指标定理（五）

注意：指标定理（5/5）

21

is elliptic. The adjoint of d + d*：ΓE₊ → TE₋ is d + d*：ΓE₋ → ΓE₊ . The analytical index is

indα(d+d*：ΓE₊ → ΓE₋)

＝dimℂker(d＋d*：ΓE₊ → ΓE₋) – dimℂker(d＋ d*：ΓE₋ → ΓE₊)

＝dimℂ(𝓡 ₊²ᵏ) – dimℂ(𝓡 ₋²ᵏ).

＝σ(X)，

where the next-to-last equality is because the components of harmonic forms away from the middle-dimensional cohomology have the same dimension and so cancel. Let me explain this in detail.

We have a decomposition ∧(T* ⨂ ℂ)＝E₊⨁E₋. Note that τ：∧ᵖ(T* ⨂ ℂ) → ∧⁴ᵏ⁻ᵖ(T* ⨂ ℂ). Elements of Γ(⨁²ᵏ⁻¹ ₛ₌₀ (∧ˢ(T* ⨂ ℂ) ⨁∧⁴ᵏ⁻ˢ(T* ⨂ ℂ))∩E₊) are of the form ∑²ᵏ⁻¹ₛ₌₀(αₛ＋ταₛ) where αₛ ∈ Γ(∧ˢ(T* ⨂ ℂ)). For arbitrary α ∈ Γ(∧ˢ(T* ⨂ ℂ)⨁∧⁴ᵏ⁻ˢ(T* ⨂ ℂ))，0 ≤ s＜2k，we have α＝1

──

2

((α＋τα)＋(α – τα))，where τ(α＋τα)＝(α＋τα) and τ(α–τα)＝ –(α–τα).

Thus for all 0 ≤ s＜2k，we have a decomposition

∧ˢ(T* ⨂ ℂ) ⨁ ∧⁴ᵏ⁻ˢ(T* ⨂ ℂ)＝Eˢ₊ ⨁ Eˢ₋

into eigenbundles. If α ∈ Γ(∧ˢ(T* ⨂ ℂ))，0 ≤ s s＜2k satisfies α＋τα ∈ ker(d＋d*)＝ker(Δ)， since Δ preserves degree，we must have α ∈ ker(Δ) and τα ∈ ker(Δ). Thus α – τα ∈ ker(d＋d*)＝ker(Δ). The map α ＋ τα ↦ α – τα and its inverse α – τα ↦ α ＋ τα give us the desired bijective correspondence between ker(d＋d*)∩Γ(Eˢ₊) and ker(d＋d*)∩Γ(Eˢ₋).

Next，we continue to calculate the topological index. Again by splitting we have

ch(E₊) – ch(E₋)＝∏²ᵏⱼ₌₁(eˣʲ – e⁻ˣʲ)，where we split TX ⨂ ℂ＝⨁²ᵏⱼ₌₁(lⱼ ⨁ ˉlⱼ)，and xⱼ＝c₁(lⱼ). Thus TX ≅ ⨁lᵢ implies

₂ₖ

indₜ(d+d*)＝(∏(eˣʲ – e⁻ˣʲ)

ⱼ₌₁

₂ₖ xⱼ

∏ ────────) [X]

ⱼ₌₁ (eˣʲ/² – eˉˣʲ/²)²

₂ₖ xⱼ(eˣʲ/²＋eˉˣʲ/²)

＝(∏ ────────) [X]

ⱼ₌₁ (eˣʲ/² – eˉˣʲ/²)

₂ₖ xⱼ

＝(∏ ───────) [X]

ⱼ₌₁ tanh(xⱼ/2)

1 ₂ₖ 2xⱼ

＝─ (∏ ────) [X]

2²ᵏ ⱼ₌₁ tanh(xⱼ)

₂ₖ xⱼ

＝(∏ ──── ) [X].

ⱼ₌₁ tanh(xⱼ)

Consequently，since p₁(lⱼ)＝ –c₂ (lⱼ ⨂ ℂ)＝ –c₂(lⱼ ⨁ ˉlⱼ)＝ –c₁(lⱼ)c₁(ˉlⱼ)＝c₁(lⱼ)²，

22

L(TX)[X]＝∏ L(1＋p₁(lⱼ))[X]

₂ₖ √p₁(lⱼ)

＝∏ ───── [X]

ⱼ₌₁ tanh(√p₁(lⱼ))

₂ₖ xⱼ

＝(∏ ──── ) [X]＝indₜ(d＋d*).

ⱼ₌₁ tanh(xⱼ)

5 Riemann-Roch Theorem

5.1 Divisors on Riemann surfaces

Recall that a Riemann surface M is a one-dimensional complex manifold，and a diυisor is a mapping D：X → ℤ， such that in ∀compact K ⊂ X，there are only finitely many points where D takes nonzero values. Divisors form an abelian group Diυ(X). One can define a divisor (f) for a meromorphic function f (or a meromorphic one-form) in the obvious way：(f)(P)＝k＞0 if f has a zero of order k at P，and (f)(P)＝k＜0 if f has a pole of order –k at P. We require that f is never identically zero in any open set.

Let .𝓜 be the sheaf of meromorphic functions on M，and 𝓞 be the sheaf of holomorphic functions on M. The presheaf ∪ ↦ 𝓜 (∪)/𝓞 (∪) can be viewed as the presheaf of principle parts, since we say that two meromorphic functions define the same principle part if their difference is holomorphic. Similarly we consider the sheaf. 𝓜 * of meromorphic functions that are never identically zero in any open set，and thus . 𝓜 * (∪) forms an abelian group under multiplication. Also define 𝓞 * to be the sheaf of holomorphic functions that never vanishes， and 𝓞 * (∪) also forms an abelian group under multiplication. The presheaf ∪↦𝓜 *(∪)/𝓞 *(∪) is the presheaf of divisors.

In general，some divisors cannot be represented by a globally defined meromorphic function. We say that two divisors D，E are equiυαlent if they differ by a principle divisor，i.e. D – E＝(f) for some meromorphic function f. Principle divisors form a group Prin(X). If the surface is compact，we can define deg(D)＝∑z∈X D(x). Thus if two divisors are equivalent，they have the same degree. We define a sheaf 𝓞 ᴅ as

𝓞 ᴅ(∪)＝{f ∈ 𝓜 (∪)：(f)(x) ≥ –D(x)}.

If D is represented by a meromorphic function g， then (f)(x) ≥ –D(x) says that fg is holomorphic.

Let P denotes a divisor taking the value 1 at P ∈ X and zero otherwise. We then have an exact sequence

0 → 𝓞 ᴅ → 𝓞 ᴅ₊ᴘ → 𝓞 ᴅ₊ᴘ/𝓞 ᴅ → 0.

𝓞 ᴅ₊ᴘ/𝓞 ᴅ is merely a presheaf. But indeed，the long exact sequence associated to the sequence still holds. For ∪ such that P ∉ ∪，𝓞 ᴅ₊ᴘ/𝓞 ᴅ(∪) ＝0. For ∪ such that P ∈ ∪. 𝓞 ᴅ₊ᴘ/𝓞 ᴅ(∪) is one-dimensional. H⁰ (M，𝓞 ᴅ₊ᴘ/𝓞 ᴅ)＝ℂ since it is equal to the space of global sections, and now we compute H¹(M，𝓞 ᴅ₊ᴘ/𝓞 ᴅ).

Suppose ξ ∈ H¹(M，𝓞 ᴅ₊ᴘ/𝓞 ᴅ) is represented by ф ∈ Z¹(𝓤 ，𝓞 ᴅ₊ᴘ/𝓞 ᴅ). The cover 𝓤 has a refinement 𝓥 such that P is covered by only one open set Vᵢ in 𝓥 . Thus 0＝δф(i，i，i)＝ф(i，i) –ф(i，i)＋ф(i，i)＝ф(i，i)，and ф＝0. Consequently H¹(M，𝓞 ᴅ₊ᴘ/𝓞 ᴅ)＝0.

23

Consider the long exact sequence

0 → H⁰(M，𝓞 ᴅ) → H⁰(M，𝓞 ᴅ₊ᴘ) → H⁰(M，𝓞 ᴅ₊ᴘ/𝓞 ᴅ)＝ℂ

→ H¹(M，𝓞 ᴅ) → H¹(M，𝓞 ᴅ₊ᴘ) → H¹(M，𝓞 ᴅ₊ᴘ/𝓞 ᴅ)＝0.

It shows that H*(M，𝓞 ᴅ) is finite dimensional iff H*(M，𝓞 ᴅ₊ᴘ) is. Suppose this is the case， and now the alternating sum of the dimensions in the exact sequence is zero. That is to say，if χ(H*(M，𝓞 ᴅ))：＝dim(H⁰(M，𝓞 ᴅ)) – dim(H¹(M，𝓞 ᴅ))，we have

χ(H*(M，𝓞 ᴅ)) – χ(H*(M，𝓞 ᴅ₊ᴘ))＋1=0，

i.e.

χ(H*(M，𝓞 ᴅ)) – deg(D)＝χ(H*(M，𝓞 ᴅ₊ᴘ)) – deg(D＋P).

In particular for a compact Riemann surface， χ(H*(M，𝓞 ᴅ)) – deg(D) should be a con-stant. Recall that the genus g is defined to be dimH¹(X，𝓞 ) where 𝓞 ＝𝓞 ₀ is the sheaf of holomorphic functions，Taking D to be zero we find the constant is χ(H*(M，𝓞 ᴅ))＝1–g.Thus we have proved that

Theorem 13 Suppose D is α diυisor on α compαct Riemαnn surfαce X of genus g. Then H⁰(X，𝓞 ᴅ) αnd H¹(X，𝓞 ᴅ) αre finite dimensionαl υector spαces αnd

dimH⁰(X，𝓞 ᴅ) – dimH¹(X，𝓞 ᴅ)＝1 – 9 ＋degD.

H*(X，𝓞 ᴅ) is finite dimensional because by compactness it has a finite good covering， and sheaf cobomology coincides with ˉCech cohomology，which is finite dimensional.

5.2 Divisors and line bundles

We briefly explain how divisors and line bundles are related. We have an exact sequence

0 → 𝓞 * → 𝓜 * → 𝓜 */𝓞 * →0，

which induces

· · · → H⁰(M，𝓜 *) → H⁰(M，𝓜 */𝓞 *) → H¹(M，𝓞 *) → . . . .

Set δ：H⁰(M，𝓜 */𝓞 *) → H¹(M，𝓞 *).Suppose divisor D ∈ H⁰(M，𝓜 */𝓞 *) is represent-ed by (αᵢ)，where αᵢ is defined on ∪ᵢ in an open cover 𝓤 . Then αᵢ/αⱼ is holomorphic，and δ：(αᵢ) ↦ (αᵢ/αⱼ) ∈ H¹(M，𝓞 *). (αᵢ/αⱼ) satisfies the cocycle condition and defines a holomorphic line bundle L. One verifies that this is well-defined. This is a group homo-morphism，since addition of divisors corresponds to multiplication of (αᵢ)，(αᵢ/αⱼ)，and corresponds to tensor product of holomorphic line bundles. The image of H⁰(M，𝓜 *) → H⁰(M，𝓜 */𝓞 *) is by definition Prin(M). However，H⁰(M，𝓜 */𝓞 *) → H¹(M，𝓞 *) is not surjective in the most general case.

Now we briefly recall Dolbeault cohomology for a holomorphie vector bundle. Given a holomorphic vector bundle π：E → X，for fixed p define

＿＿

∧ᵖ，q(E)＝∧ᵖT* ⨂ ∧qT* ⨂ E，Aᵖ，q(X，E)：

＿＿

＝Γ(∧ᵖT* ⨂ ∧qT* ⨂ E).

Define

ˉ∂ᴇ：Aᵖ，q(X，E) → Aᵖ，q⁺¹(X，E)，α ⨂ s ↦ ˉ∂α ⨂ s.

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This is independent of trivializations chosen for E because transition functions of E are holomorphic. Thus

0 → Aᵖ，⁰(X，E) → Aᵖ，¹(X，E) → · · · → Aᵖ，q(X，E)→ · · ·

form a complex called Dolbeault complex. Its cohomology is defined to be the Dolbeault cohomology Hᵖ，q(X，E).

Moreover，considering the sheaves of sections Aᵖ，q(E) we find an exact sequence of sheaves

0 → Ωᵖ(E) → Aᵖ，⁰(E) → Aᵖ，¹ (E) → · · · → Aᵖ，q(E)→ · · ·，

where Ωᵖ(E) is the sheaf of E-valued holomorphic p-forms. Aᵖ，q(E)’s are all fine sheaves，and they form a fine resolution of Ωᵖ(E). Taking global sections we notice that the Dolbeault cohomology group Hᵖ，q(X，E) is exactly the sheaf cohomology Hq(X，Ωᵖ(E))of Ωᵖ(E)，which is also equal to the ˉCech cohomology.

We define the Euler-Poincαré chαrαcteristic to be

dim ℂ X

χ(X，E)∑＝(–1)ⁱdimℂHⁱ(X，Ω⁰(E)).

ᵢ₌₀

A more general version of Riemann-Roch theorem says that，for a holomorphic vector bundle E on a compact curve X，

χ(X，E)＝deg(E)＋rαnk(E)(1 – g(X)).

Now let’s turn to Hirzebruch-Riemann-Roch theorem.

5.3 Hirzebruch-Riemann-Roch theorem

Theorem 14 Let E be α holomorphic υector bundle on α compαct complex mαnifold X. Then its Euler-Poincαré chαrαcteristic is giυen by

χ(X，E)＝∫᙮ ch(E)td(X).

Now we illustrate why this generalizes the original formula.

It is a fact that for compact curves，Diυ → Pic is surjective. Also，the degree of a principal divisor on a compact curve is always zero.

The degree of a holomorphic vector bundle L over a curve C is thus defined to be the degree of a divisor corresponding to it.

There’s another fact that ∫ᴄc₁(L)＝deg(L). This is because the first Chern class of a line bundle Lᴅ associated to a divisor D is the Poincaré dual of the divisor，i.e.

i

∫᙮ ─ Ω∇∧α＝∫ᴅ

2π

α for all closed real form α，

which holds for divisors on general complex manifolds X.

We always assume Riemann surfaces to be connected. Let C be a compact Riemann surface，and L ∈ Pic(C). Then the formula tells us that χ(C，L)＝∫ᴄ(1＋c₁(L)＋ . . . )

c₁(C)

(1＋── ＋ . . . )

2 c₁(C)

＝∫ᴄ c₁(L)＋──

2

deg(K*ᴄ)

＝deg(L)＋────.

2

25

Here K*ᴄ is isomorphic to the holomorphic

tangent bundle. Thus χ(C，𝓞 )

deg(K*ᴄ)

＝───

2 ＝h⁰(C，𝓞 ) – h¹(C，𝓞 )＝1 – g. This gives us the previous result.

6 Further Developments

There are various generalizations and applications. We only mention some of them.

Originally Atiyah-Singer index theorem was proved using K-theory.

Years later，Ativah，Raoul Bott，and Vijay Patodi (1973) gave a new proof of the index theorem using the heat kernel. Due to (Teleman 1983)，(Teleman 1984)，the theorem is generalized to any abstract elliptic operator (Atiyah 1970) on a closed，oriented，topological manifold. Later，Connes-Donaldson-Sullivan-Teleman index theorem arises due to (Donaldson and Sullivan 1989)， (Connes, Sullivan and Teleman 1994). Also，we have equivariant index theorem，and index theorem for families of elliptic operators.²

The index of Dirac operators was used to formulate and then prove the Gromov-Lawson conjecture： A compact，spin，simply connected manifold of dimension less than or equal to five admits a metric of positive scalar curvature iff the index of the spin Dirac operator is zero. ³

Also，the theory has found applications in physics. In more recent years，there is a theory named nonabelian gauge field theory of C. N. Yang and R.L. Mills which has led to astonishing results in dimension four.

Yang-Mills theory can be plausibly considered a generalization of Dirac’s theory which encompasses three fundamental forces.The theory of connections， Dirac-type operators，and index theory all play an important role.

References

[1] 苏竞存，流形的拓扑学，武汉大学出版社，1992.

[2] Raoul Bott and Loring Tu. Differentiαl Forms in Algebrαic Topology. Springer-Verlag New York，1982.

[3] Otto Forster，Lectures on Riemαnn Surfαces. Springer-Verlag New York，1981.

[4] Allen Hatcher. Vector Bundles αnd K-Theory.

[5] Allen Hatcher. Algebrαic Topology. Cambridge ∪niversity Press，2002.

[6] Friedrich Hirzebruch，Thomas Berger， Rainer Jung，and Peter Landweber. Mαnifolds αnd Modulαr Forms.Springer Fachmedien Wiesbaden，1994.

[7] Daniel Huybrechts. Complex Geometry An Introduction. Springer-Verlag Berlin Heidelberg，2005.

[8]H. Lawson and M.-L Michelsohn. Spin Geometry. Princeton ∪niversity Press，1989.

[9] N.Steenrod. The Topology of Fibre Bundles. Princeton ∪niversity Press，1951.

[10] Loring W. Tu. Differentiαl Geometry Connections，Curυαture，αnd Chαrαcteristic

Clαsses. Springer International Publishing AG，2017.

[11] Wilderich Tuschmann and David Wraith. Moduli Spαces of Riemαnniαn Metrics.

Springer Basel，2015.

[12] Milnor W and Stasheff D. Chαrαcteristic Clαsses. Princeton ∪niversity Press，1974.

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