On a complex manifold $X$, the exponential sequence $0 \to \underline{2\pi i \mathbb{Z}} \to \mathcal{O} \xrightarrow{\exp} \mathcal{O}^* \to 0$ gives:

$H^1(X, \mathcal{O}) \to H^1(X, \mathcal{O}^*) \xrightarrow{c_1} H^2(X, \mathbb{Z})$

Here $H^1(X, \mathcal{O}^*) \cong \operatorname{Pic}(X)$ classifies line bundles. So $H^1$ classifies geometric objects (line bundles, extensions, deformations).

The most fundamental computation:

• $H^0(\mathbb{P}^n, \mathcal{O}(d)) = k[x_0, \ldots, x_n]_d$ (degree-$d$ homogeneous polynomials) for $d \geq 0$
• $H^0(\mathbb{P}^n, \mathcal{O}(d)) = 0$ for $d < 0$

with $\dim H^0(\mathbb{P}^n, \mathcal{O}(d)) = \binom{n+d}{d}$ for $d \geq 0$.

Special cases:

• $H^0(\mathbb{P}^n, \mathcal{O}) = k$: the only global regular functions are constants.
• $H^0(\mathbb{P}^1, \mathcal{O}(2)) = k \cdot x^2 + k \cdot xy + k \cdot y^2$: $3$-dimensional (conics on $\mathbb{P}^1$).
• $H^0(\mathbb{P}^2, \mathcal{O}(3))$: $10$-dimensional (cubics in $\mathbb{P}^2$, i.e., elliptic curves and their degenerations).

The full computation (Theorem III.5.1 in Hartshorne):

• $H^0(\mathbb{P}^n, \mathcal{O}(d)) = k[x_0,\ldots,x_n]_d$ for $i = 0$, $d \geq 0$
• $H^i(\mathbb{P}^n, \mathcal{O}(d)) = 0$ for $0 < i < n$, all $d$
• $H^n(\mathbb{P}^n, \mathcal{O}(d)) = k[x_0^{-1},\ldots,x_n^{-1}]_{-d-n-1}$ for $d \leq -(n+1)$
• $H^i(\mathbb{P}^n, \mathcal{O}(d)) = 0$ otherwise

Key features:

• Gap: $H^i = 0$ for $0 < i < n$ (all twists). Only $H^0$ and $H^n$ can be nonzero.
• Serre duality: $H^n(\mathbb{P}^n, \mathcal{O}(d)) \cong H^0(\mathbb{P}^n, \mathcal{O}(-d-n-1))^\vee$ — the $H^n$ is dual to $H^0$ of the "opposite" twist.
• $\chi(\mathbb{P}^n, \mathcal{O}(d)) = \binom{n+d}{n}$ (the Hilbert polynomial).

On $\mathbb{P}^1$:

| $d$ | $h^0(\mathcal{O}(d))$ | $h^1(\mathcal{O}(d))$ | $\chi(\mathcal{O}(d))$ | |-----|------------------------|------------------------|------------------------| | $\leq -2$ | $0$ | $-d-1$ | $d + 1$ | | $-1$ | $0$ | $0$ | $0$ | | $0$ | $1$ | $0$ | $1$ | | $1$ | $2$ | $0$ | $2$ | | $d \geq 0$ | $d+1$ | $0$ | $d+1$ |

Note: $h^0 - h^1 = d + 1 = \chi$ always (Riemann–Roch on $\mathbb{P}^1$), and Serre duality gives $h^1(\mathcal{O}(d)) = h^0(\mathcal{O}(-d-2))$.

For a smooth projective curve $C$ of genus $g$:

• $H^0(C, \mathcal{O}_C) = k$ (connected).
• $H^1(C, \mathcal{O}_C) = k^g$ ($g$-dimensional).
• $H^i(C, \mathcal{O}_C) = 0$ for $i \geq 2$ ($\dim C = 1$).

The genus $g = h^1(\mathcal{O}_C)$ is the most basic invariant of a curve. It determines the topology ($g$ = number of "holes"):

• $g = 0$: $\mathbb{P}^1$ (rational curve, sphere).
• $g = 1$: elliptic curve (torus).
• $g \geq 2$: hyperbolic curves (Faltings: finitely many rational points over $\mathbb{Q}$).

For a line bundle $\mathcal{L}$ of degree $d$ on a genus-$g$ curve $C$:

• $d < 0$: $H^0(C, \mathcal{L}) = 0$ (no nonzero sections of negative degree).
• $d > 2g - 2$: $H^1(C, \mathcal{L}) = 0$ (by Serre duality + degree argument).
• $d = 2g - 2$: $\mathcal{L}$ could be the canonical bundle $\omega_C$; $h^0(\omega_C) = g$.
• Riemann–Roch: $h^0(\mathcal{L}) - h^1(\mathcal{L}) = d - g + 1$ always.

The "interesting range" is $0 \leq d \leq 2g - 2$, where both $H^0$ and $H^1$ can be nonzero.

Let $E$ be an elliptic curve ($g = 1$) over $k$. Then:

• $h^0(\mathcal{O}_E) = 1$, $h^1(\mathcal{O}_E) = 1$, so $\chi(\mathcal{O}_E) = 0$.
• $\omega_E \cong \mathcal{O}_E$ (trivial canonical bundle — this characterizes elliptic curves among genus-1 curves).
• For a degree-$d$ line bundle $\mathcal{L}$: $h^0(\mathcal{L}) = d$ for $d > 0$, $h^0(\mathcal{L}) = 0$ for $d < 0$, and $h^0(\mathcal{L}) = 0$ or $1$ for $d = 0$.

The Riemann–Roch space $H^0(E, \mathcal{O}(nP))$ has dimension $n$ for $n \geq 1$. Taking $n = 2$: a basis $\{1, x\}$; $n = 3$: a basis $\{1, x, y\}$. The relation in degree 6 gives the Weierstrass equation $y^2 = x^3 + ax + b$.

| Sheaf | $P(d)$ | Degree | Genus | |-------|--------|--------|-------| | $\mathcal{O}_{\mathbb{P}^n}$ | $\binom{n+d}{n}$ | $n$ | — | | $\mathcal{O}_{\mathbb{P}^1}$ | $d + 1$ | $1$ | $0$ | | $\mathcal{O}_C$ ($C \subseteq \mathbb{P}^2$, deg $e$) | $ed - \frac{e(e-3)}{2}$ | $1$ | $\frac{(e-1)(e-2)}{2}$ | | $\mathcal{O}_p$ (point) | $1$ | $0$ | — | | $\mathcal{O}_S$ ($S \subseteq \mathbb{P}^3$, deg $e$) | $\frac{e}{2}d^2 + \cdots$ | $2$ | — |

The Hilbert polynomial determines the Hilbert scheme: varieties with a given Hilbert polynomial form a projective scheme $\operatorname{Hilb}_P(\mathbb{P}^n)$.

On $\mathbb{P}^2$, for any coherent sheaf $\mathcal{F}$: $H^i(\mathcal{F}(d)) = 0$ for $i > 0$ and $d \gg 0$.

Concretely for $\mathcal{O}_C$ where $C$ is a degree-3 curve: $H^1(\mathcal{O}_C(d)) = 0$ for $d \geq 1$ (by Serre duality: $H^1(\mathcal{O}_C(d)) \cong H^0(\omega_C(-d))^\vee = H^0(\mathcal{O}_C(-d))^\vee = 0$ for $d > 0$).

On $\mathbb{P}^n$ with $\mathcal{L} = \mathcal{O}(d)$, $d > 0$: $\omega_{\mathbb{P}^n} \otimes \mathcal{O}(d) = \mathcal{O}(d - n - 1)$. Kodaira gives $H^i(\mathcal{O}(d-n-1)) = 0$ for $i > 0$, i.e., $H^i(\mathcal{O}(m)) = 0$ for $i > 0$ and $m > -(n+1)$. This reproduces part of the computation of $H^*(\mathbb{P}^n, \mathcal{O}(d))$.

Warning: Kodaira vanishing fails in characteristic $p > 0$ (Raynaud's counterexample).

For a flat projective morphism $f: X \to T$ and a coherent sheaf $\mathcal{F}$ flat over $T$, the function $t \mapsto h^i(X_t, \mathcal{F}_t)$ is upper semicontinuous. It is locally constant if and only if $R^i f_* \mathcal{F}$ is locally free.

Example: a family of elliptic curves $f: \mathcal{E} \to T$. Then $R^1 f_* \mathcal{O}_\mathcal{E}$ is a line bundle on $T$ (the Hodge bundle), since $h^1(\mathcal{O}_{E_t}) = 1$ for all $t$.