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Hodge Numbers Of Projective Spaces.pdf

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Preview of Hodge Numbers of Projective Spaces
🔗 Source: math.purdue.edu
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👤 Author: Donu Arapura
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Summary

Computation of Some Hodge Numbers
- Hodge Numbers of P^n (Projective Spaces):
- Proposition 17.1.1: Exact sequence for sheaf of differentials: 0 → Ω^1P → OP(-n-1) → OP → 0.
- Corollary 17.1.3: Exact sequence for Ω^pP: 0 → Ω^pP → OP(-p)(n+1-p) → Ω^(p-1)P → 0.
- Proposition 17.1.4: Hodge numbers H^q(P, Ω^pP) = #k if p = q ≤ n, 0 otherwise.
- Theorem 17.1.5 (Bott): Hodge numbers for Ω^pP(r) under certain conditions.
- Hodge Numbers of a Hypersurface:
- Proposition 17.2.1: Restriction map H^q(P^n, Ω^pP) → H^q(X, Ω^pX) is an isomorphism when p+q < n-1.
- Corollary 17.2.2: Hodge numbers hpq(X) = δpq when n-1 ≠ p+q < 2n-2.
- Lemma 17.2.3: Exact sequence for Ω^pP|X: 0 → Ω^pP(-d) → Ω^pP → Ω^pP|X → 0.
- Lemma 17.2.4: Exact sequence for Ω^1X: 0 → OX(-d) → Ω^1P|X → Ω^1_X → 0.

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Description

Computation of Some Hodge Numbers. Hodge theory determines Betti numbers; this chapter focuses on computing Hodge numbers for specific algebraic varieties, leveraging the GAGA theorem. Starting with projective spaces, the sheaf of differentials is determined.

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  • File Format: PDF
  • File Size: 133 KB
  • Pages: 13
  • Language: EN
  • Author: Donu Arapura
  • Total Downloads: 40
  • Last Updated: 7 days ago

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