $S^{-}$ . This count matches the parallel-spinor count for $SU (3)$ holonomy. $□$

The exercise pins down the LM §IV.11.2 statement: on Calabi-Yau manifolds, spinor cohomology equals Dolbeault cohomology, and the harmonic-spinor count equals $χ (O_{M})$ . This is the formula that makes the K3 / Calabi-Yau parallel-spinor counts of 03.09.18 agree with Hodge-number bookkeeping.

Prerequisites

03.09.16
03.09.17
03.09.18
03.09.19

Tier anchors

beginner
intermediate: Lawson-Michelsohn — Spin Geometry Ch. IV (§IV.1–§IV.12) exercise sets
master

References

TODO_REF
Lawson-Michelsohn — Spin Geometry · Ch. IV exercises (§IV.1–§IV.4 psc; §IV.9 holonomy; §IV.10–§IV.11 Witten; §IV.12 calibrations)
TODO_REF
Joyce — Compact Manifolds with Special Holonomy · Oxford University Press, 2000 — chapter exercises on G_2 and Spin(7)
TODO_REF
Harvey-Lawson — Calibrated geometries · Acta Mathematica 148 (1982) — explicit calibration examples
TODO_REF
Gromov-Lawson — Positive scalar curvature and the Dirac operator on complete Riemannian manifolds · Publ. Math. IHÉS 58 (1983) — enlargeable manifold examples and exercises

Reviewer

Tyler — Yellow per PILOT_PLAN §3 (cross-cuts §IV.1–§IV.12 of Lawson-Michelsohn)

Estimated time

intermediate: 5h