Hodge Theory

Hodge numbers

Example: Hodge numbers of a blow-up

Let \(C \subset \mathbb{P}^3\) be the twisted cubic curve, parametrized by \([s,t] \mapsto [s^3, s^2 t, s t^2, t^3]\). As a smooth rational curve, \(C \cong \mathbb{P}^1\). We compute the Hodge diamond of the blow-up \(X = \operatorname{Bl}_C(\mathbb{P}^3)\) over \(\mathbb{C}\).

The blow-up formula for Hodge numbers. For a smooth variety \(X\) of dimension \(d\) blown up along a smooth subvariety \(Z\) of codimension \(c \geq 2\), the Hodge numbers satisfy

\[ h^{p,q}(\operatorname{Bl}_Z(X)) = h^{p,q}(X) + \sum_{i=1}^{c-1} h^{p-i,q-i}(Z). \]

Here \(Z = C \cong \mathbb{P}^1\) has codimension \(c = 2\) in \(\mathbb{P}^3\). The formula gives

\[ h^{p,q}(\operatorname{Bl}_C(\mathbb{P}^3)) = h^{p,q}(\mathbb{P}^3) + h^{p-1,q-1}(\mathbb{P}^1). \]

Hodge numbers of the ingredients. For projective spaces, \(h^{p,p}(\mathbb{P}^n) = 1\) for \(0 \le p \le n\) and all other Hodge numbers vanish. In tabular form:

\((p,q)\)	\(h^{p,q}(\mathbb{P}^3)\)	\(h^{p-1,q-1}(\mathbb{P}^1)\)	\(h^{p,q}(\operatorname{Bl}_C(\mathbb{P}^3))\)
\((0,0)\)	\(1\)	\(0\)	\(1\)
\((1,1)\)	\(1\)	\(1\)	\(2\)
\((2,2)\)	\(1\)	\(1\)	\(2\)
\((3,3)\)	\(1\)	\(0\)	\(1\)
all other	\(0\)	\(0\)	\(0\)

The entry \(h^{p-1,q-1}(\mathbb{P}^1)\) contributes when \((p-1,q-1) \in \{(0,0),(1,1)\}\), i.e., at \((p,q) = (1,1)\) and \((2,2)\). For \(p \neq q\), both summands vanish since \(\mathbb{P}^3\) and \(\mathbb{P}^1\) have \(h^{p,q} = 0\) for \(p \neq q\).

The Hodge diamond of \(\operatorname{Bl}_C(\mathbb{P}^3)\). Assembling the results:

\[ \begin{array}{ccccccc} & & & 1 & & & \\ & & 0 & & 0 & & \\ & 0 & & 2 & & 0 & \\ 0 & & 0 & & 0 & & 0 \\ & 0 & & 2 & & 0 & \\ & & 0 & & 0 & & \\ & & & 1 & & & \end{array} \]

The Betti numbers are \(b_0 = b_6 = 1\), \(b_1 = b_5 = 0\), \(b_2 = b_4 = 2\), \(b_3 = 0\), giving Euler characteristic \(\chi = 1 + 2 + 2 + 1 = 6\).

The exceptional divisor. The exceptional divisor \(E = \mathbb{P}(N_{C/\mathbb{P}^3})\) is the projectivization of the normal bundle of \(C\) in \(\mathbb{P}^3\). For the twisted cubic, the normal bundle is \(N_{C/\mathbb{P}^3} \cong \mathcal{O}_{\mathbb{P}^1}(5) \oplus \mathcal{O}_{\mathbb{P}^1}(1)\), which can be computed from the conormal exact sequence

\[ 0 \to N_{C/\mathbb{P}^3}^{\vee} \to \Omega_{\mathbb{P}^3}|_C \to \Omega_C \to 0 \]

using \(\Omega_C \cong \mathcal{O}_{\mathbb{P}^1}(-2)\) and \(c_1(\Omega_{\mathbb{P}^3}|_C) = -4 \cdot 3 = -12\) (restricting \(\mathcal{O}(-4)\) along the degree 3 embedding), so \(\deg(N_{C/\mathbb{P}^3}^{\vee}) = -12 + 2 = -10\) and \(\deg(N_{C/\mathbb{P}^3}) = 10 - 4 = 6\). The splitting type \(\mathcal{O}(5) \oplus \mathcal{O}(1)\) is determined by the fact that \(H^0(N_{C/\mathbb{P}^3}(-2))\) is nonzero (from the deformation theory of the twisted cubic), forcing the minimum summand degree to be at least 1. Since \(5 + 1 = 6\), this determines the splitting.

The surface \(E = \mathbb{P}(\mathcal{O}(5) \oplus \mathcal{O}(1))\) is the Hirzebruch surface \(\mathbb{F}_4\). All Hirzebruch surfaces have the same Hodge numbers: \(h^{0,0} = h^{2,2} = 1\), \(h^{1,1} = 2\), and all other \(h^{p,q} = 0\).

Verification via \(\chi^{p,q}\)-additivity. The Hodge--Euler numbers \(\chi^{p,q} = (-1)^{p+q} h^{p,q}\) satisfy the additivity relation

\[ \chi^{p,q}(\operatorname{Bl}_Z(X)) = \chi^{p,q}(X) + \chi^{p,q}(E) - \chi^{p,q}(Z) \]

for a blow-up with exceptional divisor \(E\) lying over center \(Z\). Checking at \((p,q) = (1,1)\): \(\chi^{1,1} = h^{1,1}\), and the formula gives \(1 + 2 - 1 = 2\), consistent with the direct computation. At \((p,q) = (2,2)\): \(1 + 1 - 0 = 2\) (since \(\mathbb{P}^1\) has no \((2,2)\)-class but \(\mathbb{P}^3\) contributes \(h^{2,2} = 1\) and \(E \cong \mathbb{F}_4\) contributes \(h^{2,2} = 1\)), also consistent.

Geometric interpretation. The increase of \(h^{1,1}\) from 1 to 2 reflects the new algebraic cycle class \([E] \in H^{1,1}(\operatorname{Bl}_C(\mathbb{P}^3))\) contributed by the exceptional divisor. Since \(C\) is rational, no new odd-dimensional cohomology is introduced, and the Hodge structure remains pure of type \((p,p)\).

Remark: Reference

The computations in this section follow the methods detailed in Donu Arapura's notes, Computations of some Hodge numbers, which provide a systematic treatment of Hodge number calculations for blow-ups, fibrations, and related constructions. In particular, Chapter 17 of those notes covers the additivity of Hodge--Euler characteristics and the blow-up formula for Hodge numbers. See also Voisin, Hodge Theory and Complex Algebraic Geometry I, Section 7.3.3, for the general blow-up formula.