Hⁱ(X,F) derived functors of Γ. Cech cohomology. Serre vanishing: ample L ⇒ Hⁱ(X,F⊗Lⁿ)=0 for i>0, n≫0. Riemann–Roch on curves.
graph TD
D1["Def: Hⁱ(X,F)\nderived of Γ"]
D2["Def: Ample line bundle\nLⁿ very ample n≫0"]
D3["Def: Degree deg D\ncurve divisors"]
D4["Def: Genus g\nh¹O_X"]
T1["Thm: Long exact\n0→Γ→H⁰→H¹→..."]
T2["Thm: Serre vanishing\n ample L, i>0 ⇒ HⁱF⊗Lⁿ=0"]
T3["Thm: Serre duality\nHⁱ ≅ Hⁿ⁻ⁱ(K⊗·)*"]
T4["Thm: Riemann–Roch\nχ(L)=deg L + 1 − g"]
T5["Thm: Kodaira vanishing\nK ample, char 0"]
L1["Lemma: Čech = derived"]
D1 --> T1
D2 --> T2
D3 --> T4
D4 --> T4
D1 --> L1
T2 --> T5
T3 --> T4
classDef definition fill:#b197fc,color:#fff
classDef theorem fill:#51cf66,color:#fff
classDef lemma fill:#74c0fc,color:#fff
class D1,D2,D3,D4 definition
class T1,T2,T3,T4,T5 theorem
class L1 lemma
Process Statistics
- Nodes: 15
- Edges: 12
- Definitions: 4
- Theorems: 5
- Lemmas: 1
Frontier: Derived categories, intersection cohomology, D-modules. math.AG