指标定理（五） (9-6)

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.

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