Cohomology of Projective Space

This post was machine-translated from the Korean original by Marvin (via Kimi). It may contain errors or awkward phrasing — the Korean original is the source of truth.

We previously defined the line bundle \(\mathcal{O}(d)\) in §Line Bundles and Vector Bundles, ⁋Example 12, and verified through the computation immediately following §Line Bundles and Vector Bundles, ⁋Example 16 that its global sections \(H^0(\mathbb{P}^n, \mathcal{O}(d))\) are isomorphic to homogeneous polynomials of degree \(d\). However, the sheaf cohomology introduced in our previous post, §Sheaf Cohomology, ⁋Definition 1, is a richer invariant that includes not only \(H^0\) but also the higher cohomology groups \(H^1, H^2, \ldots\). Thus, we shall now extract all information about \(\mathcal{O}(d)\) using not only \(H^0\) but also the higher cohomology groups.

Bott’s Formula

Since \(\mathcal{O}(d)\) is a line bundle, it is a quasi-coherent sheaf, and therefore to compute its sheaf cohomology it suffices to use Čech cohomology with the standard affine cover \(\mathcal{U}=\{U_0,\ldots, U_n\}\). The result of this computation is as follows.

Proposition 1 (Bott) The cohomology of the line bundle \(\mathcal{O}(d)\) on \(\mathbb{P}^n\) is given by

\[H^q(\mathbb{P}^n, \mathcal{O}(d)) = \begin{cases} \mathbb{K}[\x_0, \ldots, \x_n]_d & q = 0, d \geq 0 \\ \mathbb{K}[\x_0^{-1}, \ldots, \x_n^{-1}]_{-d-n-1} & q = n, d \leq -n-1 \\ 0 & \text{otherwise} \end{cases}\]

Proof

As explained above, we use Čech cohomology. First, recall that on each open set the sections \(\mathcal{O}(d)(U_i)\) are

\[\x_i^d \cdot \mathbb{K}[\x_0/\x_i, \ldots, \widehat{\x_i/\x_i}, \ldots, \x_n/\x_i]\]

(§Line Bundles and Vector Bundles, ⁋Example 12). Then a Čech cochain \(f \in \check{C}^p(\mathcal{U}, \mathcal{O}(d))\) assigns to each \((p+1)\)-tuple \((i_0, \ldots, i_p)\) a regular section over the open set \(U_{i_0}\cap\cdots\cap U_{i_p}\). For a section to be regular on the intersection \(U_{i_0}\cap\cdots\cap U_{i_p}\), only those coordinates that do not vanish (namely \(\x_{i_0}, \ldots, \x_{i_p}\)) may appear in the denominator; the remaining coordinates may not. Thus the sections are generated by monomials of the form

\[f_{i_0 \cdots i_p} = \x_0^{a_0} \cdots \x_n^{a_n},\qquad \sum_{j=0}^n a_j=d,\quad a_j\geq 0\text{ for $j\not\in \{i_0, \ldots, i_p\}$}.\]

For the coboundary map \(\delta : \check{C}^p \rightarrow \check{C}^{p+1}\), we have

\[(\delta f)_{i_0 \cdots i_{p+1}} = \sum_{k=0}^{p+1} (-1)^k f_{i_0 \cdots \hat{i_k} \cdots i_{p+1}}.\]

That is, we take the alternating sum of the sections corresponding to \(p\)-tuples obtained by omitting one index from each \((p+1)\)-tuple.

Now let us use this data to compute the cohomology groups. Beginning with the case of \(\mathbb{P}^1\), the Čech complex is

\[0 \longrightarrow \check{C}^0\overset{\delta}{\longrightarrow}\check{C}^1\longrightarrow 0,\]

where

\[\check{C}^0=\mathcal{O}(d)(U_0)\oplus \mathcal{O}(d)(U_1),\qquad \check{C}^1=\mathcal{O}(d)(U_0\cap U_1),\]

and the respective section spaces are

\[\mathcal{O}(d)(U_0) = \x_0^d \cdot \mathbb{K}[\x_1/\x_0], \qquad \mathcal{O}(d)(U_1) = \x_1^d \cdot \mathbb{K}[\x_0/\x_1], \qquad \mathcal{O}(d)(U_0 \cap U_1) = \mathbb{K}[\x_0^{\pm 1}, \x_1^{\pm 1}]_d.\]

First, to compute the cohomology in \(\check{C}^0\), let us analyze \(\ker\delta\). Since \(H^0(\mathbb{P}^n, \mathcal{O}(d))=\Gamma(\mathbb{P}^n, \mathcal{O}(d))\), this is in fact nothing more than re-verifying the computation immediately following §Line Bundles and Vector Bundles, ⁋Example 16; however, rather than treating it as a separate example, we shall carry out the Čech cohomology computation here in this proof.

By definition, a cochain \((f_0, f_1) \in \check{C}^0\) lies in \(\ker \delta\) precisely when \(f_0 = f_1\) holds in \(\mathcal{O}(d)(U_0 \cap U_1)\). Looking first at the \(U_0\) part, we know that any monomial belonging to \(\mathcal{O}(d)(U_0)\) must be of the form \(\x_0^{d-a}\x_1^a\) for some \(a\geq 0\). Similarly, any monomial belonging to \(\mathcal{O}(d)(U_1)\) must be of the form \(\x_0^b\x_1^{d-b}\) for some \(b\geq 0\). Now for a given cocycle \((f_0,f_1)\) to lie in \(\ker\delta\), we must have \(f_0=f_1\), and hence only monomials satisfying \(a+b=d\) can belong to \(\ker\delta\). That is, the monomials

\[\x_0^d, \quad\x_0^{d-1}\x_1,\quad\ldots, \quad\x_0\x_1^{d-1},\quad \x_1^d\]

form a basis of \(H^0\), which gives the desired result. If \(d<0\), then \(a,b\geq 0\) cannot satisfy this equation, so \(H^0\) becomes \(0\).

Now let us compute \(H^1\). We must compute \(\coker\delta\). From the computation above, we know that the image of \(\delta\) consists of elements of the form

\[f_1-f_0=\sum_{i\geq 0}a_i \x_0^{d-i}\x_1^i-\sum_{j\geq 0}b_j\x_0^j\x_1^{d-j}\tag{$\ast$}\]

for suitable constants \(a_i,b_j\). On the other hand, \(\check{C}^1=\mathcal{O}(d)(U_0\cap U_1)=\mathbb{K}[\x_0^{\pm 1}, \x_1^{\pm 1}]_d\) is an infinite-dimensional space with basis consisting of degree-\(d\) monomials with no restriction on the exponents,

\[\x_0^a\x_1^{d-a},\qquad a\in\mathbb{Z}\tag{$\ast\ast$}\]

The two sums in (\(\ast\)) give respectively the monomials with \(a\leq d\) and those with \(a\geq 0\), so the image of \(\delta\) is the subspace generated by those monomials in (\(\ast\ast\)) for which \(a\geq 0\) or \(d-a\geq 0\). Hence a basis for \(\coker\delta\) consists of those monomials for which both exponents are negative, that is, those corresponding to \(a\) satisfying \(d+1\leq a\leq -1\). If \(d\geq -1\), no such \(a\) exists and \(\coker\delta=0\); if \(d\leq -2\), then the \(-d-1\) elements

\[\x_0^{-1}\x_1^{d+1}, \quad \x_0^{-2}\x_1^{d+2},\quad\ldots,\quad \x_0^{d+1}\x_1^{-1}\]

form a basis of \(\coker \delta\). We shall explain the notation in the statement separately after the proof is complete.

We now finish the proof for general \(n\). In the \(\mathbb{P}^1\) computation above, we examined for each monomial whether it lay in the kernel or the image and drew a conclusion; this is no accident. Since the coboundary map sends monomials to monomials anyway, the entire Čech complex decomposes as a direct sum of subcomplexes, one for each Laurent monomial \(\x^a=\x_0^{a_0}\cdots\x_n^{a_n}\) (\(a\in\mathbb{Z}^{n+1}\), \(\sum_ja_j=d\)), and therefore computing the subcomplex for each individual \(a\) yields all \(q\) at once.

Fix a multi-index \(a\), and partition the indices into two sets according to the sign of the exponents: \(N_{<0}(a)=\{j\mid a_j<0\}\) and \(N_{\geq 0}(a)=\{j\mid a_j\geq 0\}\). By the monomial condition seen above, the condition that \(\x^a\) is regular on \(U_{i_0}\cap\cdots\cap U_{i_p}\) is equivalent to

\[N_{<0}(a)\subseteq\{i_0,\ldots, i_p\}.\]

Hence the \(p\)-th term of the subcomplex corresponding to \(a\) is the vector space spanned by those \((p+1)\)-element index sets \(I=\{i_0,\ldots, i_p\}\) containing \(N_{<0}(a)\), and its differential is the alternating sum over index sets obtained by removing one index from \(I\). Here, since \(I\supseteq N_{<0}(a)\), we can write \(I=N_{<0}(a)\sqcup J\) uniquely, where \(J\) is a subset of \(N_{\geq 0}(a)\), and \(\lvert I\rvert=p+1\) is the same as \(\lvert J\rvert=p+1-\lvert N_{<0}(a)\rvert\). That is, the \(N_{<0}(a)\) part of \(I\) is forced and only \(J\) is free, so this subcomplex is the complex built from subsets of \(N_{\geq 0}(a)\),

\[K^q=\bigoplus_{\substack{J\subseteq N_{\geq 0}(a) \\ \lvert J\rvert=q}}\mathbb{K}\cdot e_J, \qquad \delta(e_J)=\sum_{v\in N_{\geq 0}(a)\setminus J}\pm e_{J\cup\{v\}},\]

shifted in degree by \(\lvert N_{<0}(a)\rvert-1\).

For this complex, observe that if \(N_{\geq 0}(a)\) is nonempty then \(K^\bullet\) is exact. Indeed, fixing \(v\in N_{\geq 0}(a)\) and setting

\[h(e_J)=\begin{cases} e_{J\setminus\{v\}} & v\in J \\ 0 & v\notin J\end{cases}\]

we have \(\delta h+h\delta=\mathrm{id}\) with suitable sign choices. Then the cohomology of the complex with \(K^0\) removed,

\[0 \rightarrow K^1 \rightarrow K^2 \rightarrow \cdots \rightarrow 0,\]

is \(\mathbb{K}\) at \(q=1\) because

\[\ker(K^1 \rightarrow K^2)=\im(K^0 \rightarrow K^1)\cong K^0=\mathbb{K},\]

and \(0\) in all other degrees. That is, if \(N_{\geq 0}(a)\) is nonempty and the above subcomplex is the whole \(K^\bullet\), then the cohomology vanishes in every degree; therefore, for something to remain, one of these two conditions must fail. As we saw above, the latter condition concerns whether the term corresponding to the empty set \(J=\emptyset\), namely \(K^0\), is present: if \(N_{<0}(a)\neq\emptyset\), then \(I=N_{<0}(a)\) itself satisfies the condition at \(p=\lvert N_{<0}(a)\rvert-1\geq 0\), so that term is included, whereas if \(N_{<0}(a)=\emptyset\), then \(I\) cannot be empty and it is omitted.

The conclusion now falls into three cases according to \(N_{<0}(a)\). First, if \(N_{<0}(a)=\emptyset\), that is, all \(a_j\geq 0\), then as seen above \(N_{\geq 0}(a)\) is the whole set and \(K^0\) is omitted, so the cohomology is \(\mathbb{K}\) only at \(q=1\), i.e. \(p=0\), and such \(\x^a\) form a basis of \(H^0(\mathbb{P}^n, \mathcal{O}(d))\). These are the degree-\(d\) monomials with all exponents nonnegative, so they exist only when \(d\geq 0\) and give \(\mathbb{K}[\x_0,\ldots, \x_n]_d\). Next, if \(N_{<0}(a)=\{0,\ldots, n\}\), that is, all \(a_j<0\), then \(N_{\geq 0}(a)=\emptyset\), so the only index set \(I\) satisfying the condition is the whole set, and the complex consists of a single \(\mathbb{K}\) at \(p=n\); such \(\x^a\) form a basis of \(H^n(\mathbb{P}^n, \mathcal{O}(d))\). This time all exponents are at most \(-1\), so they exist only when their sum is at most \(-n-1\), i.e. when \(d\leq -n-1\). Finally, if \(N_{<0}(a)\) is neither empty nor the whole set, then \(N_{\geq 0}(a)\) is nonempty and \(K^0\) is included, so \(K^\bullet\) itself is exact and contributes nothing in any degree. That is, the cohomology vanishes for \(0<q<n\).

In the proof above, we showed that for each variable \(\x_0,\cdots, \x_n\) and for \(d\leq -n-1\), the group \(H^n(\mathbb{P}^n, \mathcal{O}(d))\) is generated by the monomials

\[\x_0^{a_0} \cdots \x_n^{a_n},\qquad a_i \leq -1, \quad \sum a_i=d.\]

(Note that \(d\) is negative.) We may think of this as the space generated by the expressions

\[\y_0^{\lvert a_0\rvert}\cdots \y_n^{\lvert a_n\rvert}\qquad \lvert a_i\rvert\geq 1,\quad \sum \lvert a_i\rvert=\lvert d\rvert\]

where we introduce new variables \(\y_i=\x_i^{-1}\). Here, since each \(a_i\) and \(d\) are negative, we have \(\lvert a_i\rvert=-a_i\) and \(\lvert d\rvert=-d\). This space is almost like the space of homogeneous polynomials of degree \(\lvert d\rvert\), except that none of the \(\lvert a_i\rvert\) may be zero. Thus, substituting \(b_i=\lvert a_i\rvert-1\), we may regard this space as the space of

\[\y_0^{b_0}\cdots \y_n^{b_n},\qquad b_i\geq 0,\quad \sum b_i=\lvert d\rvert-(n+1).\]

That is, we may think of this space as the space of “negative degree” monomials of degree \(-d-n-1\), and for this reason we denote this space by

\[\mathbb{K}[\x_0^{-1}, \ldots, \x_n^{-1}]_{-d-n-1}.\]

For later use, we define the Euler characteristic.

Definition 2 For an \(n\)-dimensional projective variety \(X\) and a coherent sheaf \(\mathcal{F}\) defined on it, the Euler characteristic of \(\mathcal{F}\) is defined by the formula

\[\rchi(X, \mathcal{F}) = \sum_{i=0}^{n} (-1)^i \dim H^i(X, \mathcal{F}).\]

In the special case where \(X=\mathbb{P}^n\) and \(\mathcal{F}=\mathcal{O}(d)\), since in any case the intermediate cohomology groups all vanish and we need only consider the cohomology at the two ends, the following corollary is easily proved.

Corollary 3 The Euler characteristic of \(\mathcal{O}(d)\) on \(\mathbb{P}^n\) is given by the formula

\[\rchi(\mathbb{P}^n, \mathcal{O}(d)) = \binom{n+d}{n}.\]

Proof

By Proposition 1 (Bott), the cohomology falls into three cases.

First, if \(d \geq 0\), then only \(H^0\) is non-zero, so

\[\rchi(\mathcal{O}(d)) = \dim H^0(\mathbb{P}^n, \mathcal{O}(d)) = \dim \mathbb{K}[\x_0, \ldots, \x_n]_d = \binom{n+d}{n}.\]

Second, if \(-n \leq d \leq -1\), then all cohomology vanishes, so \(\rchi(\mathcal{O}(d)) = 0\), and in this case we usually define \(\binom{n+d}{n}=0\), which agrees with the convention.

Finally, consider the case \(d \leq -n-1\). In this case, only \(H^n\) is non-zero, so

\[\rchi(\mathcal{O}(d)) = (-1)^n \dim \mathbb{K}[\x_0^{-1}, \ldots, \x_n^{-1}]_{-d-n-1}.\]

By the explanation immediately following Proposition 1 (Bott), we know that the dimension of this space is

\[\binom{-d-1}{n}=(-1)^n\binom{n+d}{n}.\]

Here \(\binom{n+d}{n}\) follows the usual convention for binomial coefficient notation, just as in the case above.

The Euler characteristic has the important property of additivity with respect to short exact sequences. That is, for a short exact sequence

\[0 \rightarrow \mathcal{F} \rightarrow \mathcal{G} \rightarrow \mathcal{H} \rightarrow 0\]

we have \(\rchi(\mathcal{G}) = \rchi(\mathcal{F}) + \rchi(\mathcal{H})\). Thus the Euler characteristic becomes an invariant that is much easier to compute and manipulate, at the cost of losing information about the individual cohomology groups.

Serre Vanishing

By Proposition 1 (Bott), on \(\mathbb{P}^n\) the higher cohomology of \(\mathcal{O}(d)\) vanishes for sufficiently large \(d\). Since every line bundle on \(\mathbb{P}^n\) is of the form \(\mathcal{O}(d)\) for some \(d\), this means that for any line bundle \(\mathcal{L}\) on \(\mathbb{P}^n\), the twisted line bundle

\[\mathcal{L}\otimes \mathcal{O}(d)\]

using a sufficiently large \(d\gg 0\) necessarily has vanishing higher cohomology.

More generally, we can extend this to arbitrary projective varieties and arbitrary coherent sheaves defined on them. For this we first need something to play the role of \(\mathcal{O}(1)\); in our definition, a projective variety \(X\) is always given by an embedding \(X\hookrightarrow\mathbb{P}^N\), so it suffices to pull back \(\mathcal{O}(1)\) from \(\mathbb{P}^N\).

Meanwhile, the key ingredient in proving this is the fact that a coherent sheaf becomes globally generated upon sufficient twisting. To gain intuition for this concept, let us first consider the case of line bundles. That a line bundle \(\mathcal{L}\) is basepoint-free, as defined in §Linear Systems, ⁋Definition 5, means that for every point \(p \in X\) there exists a global section \(s \in H^0(X, \mathcal{L})\) with \(s(p) \neq 0\). That is, the base locus is empty, and the linear system \(\lvert\mathcal{L}\rvert\) provides a nonzero value at each point. This is equivalent to the evaluation map

\[H^0(X, \mathcal{L}) \otimes \mathcal{O}_X \rightarrow \mathcal{L}\]

being surjective, and the following definition generalizes this condition to arbitrary coherent sheaves.

Definition 4 A coherent sheaf \(\mathcal{F}\) is said to be globally generated if the evaluation map

\[H^0(X, \mathcal{F}) \otimes \mathcal{O}_X \rightarrow \mathcal{F}\]

is surjective. That is, the stalks can all be generated by global sections.

In particular, for line bundles, being globally generated is equivalent to being basepoint-free. We first verify that twisting on \(\mathbb{P}^N\) actually produces enough global sections.

Lemma 5 For a coherent sheaf \(\mathcal{G}\) on \(\mathbb{P}^N\), \(\mathcal{G}(n)\) is globally generated for sufficiently large \(n\).

Proof

Let \(S = \mathbb{K}[\x_0, \ldots, \x_N]\) and let \(M = \bigoplus_{n \in \mathbb{Z}} \Gamma(\mathbb{P}^N, \mathcal{G}(n))\) be a graded \(S\)-module. On each standard affine open set \(D_+(\x_j)\), we have \(\Gamma(D_+(\x_j), \mathcal{G}) = M_{(\x_j)}\), which is a finitely generated \(S_{(\x_j)}\)-module. If we write generators of these as \(\overline{m}_1, \ldots, \overline{m}_{r_j} \in M_{(\x_j)}\), then we can write \(\overline{m}_k = m_k / \x_j^{d_k}\) for homogeneous elements \(m_k \in M\). Setting \(d_0 = \max_{j,k} d_k\) and multiplying each generator by \(\x_j^{d_0 - d_k}\), we obtain homogeneous elements \(m_k \cdot \x_j^{d_0 - d_k} \in M_{d_0}\). These are elements of \(\Gamma(\mathbb{P}^N, \mathcal{G}(d_0))\), and since multiplying by \(\x_j^{d_0}\) gives an isomorphism between \(\mathcal{G}\) and \(\mathcal{G}(d_0)\) on \(D_+(\x_j)\), the global sections obtained in this way generate the stalks of \(\mathcal{G}(d_0)\) on \(D_+(\x_j)\). As the \(D_+(\x_j)\) cover \(\mathbb{P}^N\), we conclude that \(\mathcal{G}(d_0)\) is globally generated. On the other hand, for \(k \geq 0\) the monomials \(\x_0^k, \ldots, \x_N^k\) generate the stalks of \(\mathcal{O}(k)\), so \(\mathcal{O}(k)\) is also globally generated, and taking the tensor product of the two evaluation maps shows that \(\mathcal{G}(n) = \mathcal{G}(d_0) \otimes \mathcal{O}(n - d_0)\) is globally generated for all \(n \geq d_0\).

The usefulness of Lemma 5 is that it allows us to write an arbitrary coherent sheaf as a quotient of a direct sum of line bundles, and from this we can extend the computation of Proposition 1 (Bott) to arbitrary coherent sheaves.

Lemma 6 For a coherent sheaf \(\mathcal{G}\) on \(\mathbb{P}^N\), for all sufficiently large \(n\),

\[H^i(\mathbb{P}^N, \mathcal{G}(n)) = 0 \quad (i > 0)\]

holds.

Proof

By Lemma 5 we can choose \(a \gg 0\) such that \(\mathcal{G}(a)\) is globally generated, and from the resulting surjection \(\mathcal{O}^{\oplus r} \twoheadrightarrow \mathcal{G}(a)\) we obtain, by tensoring with \(\mathcal{O}(-a)\),

\[\mathcal{E} := \mathcal{O}(-a)^{\oplus r} \twoheadrightarrow \mathcal{G}.\]

Thus any coherent sheaf is a quotient of a direct sum of line bundles, and the kernel \(\mathcal{K}\) of this surjection is also coherent.

We now prove the claim by descending induction on \(i\). For the base step, for \(i\) larger than \(N\) the standard affine open cover of \(\mathbb{P}^N\) consists of \(N+1\) open sets, so the cohomology of any quasi-coherent sheaf vanishes even without twisting. So assume \(0 < i \leq N\) and that the claim holds for \(i+1\) for all coherent sheaves. The long exact sequence associated to tensoring the short exact sequence \(0 \rightarrow \mathcal{K} \rightarrow \mathcal{E} \rightarrow \mathcal{G} \rightarrow 0\) with \(\mathcal{O}(n)\) gives

\[H^i(\mathcal{E}(n)) \rightarrow H^i(\mathcal{G}(n)) \rightarrow H^{i+1}(\mathcal{K}(n)).\]

Since \(\mathcal{E}(n) = \mathcal{O}(n-a)^{\oplus r}\), the left term vanishes for \(n - a \geq -N\) by Proposition 1 (Bott), and by the inductive hypothesis on \(\mathcal{K}\) the right term vanishes for all sufficiently large \(n\). Hence \(H^i(\mathcal{G}(n)) = 0\) for all sufficiently large \(n\). Since there are only finitely many values of \(i\) from \(1\) to \(N\), taking the maximum of the lower bounds obtained for each yields an \(n_0\) such that vanishing holds simultaneously for all \(i > 0\).

What remains is to embed an arbitrary projective variety into \(\mathbb{P}^N\) and apply Lemma 6.

Proposition 7 (Serre Vanishing) Let \(X\) be a projective variety, \(\mathcal{L}\) an ample line bundle, and \(\mathcal{F}\) a coherent sheaf. Then for sufficiently large \(m\),

\[H^i(X, \mathcal{F} \otimes \mathcal{L}^{\otimes m}) = 0 \quad (i > 0)\]

holds.

Proof

Since \(\mathcal{L}\) is ample, for sufficiently large \(m_0\) the sheaf \(\mathcal{L}^{\otimes m_0}\) is very ample. That is, there exists a suitable embedding \(i \colon X \hookrightarrow \mathbb{P}^N\) such that \(\mathcal{L}^{\otimes m_0} = i^\ast\mathcal{O}(1)\). Restricting the standard affine cover \(\{U_i\}\) of \(\mathbb{P}^N\) to \(X\) yields an affine open cover \(\{X \cap U_i\}\). Since a finite intersection \(U_{i_0} \cap \cdots \cap U_{i_p}\) is affine, so is \((X \cap U_{i_0}) \cap \cdots \cap (X \cap U_{i_p}) = X \cap (U_{i_0} \cap \cdots \cap U_{i_p})\). Therefore the two Čech complexes are literally the same, so

\[\check{H}^i(\{X \cap U_j\}, \mathcal{F}) = \check{H}^i(\{U_j\}, i_\ast\mathcal{F})\]

holds. On the other hand, by §Sheaf Cohomology, ⁋Theorem 11 (Leray), the cohomology of a quasi-coherent sheaf on a variety is computed by Čech cohomology with respect to an affine open cover, so the following identity

\[H^i(X, \mathcal{F}) = \check{H}^i(\{X \cap U_j\}, \mathcal{F}) = \check{H}^i(\{U_j\}, i_\ast\mathcal{F}) = H^i(\mathbb{P}^N, i_\ast\mathcal{F})\]

holds. We apply this identity with \(\mathcal{F} \otimes \mathcal{L}^{\otimes m}\) in place of \(\mathcal{F}\). Writing a given \(m\) as \(m = m_0 n + r\) with \(0 \leq r < m_0\), we have \(\mathcal{L}^{\otimes m} = \mathcal{L}^{\otimes r} \otimes i^\ast\mathcal{O}(n)\), so by the projection formula

\[i_\ast(\mathcal{F} \otimes \mathcal{L}^{\otimes m}) = i_\ast(\mathcal{F} \otimes \mathcal{L}^{\otimes r}) \otimes \mathcal{O}(n) = \mathcal{G}_r(n), \qquad \mathcal{G}_r := i_\ast(\mathcal{F} \otimes \mathcal{L}^{\otimes r})\]

and therefore \(H^i(X, \mathcal{F} \otimes \mathcal{L}^{\otimes m}) = H^i(\mathbb{P}^N, \mathcal{G}_r(n))\). Here the \(\mathcal{G}_r\) are finitely many coherent sheaves on \(\mathbb{P}^N\) determined by \(r = 0, \ldots, m_0-1\), and if \(m\) is sufficiently large then \(n = (m-r)/m_0\) is also sufficiently large, so applying Lemma 6 to each \(\mathcal{G}_r\) and taking the maximum of the finitely many lower bounds obtained yields the desired conclusion for all sufficiently large \(m\).

Generally, vanishing theorems of this kind play a major role in cohomology computations. If Proposition 7 (Serre Vanishing) said that sufficient twisting can make all higher cohomology zero, the next proposition says that cohomology in sufficiently high degree is always zero even without twisting.

Proposition 8 (Grothendieck Vanishing) For an \(n\)-dimensional projective variety \(X\) and a coherent sheaf \(\mathcal{F}\) on it, if \(i > n\) then

\[H^i(X, \mathcal{F}) = 0\]

holds.

Proof

The homogeneous coordinate ring \(S(X)\) of \(X \subseteq \mathbb{P}^M\) has Krull dimension \(n+1\), so by [Commutative Algebra] §Noether Normalization, ⁋Theorem 1 there exists a polynomial subring over which \(S(X)\) is a finitely generated module, and since \(\mathbb{K}\) is algebraically closed it is an infinite field, so by the second case of [Commutative Algebra] §Noether Normalization, ⁋Lemma 2 we can choose its generators to be linear forms \(\y_0, \ldots, \y_n\). These have no common zero on \(X\), so they define a morphism

\[\pi \colon X \rightarrow \mathbb{P}^n, \qquad p \mapsto [\y_0(p) : \cdots : \y_n(p)]\]

and on each \(D_+(\y_j)\) the coordinate ring of \(\pi^{-1}(D_+(\y_j))\) is \(S(X)_{(\y_j)}\), which is a finitely generated module over \(\mathbb{K}[D_+(\y_j)]\), so \(\pi\) is finite in the sense of §Dimension, ⁋Definition 11, and surjective by lying over for integral extensions. Geometrically, this is the projection with center the linear subspace \(V(\y_0, \ldots, \y_n)\) disjoint from \(X\).

Since a finite morphism has affine preimages of affine open sets, the pull-back \(\{\pi^{-1}(U_j)\}\) of the standard affine cover \(\{U_j\}\) of \(\mathbb{P}^n\) is an affine open cover of \(X\), and its finite intersections are also affine. Then, just as in the proof of Proposition 7 (Serre Vanishing), the two Čech complexes agree termwise as

\[\check{C}^p(\{\pi^{-1}(U_j)\}, \mathcal{F}) = \prod_{j_0 < \cdots < j_p} \mathcal{F}(\pi^{-1}(U_{j_0} \cap \cdots \cap U_{j_p})) = \check{C}^p(\{U_j\}, \pi_\ast\mathcal{F}),\]

and by §Sheaf Cohomology, ⁋Theorem 11 (Leray) the Čech cohomology on each side is \(H^i(X, \mathcal{F})\) and \(H^i(\mathbb{P}^n, \pi_\ast\mathcal{F})\) respectively, so these are equal. Now \(\{U_j\}\) consists of \(n+1\) open sets, so \(\check{C}^p(\{U_j\}, \pi_\ast\mathcal{F}) = 0\) for \(p > n\), and therefore the right-hand side vanishes when \(i > n\).

Regularity

Proposition 7 (Serre Vanishing) gave a qualitative result that higher cohomology vanishes after sufficiently large twisting. Regularity quantifies this, measuring specifically how much twisting is needed.

Intuitively, higher cohomology arises from failures in lower-degree cohomology, so this twisting is “less” necessary in high degrees. With this in mind, the following definition is natural.

Definition 9 Fix a projective variety \(X\) and a very ample line bundle \(\mathcal{L}\) on it. We say that a coherent sheaf \(\mathcal{F}\) on \(X\) is \(m\)-regular if for all \(i > 0\),

\[H^i(X, \mathcal{F} \otimes \mathcal{L}^{\otimes m - i}) = 0\]

holds.

In general it is almost impossible to compute all cohomology groups of a coherent sheaf, but the basic idea is that higher cohomology vanishes after sufficient twisting. Regularity goes further, measuring specifically how much twisting is needed.

To define regularity in general, we first need the notion of a twist. On \(\mathbb{P}^n\) we use \(\mathcal{O}(1)\) as the basic twist, so we write \(\mathcal{F}(d) := \mathcal{F} \otimes \mathcal{O}(d)\). On an arbitrary projective variety \(X\) we choose an ample line bundle \(\mathcal{L}\) and define \(\mathcal{F}(d) := \mathcal{F} \otimes \mathcal{L}^{\otimes d}\). Twisting satisfies the following properties. By associativity of the tensor product, \(\mathcal{F}(d)(e) = \mathcal{F}(d+e)\) holds. Moreover, the tensor product functor \(- \otimes \mathcal{L}^{\otimes d}\) is exact because \(\mathcal{L}^{\otimes d}\) is a line bundle, so for a short exact sequence

\[0 \rightarrow \mathcal{F} \rightarrow \mathcal{G} \rightarrow \mathcal{H} \rightarrow 0\]

the sequence

\[0 \rightarrow \mathcal{F}(d) \rightarrow \mathcal{G}(d) \rightarrow \mathcal{H}(d) \rightarrow 0\]

is also short exact.

Proposition 10 (Castelnuovo-Mumford Regularity) Let \(X\) be a projective variety, \(\mathcal{L}\) a very ample line bundle, and \(\mathcal{F}\) a coherent sheaf. If \(\mathcal{F}\) is \(m\)-regular with respect to \(\mathcal{L}\), then the following hold.

1. \(\mathcal{F} \otimes \mathcal{L}^{\otimes m}\) is globally generated.
2. \(\mathcal{F}\) is \((m+p)\)-regular with respect to \(\mathcal{L}\) for all \(p \geq 0\).

Proof

Set \(n = \dim X\) and argue by induction on the dimension of \(X\). If \(\dim X = 0\), then \(X\) is a point and a coherent sheaf \(\mathcal{F}\) is a finite-dimensional vector space, so all cohomology except \(H^0\) vanishes automatically. Now assume \(\dim X \geq 1\).

The key is to use the restriction exact sequence for an effective divisor \(D\) defined by a global section \(s \in H^0(X, \mathcal{L})\). Since \(\mathcal{L}\) is very ample, \(D\) is a hyperplane section of \(X\) embedded in projective space, and hence by choosing a general \(s\), §Linear Systems, ⁋Proposition 11 (Bertini’s theorem) implies that \(D\) is smooth away from the singular points of \(X\). We then obtain the following short exact sequence.

\[0 \rightarrow \mathcal{F} \otimes \mathcal{L}^{\otimes k-1} \xrightarrow{\cdot s} \mathcal{F} \otimes \mathcal{L}^{\otimes k} \rightarrow \mathcal{F} \otimes \mathcal{L}^{\otimes k}\vert_D \rightarrow 0\]

The long exact sequence in cohomology of this sequence yields

\[\cdots \rightarrow H^i(\mathcal{F} \otimes \mathcal{L}^{\otimes k-1}) \rightarrow H^i(\mathcal{F} \otimes \mathcal{L}^{\otimes k}) \rightarrow H^i(\mathcal{F} \otimes \mathcal{L}^{\otimes k}\vert_D) \rightarrow H^{i+1}(\mathcal{F} \otimes \mathcal{L}^{\otimes k-1}) \rightarrow \cdots\]

In the special case of \(\mathbb{P}^n\), we have \(\mathcal{L} = \mathcal{O}(1)\), \(s\) is a general linear form, and \(D\) becomes a hyperplane \(H\) isomorphic to \(\mathbb{P}^{n-1}\).

First, we show that \(\mathcal{F}\vert_D\) is \(m\)-regular with respect to \(\mathcal{L}\vert_D\). Since \(\mathcal{F}\) is \(m\)-regular with respect to \(\mathcal{L}\), we have \(H^i(\mathcal{F} \otimes \mathcal{L}^{\otimes m-i}) = 0\) for \(i > 0\). Substituting \(k = m - i\) in the restriction sequence (\(0 < i \leq n-1\))

\[0 \rightarrow \mathcal{F} \otimes \mathcal{L}^{\otimes m-i-1} \rightarrow \mathcal{F} \otimes \mathcal{L}^{\otimes m-i} \rightarrow \mathcal{F}\vert_D \otimes (\mathcal{L}\vert_D)^{\otimes m-i} \rightarrow 0\]

and from its long exact sequence

\[H^i(\mathcal{F} \otimes \mathcal{L}^{\otimes m-i}) \rightarrow H^i(\mathcal{F}\vert_D \otimes (\mathcal{L}\vert_D)^{\otimes m-i}) \rightarrow H^{i+1}(\mathcal{F} \otimes \mathcal{L}^{\otimes m-i-1}).\]

By \(m\)-regularity, \(H^i(\mathcal{F} \otimes \mathcal{L}^{\otimes m-i}) = 0\), and \(H^{i+1}(\mathcal{F} \otimes \mathcal{L}^{\otimes m-i-1}) = 0\) (\(i+1 > 0\)), so we obtain

\[H^i(\mathcal{F}\vert_D \otimes (\mathcal{L}\vert_D)^{\otimes m-i}) = 0\]

for \(0 < i \leq n-1\). This means that \(\mathcal{F}\vert_D\) is \(m\)-regular with respect to \(\mathcal{L}\vert_D\).

Now apply the inductive hypothesis to \(D\). The divisor \(D\) is a projective variety with \(\dim D < \dim X\), and \(\mathcal{L}\vert_D\) is an ample line bundle. Since \(\mathcal{F}\vert_D\) is \(m\)-regular, the inductive hypothesis implies that \(\mathcal{F}\vert_D \otimes (\mathcal{L}\vert_D)^{\otimes m}\) is globally generated on \(D\).

We now show that \(\mathcal{F} \otimes \mathcal{L}^{\otimes m}\) is globally generated. It suffices to verify that for any point \(x \in X\), the fiber \((\mathcal{F} \otimes \mathcal{L}^{\otimes m})_x\) is generated by the images of global sections. Choose a general divisor \(D\) passing through \(x\), and substitute \(k = m\) in the restriction sequence:

\[0 \rightarrow \mathcal{F} \otimes \mathcal{L}^{\otimes m-1} \rightarrow \mathcal{F} \otimes \mathcal{L}^{\otimes m} \rightarrow \mathcal{F}\vert_D \otimes (\mathcal{L}\vert_D)^{\otimes m} \rightarrow 0\]

From \(m\)-regularity with \(i = 1\), we have \(H^1(\mathcal{F} \otimes \mathcal{L}^{\otimes m-1}) = 0\), so

\[H^0(\mathcal{F} \otimes \mathcal{L}^{\otimes m}) \rightarrow H^0(\mathcal{F}\vert_D \otimes (\mathcal{L}\vert_D)^{\otimes m})\]

is surjective. By the inductive hypothesis, \(\mathcal{F}\vert_D \otimes (\mathcal{L}\vert_D)^{\otimes m}\) is globally generated on \(D\), so its fiber at \(x\) is generated by the image of \(H^0(\mathcal{F}\vert_D \otimes (\mathcal{L}\vert_D)^{\otimes m})\). Since the restriction map is surjective, the global sections of \(\mathcal{F} \otimes \mathcal{L}^{\otimes m}\) also generate the fiber at \(x\). Therefore \(\mathcal{F} \otimes \mathcal{L}^{\otimes m}\) is globally generated.

Finally, we show that \(\mathcal{F}\) is \((m+p)\)-regular by induction on \(p\). For \(p = 0\), \(\mathcal{F}\) being \(m\)-regular is by definition. Assume \(p \geq 1\), and we show that \(\mathcal{F}\) is \((m+p)\)-regular, i.e., \(H^i(\mathcal{F} \otimes \mathcal{L}^{\otimes m+p-i}) = 0\) (\(i > 0\)). For \(i = 1\), substituting \(k = m + p - 1\) in the restriction sequence gives

\[H^0(\mathcal{F}\vert_D \otimes (\mathcal{L}\vert_D)^{\otimes m+p-1}) \rightarrow H^1(\mathcal{F} \otimes \mathcal{L}^{\otimes m+p-2}) \rightarrow H^1(\mathcal{F} \otimes \mathcal{L}^{\otimes m+p-1}) \rightarrow H^1(\mathcal{F}\vert_D \otimes (\mathcal{L}\vert_D)^{\otimes m+p-1})\]

By the inductive hypothesis (for \(p-1\)), \(H^1(\mathcal{F} \otimes \mathcal{L}^{\otimes m+p-2}) = 0\). Also, as shown above, \(\mathcal{F}\vert_D\) is \(m\)-regular, so by the inductive hypothesis on dimension, \(\mathcal{F}\vert_D\) is \((m+p)\)-regular, and hence \(H^1(\mathcal{F}\vert_D \otimes (\mathcal{L}\vert_D)^{\otimes m+p-1}) = 0\). From the exact sequence, \(H^1(\mathcal{F} \otimes \mathcal{L}^{\otimes m+p-1})\) embeds into \(H^1(\mathcal{F}\vert_D \otimes (\mathcal{L}\vert_D)^{\otimes m+p-1})\), so we obtain \(H^1(\mathcal{F} \otimes \mathcal{L}^{\otimes m+p-1}) = 0\). For \(i \geq 2\), from the same restriction sequence

\[H^i(\mathcal{F} \otimes \mathcal{L}^{\otimes m+p-i-1}) \rightarrow H^i(\mathcal{F} \otimes \mathcal{L}^{\otimes m+p-i}) \rightarrow H^i(\mathcal{F}\vert_D \otimes (\mathcal{L}\vert_D)^{\otimes m+p-i})\]

the left term vanishes by the inductive hypothesis (for \(p-1\)). For the right term, if \(i \leq n-1\), then by the inductive hypothesis on dimension, \(\mathcal{F}\vert_D\) is \((m+p)\)-regular, so \(H^i(\mathcal{F}\vert_D \otimes (\mathcal{L}\vert_D)^{\otimes m+p-i}) = 0\); if \(i \geq n\), then \(\dim D = n-1\), so this term vanishes automatically by Proposition 8 (Grothendieck Vanishing). In either case we obtain \(H^i(\mathcal{F} \otimes \mathcal{L}^{\otimes m+p-i}) = 0\).

Example 11 Let us compute the regularity of the line bundle \(\mathcal{O}(d)\) on \(\mathbb{P}^n\). Here \(\mathcal{L} = \mathcal{O}(1)\), so the twist is \(\mathcal{O}(d) \otimes \mathcal{O}(m) = \mathcal{O}(d+m)\). The \(m\)-regularity condition is \(H^i(\mathbb{P}^n, \mathcal{O}(d+m-i)) = 0\) (\(i > 0\)). If \(d \geq 0\) and we choose \(m = 0\), we need to check \(H^i(\mathcal{O}(d-i))\): for \(i = 1\), \(H^1(\mathcal{O}(d-1))\) is \(0\) when \(d \geq 1\), and when \(d = 0\), \(H^1(\mathcal{O}(-1)) = 0\) (by Bott’s formula, since \(-1 \geq -n\), all cohomology vanishes). In general, if \(d \geq 0\) and \(i > 0\), then \(d - i \geq -n\) implies \(H^i(\mathcal{O}(d-i)) = 0\), and if \(d - i < -n\), i.e., \(i > d + n\), then \(i > n\) so \(H^i = 0\) anyway. Thus \(\mathcal{O}(d)\) is \(0\)-regular with respect to \(\mathcal{L} = \mathcal{O}(1)\). On the other hand, if \(d < 0\), then \(\mathcal{O}(d)\) is \((-d)\)-regular. By Proposition 10 (Castelnuovo-Mumford Regularity), \(\mathcal{O}(d) \otimes \mathcal{L}^{\otimes 0} = \mathcal{O}(d)\) is globally generated when \(d \geq 0\), which agrees with the calculation immediately following §Line Bundles and Vector Bundles, ⁋Example 16.

Properties of Very Ample and Ample

The above Proposition 7 (Serre Vanishing) and Proposition 10 (Castelnuovo-Mumford Regularity) are representative results on the properties of ample line bundles. We conclude this post by examining additional properties of ample and very ample line bundles.

Proposition 12 If \(\mathcal{L}\) is very ample and \(\mathcal{M}\) is a globally generated line bundle, then \(\mathcal{L} \otimes \mathcal{M}\) is very ample.

Proof

Since \(\mathcal{L}\) is very ample, there exists a projective embedding \(i: X \hookrightarrow \mathbb{P}^N\) such that \(\mathcal{L} = i^\ast\mathcal{O}_{\mathbb{P}^N}(1)\). On the other hand, since \(\mathcal{M}\) is globally generated, global sections \(s_0, \ldots, s_n \in H^0(X, \mathcal{M})\) generate the stalk at every point, and from these we can define a morphism \(\phi: X \rightarrow \mathbb{P}^n\).

Now consider the closed embedding \((i, \phi): X \rightarrow \mathbb{P}^N \times \mathbb{P}^n\). Composing with the Segre embedding (§Projective Varieties, ⁋Example 16)

\[\sigma: \mathbb{P}^N \times \mathbb{P}^n \hookrightarrow \mathbb{P}^{Nn+N+n}\]

we have \(\sigma^\ast\mathcal{O}(1) = \pi_1^\ast\mathcal{O}(1) \otimes \pi_2^\ast\mathcal{O}(1)\), so

\[(\sigma \circ (i, \phi))^\ast\mathcal{O}(1) = i^\ast\mathcal{O}(1) \otimes \phi^\ast\mathcal{O}(1) = \mathcal{L} \otimes \mathcal{M}\]

That is, \(\mathcal{L} \otimes \mathcal{M}\) is very ample.

That is, although the explanation was somewhat involved, the key point is that the morphism \(\phi:X\rightarrow \mathbb{P}^n\) defined by a globally generated line bundle \(\mathcal{M}\) need not be a closed embedding, but by tensoring with \(\mathcal{L}\) and embedding into projective space in the form \((i,\phi)\), the first component \(i\) makes this map a closed embedding. From this, the following useful result can also be proved.

Proposition 13 For an ample line bundle \(\mathcal{L}\) and an arbitrary line bundle \(\mathcal{M}\) defined on a projective variety \(X\), the tensor product \(\mathcal{M} \otimes \mathcal{L}^{\otimes n}\) is very ample for all sufficiently large \(n\).

Proof

First, since \(\mathcal{L}\) is ample, \(\mathcal{L}^{\otimes m}\) is very ample for some \(m > 0\). On the other hand, by Proposition 7 (Serre Vanishing) there exists \(k_0\) such that \(H^i(X, \mathcal{M} \otimes \mathcal{L}^{\otimes k}) = 0\) for all \(k \geq k_0\) and \(i > 0\). Now set \(k = k_0 + m\dim X\); then for \(1 \leq i \leq \dim X\) we have \(k - mi \geq k_0\), so

\[H^i(\mathcal{M} \otimes \mathcal{L}^{\otimes k} \otimes (\mathcal{L}^{\otimes m})^{\otimes -i}) = H^i(\mathcal{M} \otimes \mathcal{L}^{\otimes k-mi}) = 0\]

and for \(i > \dim X\), this cohomology vanishes automatically by Proposition 8 (Grothendieck Vanishing). That is, \(\mathcal{M} \otimes \mathcal{L}^{\otimes k}\) is \(0\)-regular with respect to the very ample line bundle \(\mathcal{L}^{\otimes m}\), so by Proposition 10 (Castelnuovo-Mumford Regularity) it is globally generated. Now by Proposition 12,

\[(\mathcal{M} \otimes \mathcal{L}^{\otimes k}) \otimes \mathcal{L}^{\otimes m} = \mathcal{M} \otimes \mathcal{L}^{\otimes (k+m)}\]

is very ample, and setting \(n = k + m\) completes the proof.

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References

[Hart] R. Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics, Springer, 1977.
[Bot] R. Bott, Homogeneous vector bundles, Annals of Mathematics, 1957.
[Laz] R. Lazarsfeld, Positivity in Algebraic Geometry I, Ergebnisse der Mathematik, Springer, 2004.
[Mum] D. Mumford, Lectures on Curves on an Algebraic Surface, Annals of Mathematics Studies, Princeton, 1966.