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 of §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 we introduced in earlier posts (§Sheaf Cohomology, ⁋Definition 1) is a richer invariant that includes not only \(H^0\) but also higher cohomology groups \(H^1, H^2, \ldots\). Thus, we now aim to extract all information about \(\mathcal{O}(d)\) using not just \(H^0\) 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 following is the result of that computation.

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 the sections \(\mathcal{O}(d)(U_i)\) over each open set are given by

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

(§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 \(U_{i_0}\cap\cdots\cap U_{i_p}\). In this case, for a section to be regular over the intersection \(U_{i_0}\cap\cdots\cap U_{i_p}\), it must be generated by monomials in which only the non-vanishing coordinates, namely \(\x_{i_0}, \ldots, \x_{i_p}\), are allowed in the denominator, while the others are not:

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

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

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

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

Now let us compute each cohomology group using these data. We begin 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 of \(\check{C}^0\), we analyze \(\ker\delta\). Since \(H^0(\mathbb{P}^n, \mathcal{O}(d))=\Gamma(\mathbb{P}^n, \mathcal{O}(d))\), this is in fact nothing but rechecking the computation of §Line Bundles and Vector Bundles, ⁋Example 16; however, instead of treating it as a separate example, we shall carry out the Čech cohomology computation within this proof.

By definition, a cochain \((f_0, f_1) \in \check{C}^0\) lies in \(\ker \delta\) if and only if \(f_0 = f_1\) 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 particular cocycle \((f_0,f_1)\) to lie in \(\ker\delta\), we must have \(f_0=f_1\), and therefore only monomials satisfying \(a+b=d\) can belong to \(\ker\delta\). That is, the following 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 \(H^0\), yielding the desired result. If \(d<0\), then \(a,b\geq 0\) cannot satisfy this equation, so \(H^0\) becomes \(0\).

Now we compute \(H^1\). That is, we must compute \(\coker\delta\). From the above computation, 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, elements of \(\check{C}^1\) are generated by degree-\(d\) monomials

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

If \(d\geq 0\), then each of these can be obtained explicitly from (\(\ast\)). For instance, \(\x_0^{2d}\x_1^{-d}\) comes from the \(j=2d\) term in the \(f_1\) component, and \(\x_0^{-d}\x_1^{2d}\) comes from the \(i=2d\) term in the \(f_0\) component. Hence in this case \(\coker\delta=0\). However, if \(d<0\), monomials that cannot be represented in the image of \(\delta\) appear; this is because, analyzing each term in (\(\ast\)), at least one exponent is greater than or equal to \(0\). On the other hand, in (\(\ast\ast\)) there appear monomials in which both exponents are negative:

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

and these generate \(\coker \delta\). We will explain the notation in the claim separately after the proof is finished.

Now let us use induction for the general case. To do so, we use the fact that the hyperplane \(H=\{\x_n=0\}\) in \(\mathbb{P}^n\) is isomorphic to \(\mathbb{P}^{n-1}\), and consider the following short exact sequence

\[0 \longrightarrow \mathcal{O}(d-1)\overset{\times \x_n}{\longrightarrow} \mathcal{O}(d)\longrightarrow \mathcal{O}(d)\vert_H\longrightarrow 0\]

From this we obtain the long exact sequence

\[\cdots \to H^{i-1}(\mathbb{P}^{n-1}, \mathcal{O}(d)) \to H^i(\mathbb{P}^n, \mathcal{O}(d-1)) \to H^i(\mathbb{P}^n, \mathcal{O}(d)) \to H^i(\mathbb{P}^{n-1}, \mathcal{O}(d)) \to \cdots\]

and by the inductive process we already know the claim for \(\mathbb{P}^{n-1}\). Then in particular, for \(0<i<n\) we have

\[H^{i-1}(\mathbb{P}^{n-1}, \mathcal{O}(d))=H^i(\mathbb{P}^{n-1}, \mathcal{O}(d))=0\]

and from this we obtain \(H^i(\mathbb{P}^n, \mathcal{O}(d-1)) \cong H^i(\mathbb{P}^n, \mathcal{O}(d))\). That is, \(H^i(\mathbb{P}^n, \mathcal{O}(d))\) is isomorphic for all \(d\), and in particular, since we can immediately show that for the trivial sheaf \(\mathcal{O}\)

\[H^i(\mathbb{P}^n, \mathcal{O}(d))=H^i(\mathbb{P}^n, \mathcal{O})=0\]

we know that

\[H^i(\mathbb{P}^n, \mathcal{O}(d))=0\]

holds for all \(0<i<n\) and all \(d\).

Now consider the top cohomology part

\[\cdots\rightarrow H^{n-1}(\mathbb{P}^{n-1}, \mathcal{O}(d))\rightarrow H^n(\mathbb{P}^n, \mathcal{O}(d-1))\rightarrow H^n(\mathbb{P}^n, \mathcal{O}(d))\rightarrow H^n(\mathbb{P}^{n-1}, \mathcal{O}(d))=0\]

If \(d\geq -n\), then by the inductive hypothesis \(H^{n-1}(\mathbb{P}^{n-1}, \mathcal{O}(d))=0\), so by the same argument as above we can show that \(H^n(\mathbb{P}^n, \mathcal{O}(d))\) holds for all \(d>-n-1\). For the case \(d\leq -n-1\), we must compute \(H^n\) directly from the Čech complex; to do so, computing \(\check{C}^n(\mathbb{P}^n, \mathcal{O}(d))\) directly, we know that an \(n\)-cochain is an element of

\[\mathbb{K}[\x_0^{\pm 1}, \ldots, \x_n^{\pm 1}]_d\]

and the monomials not appearing in the image of an \((n-1)\)-cochain are, similarly to the computation for \(\mathbb{P}^1\) above, degree-\(d\) monomials in which all exponents are less than \(-1\), from which we obtain the desired result. As for \(H^0\), although we computed it directly above, as already mentioned this is nothing but rechecking §Line Bundles and Vector Bundles, ⁋Example 16, so we will not repeat it here.

In the proof above, we showed that for each variable \(\x_0,\cdots, \x_n\) and for \(d\leq -n-1\), \(H^n(\mathbb{P}^n, \mathcal{O}(d))\) is generated by the following 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.) Thinking of each \(\x_i^{-1}\) as a new variable \(\y_i=\x_i^{-1}\), this is the space generated by expressions of the form

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

Since each \(a_i\) and \(d\) are all 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 \(\lvert d\rvert\), but with the difference that the \(\lvert a_i\rvert\) cannot be \(0\). Thus, substituting \(b_i=\lvert a_i\rvert-1\), we can think of this space as the space of

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

That is, this space can be thought of 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 a variety \(X\) and a coherent sheaf \(\mathcal{F}\) defined on it, we define the Euler characteristic of \(\mathcal{F}\) by the formula

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

In particular, when \(X=\mathbb{P}^n\) and \(\mathcal{F}=\mathcal{O}(d)\), since in any case the intermediate cohomologies all vanish and only the cohomologies at both ends matter, we can easily prove the following corollary.

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 such cases we usually define \(\binom{n+d}{n}=0\), which is consistent 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 general convention for binomial coefficient notation, just as in the cases above.

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

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

we have \(\rchi(\mathcal{G}) = \rchi(\mathcal{F}) + \rchi(\mathcal{H})\). Thus, the Euler characteristic becomes a much more computable and manageable invariant, at the cost of losing information about individual cohomology groups.

Serre Vanishing

According to Proposition 1 (Bott), on \(\mathbb{P}^n\) the higher cohomology of \(\mathcal{O}(d)\) vanishes for sufficiently large \(d\). Since any 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 \(\mathbb{P}^n\) and for sufficiently large \(d\gg 0\), the twisted line bundle

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

must have higher cohomology equal to \(0\).

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

Proposition 4 (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 \(\mathcal{L}^{\otimes m_0} = i^\ast\mathcal{O}(1)\) holds. Restricting the standard affine cover \(\{U_i\}\) of \(\mathbb{P}^N\) to \(X\), we obtain an affine open cover \(\{X \cap U_i\}\). Since a finite intersection \(U_{i_0} \cap \cdots \cap U_{i_p}\) is affine, \((X \cap U_{i_0}) \cap \cdots \cap (X \cap U_{i_p}) = X \cap (U_{i_0} \cap \cdots \cap U_{i_p})\) is also affine. Therefore, since the two Čech complexes are literally the same,

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

holds. Since \(X\) and \(\mathbb{P}^N\) are separated schemes (§Sheaf Cohomology, ⁋Theorem 11 (Leray)), for quasi-coherent sheaves Čech cohomology equals sheaf cohomology:

\(H^i(X, \mathcal{F}) = \check{H}^i(\{X \cap U_j\}, \mathcal{F}) = \check{H}^i(\{U_j\}, i_\ast\mathcal{G}) = H^i(\mathbb{P}^N, i_\ast\mathcal{G})\) Thus it suffices to show the following: for a coherent sheaf \(\mathcal{G}\) on \(\mathbb{P}^N\), for sufficiently large \(n\) we have \(H^i(\mathbb{P}^N, \mathcal{G}(n)) = 0\) (\(i > 0\)). Here \(\mathcal{G}(n) = \mathcal{G} \otimes \mathcal{O}_{\mathbb{P}^N}(n)\).

Key Lemma. We show that \(\mathcal{G}(n)\) is globally generated for sufficiently large \(n\). (See Definition 6 below.)

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 the associated graded \(S\)-module. On each standard affine open set \(D_+(\x_j)\), \(\Gamma(D_+(\x_j), \mathcal{G})\) is the degree-0 localization \(M_{(\x_j)}\), which is a finitely generated module over \(S_{(\x_j)}\). Choose generators \(\bar{m}_1, \ldots, \bar{m}_{r_j} \in M_{(\x_j)}\). Each \(\bar{m}_k\) can be written in the form \(m_k / \x_j^{d_k}\), where \(m_k \in M\) is a homogeneous element. Setting \(d_0 = \max_j \max_k d_k\), multiplying each generator by \(\x_j^{d_0 - d_k}\) yields a homogeneous element \(m_k \cdot \x_j^{d_0 - d_k} \in M_{d_0}\). This is an element of \(\Gamma(\mathbb{P}^N, \mathcal{G}(d_0))\), and one can check that it generates the stalk of \(\mathcal{G}\) over \(D_+(\x_j)\). Taking the maximum over \(j\), we obtain that \(\mathcal{G}(d_0)\) is globally generated.

Vanishing. We now show \(H^i(\mathbb{P}^N, \mathcal{G}(n)) = 0\) (\(i > 0\), \(n \gg 0\)).

If \(N = 0\), then \(\mathbb{P}^0\) is a point, so this is trivial. Assume \(N \geq 1\). By the lemma above, \(\mathcal{G}(n_0)\) is globally generated (\(n_0 \gg 0\)), so there exists a suitable surjection

\[\mathcal{O}_{\mathbb{P}^N}^{\oplus r_0} \twoheadrightarrow \mathcal{G}(n_0)\]

The kernel \(\mathcal{K}_0\) is coherent. From the long exact sequence of the short exact sequence

\[0 \to \mathcal{K}_0 \to \mathcal{O}^{\oplus r_0} \to \mathcal{G}(n_0) \to 0\]

since \(H^j(\mathbb{P}^N, \mathcal{O}^{\oplus r_0}) = 0\) (\(j > 0\)) by Proposition 1 (Bott), we obtain

\[H^j(\mathcal{G}(n_0)) \cong H^{j+1}(\mathcal{K}_0) \quad (j \geq 1)\]

Now we repeat the same process for \(\mathcal{K}_0\). That is, choose \(n_1 \gg 0\) such that \(\mathcal{K}_0(n_1)\) is globally generated, and for the kernel \(\mathcal{K}_1\) of the surjection

\[\mathcal{O}^{\oplus r_1} \twoheadrightarrow \mathcal{K}_0(n_1)\]

from the long exact sequence of

\[0 \to \mathcal{K}_1 \to \mathcal{O}^{\oplus r_1} \to \mathcal{K}_0(n_1) \to 0\]

we obtain

\[H^{j+1}(\mathcal{K}_0(n_1)) \cong H^{j+2}(\mathcal{K}_1) \quad (j \geq 1)\]

Repeating this process \(N\) times, we get

\[H^j(\mathcal{G}(n_0)) \cong H^{j+N}(\mathcal{K}_{N-1})\]

Since the cohomological dimension of \(\mathbb{P}^N\) is \(N\), we have \(H^{j+N} = 0\) (\(j \geq 1\), \(j + N \geq N+1 > N\)), and therefore \(H^j(\mathcal{G}(n_0)) = 0\).

Finally, since \(\mathcal{G}(n_0)\) is globally generated, \(\mathcal{G}(n) = \mathcal{G}(n_0) \otimes \mathcal{O}(n - n_0)\) is also globally generated for \(n \geq n_0\), and thus the same resolution argument can be constructed for \(\mathcal{G}(n)\) as well, so the vanishing holds for all \(n \geq n_0\).

Regularity

Proposition 4 (Serre Vanishing) gave a qualitative result that higher cohomology vanishes after sufficiently large twisting. Regularity quantifies this by measuring exactly how much twisting is needed.

Intuitively, higher cohomology arises from the failure of lower-degree cohomology, so this twisting is “less” necessary at higher degrees. Keeping this in mind, the following definition is natural.

Definition 5 Let a projective variety \(X\) and an ample line bundle \(\mathcal{L}\) on it be fixed. Then a coherent sheaf \(\mathcal{F}\) on \(X\) is said to be \(m\)-regular if for all \(i>0\),

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

holds.

In general, computing all cohomology of a coherent sheaf is almost impossible, but the basic idea is that higher cohomology vanishes after sufficient twisting. Regularity goes further by measuring exactly how much twisting is needed.

One of the key reasons such twisting becomes useful when dealing with coherent sheaves is that sufficiently twisted coherent sheaves become globally generated. To gain intuition for this concept, let us first consider the case of line bundles. A line bundle \(\mathcal{L}\) being 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, so the linear system \(\lvert \mathcal{L} \rvert\) provides a non-zero value at each point. This is equivalent to the evaluation map

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

being surjective. Globally generated generalizes this condition to arbitrary coherent sheaves: a coherent sheaf \(\mathcal{F}\) is globally generated if the same type of evaluation map is surjective, meaning that the stalk at each point can be generated by global sections. In particular, for line bundles, being globally generated is equivalent to being basepoint-free. This property played a key role in the proof of Proposition 4 (Serre Vanishing).

Definition 6 A coherent sheaf \(\mathcal{F}\) is said to be globally generated if the evaluation map

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

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

To define regularity in general, we first need the notion of twist. On \(\mathbb{P}^n\) we use \(\mathcal{O}(1)\) as the base, 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 \(\mathcal{F}(d) := \mathcal{F} \otimes \mathcal{L}^{\otimes d}\). Twist satisfies the following properties. By the associativity of tensor product, \(\mathcal{F}(d)(e) = \mathcal{F}(d+e)\) holds. Also, since the tensor product functor \(- \otimes \mathcal{L}^{\otimes d}\) is a line bundle, it is exact, and therefore for a short exact sequence

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

the sequence

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

is also a short exact sequence.

Proposition 7 (Castelnuovo-Mumford Regularity) Let \(X\) be a projective variety, \(\mathcal{L}\) an 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. \(\mathcal{F} \otimes \mathcal{L}^{\otimes p}\) is \((m+p)\)-regular with respect to \(\mathcal{L}\) for all \(p \geq 0\).

Proof

We prove 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 \(H^0\) automatically vanishes. Now assume \(\dim X \geq 1\).

The key is to use the restriction exact sequence for the effective divisor \(D\) defined by a global section \(s \in H^0(X, \mathcal{L})\) of \(\mathcal{L}\). Choosing a general \(s\), by Bertini’s theorem \(D\) is smooth, and we obtain the following short exact sequence.

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

The cohomology long exact sequence of this sequence gives the following.

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

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 \(\mathbb{P}^{n-1}\).

Step 1: \(m\)-regularity of \(\mathcal{F}\vert_D\). Since \(\mathcal{F}\) is \(m\)-regular with respect to \(\mathcal{L}\), \(H^i(\mathcal{F} \otimes \mathcal{L}^{\otimes m-i}) = 0\) holds for \(i > 0\). Let us show that \(\mathcal{F}\vert_D\) is \(m\)-regular with respect to \(\mathcal{L}\vert_D\). Substituting \(k = m - i\) in the restriction sequence (\(0 < i \leq n-1\))

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

and from its long exact sequence

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

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 \(\mathcal{L}\vert_D\).

Step 2: \(\mathcal{F} \otimes \mathcal{L}^{\otimes m}\) is globally generated. We apply the inductive hypothesis to \(D\). \(D\) 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, by the inductive hypothesis \(\mathcal{F}\vert_D \otimes (\mathcal{L}\vert_D)^{\otimes m}\) is globally generated on \(D\).

Now we show that \(\mathcal{F} \otimes \mathcal{L}^{\otimes m}\) is globally generated. It suffices to check that the fiber \((\mathcal{F} \otimes \mathcal{L}^{\otimes m})_p\) at an arbitrary point \(p \in X\) is generated by the image of global sections. Choose a general divisor \(D\) passing through \(p\), and substitute \(k = m\) in the restriction sequence:

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

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

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

is surjective. By the inductive hypothesis, \(\mathcal{F}\vert_D \otimes (\mathcal{L}\vert_D)^{\otimes m}\) is globally generated on \(D\), so this fiber at \(p\) 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 \(p\). Therefore \(\mathcal{F} \otimes \mathcal{L}^{\otimes m}\) is globally generated.

Step 3: \(\mathcal{F} \otimes \mathcal{L}^{\otimes p}\) is \((m+p)\)-regular. Since \(\mathcal{F} \otimes \mathcal{L}^{\otimes m}\) is globally generated, there exists a surjection

\[\mathcal{O}_X^{\oplus r_0} \twoheadrightarrow \mathcal{F} \otimes \mathcal{L}^{\otimes m}\]

for suitable \(r_0\). Tensoring this with \(\mathcal{L}^{\otimes p}\), we obtain

\[\mathcal{L}^{\oplus r_0} \twoheadrightarrow \mathcal{F} \otimes \mathcal{L}^{\otimes m+p}\]

Thus, if \(H^i(X, \mathcal{L}^{\otimes p}) = 0\) for any \(i > 0\) and \(p \geq 0\), then \(H^i(\mathcal{F} \otimes \mathcal{L}^{\otimes m+p}) = 0\) holds. For \(p = 0\), \(H^i(\mathcal{F} \otimes \mathcal{L}^{\otimes m}) = 0\) (\(i > 0\)) is exactly the definition of \(m\)-regularity of \(\mathcal{F}\). For \(p \geq 1\), since \(\mathcal{L}\) is ample, by Proposition 4 (Serre Vanishing) we have \(H^i(\mathcal{L}^{\otimes p}) = 0\) for sufficiently large \(p\), but for small \(p\) this factor may not vanish.

To resolve this issue, we use induction on \(p\). For \(p = 0\), the fact that \(\mathcal{F}(m)\) is \(m\)-regular is the definition. Assume \(p \geq 1\), and we show that \(\mathcal{F}(m+p)\) 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

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

By the inductive hypothesis (for \(p-1\)), \(H^1(\mathcal{F} \otimes \mathcal{L}^{\otimes m+p-2}) = 0\). Also, since \(\mathcal{F}\vert_D\) is \(m\)-regular (Step 2), by the inductive hypothesis on dimension \(\mathcal{F}\vert_D \otimes (\mathcal{L}\vert_D)^{\otimes p}\) is \((m+p)\)-regular, and therefore \(H^1(\mathcal{F}\vert_D \otimes (\mathcal{L}\vert_D)^{\otimes m+p-1}) = 0\). In the exact sequence, \(H^1(\mathcal{F} \otimes \mathcal{L}^{\otimes m+p-1})\) is embedded in \(H^1(\mathcal{F}\vert_D \otimes (\mathcal{L}\vert_D)^{\otimes m+p-1})\), so we obtain \(H^1(\mathcal{F} \otimes \mathcal{L}^{\otimes m+p-1}) = 0\). For \(i \geq 2\), from the same restriction sequence

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

By the inductive hypothesis, \(H^i(\mathcal{F} \otimes \mathcal{L}^{\otimes m+p-i-1}) = 0\) (hypothesis for \(p' = p-1\), \(j = i\)), and by the inductive hypothesis on dimension for \(\mathcal{F}\vert_D\), \(H^{i-1}(\mathcal{F}\vert_D \otimes (\mathcal{L}\vert_D)^{\otimes m+p-i}) = 0\) and \(H^i(\mathcal{F}\vert_D \otimes (\mathcal{L}\vert_D)^{\otimes m+p-i}) = 0\) hold for \(i-1 \geq 1\), \(i \leq n-1\). Therefore we obtain \(H^i(\mathcal{F} \otimes \mathcal{L}^{\otimes m+p-i}) = 0\).

Example 8 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 must check \(H^i(\mathcal{O}(d-i))\); when \(i = 1\), \(H^1(\mathcal{O}(d-1))\) is \(0\) if \(d \geq 1\), and if \(d = 0\) then \(H^1(\mathcal{O}(-1)) = 0\) (by Bott’s formula, since \(-1 \geq -n\), all cohomology is \(0\)). 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 anyway \(H^i = 0\). Therefore \(\mathcal{O}(d)\) 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 7 (Castelnuovo-Mumford Regularity), \(\mathcal{O}(d) \otimes \mathcal{L}^{\otimes 0} = \mathcal{O}(d)\) is globally generated when \(d \geq 0\), which is consistent with what we checked in §Line Bundles and Vector Bundles, ⁋Example 16.

Properties of Very Ample and Ample

The above Proposition 4 (Serre Vanishing) and Proposition 7 (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 9 If \(\mathcal{L}\) is very ample and \(\mathcal{M}\) is globally generated, 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, there exist global sections \(s_0, \ldots, s_n \in H^0(X, \mathcal{M})\) that generate the stalk at every point, and from these we can define a morphism \(\phi: X \to \mathbb{P}^n\).

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

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

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

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

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

That is, although the explanation was somewhat complicated, the key point is that the morphism \(\phi:X\rightarrow \mathbb{P}^n\) defined by a globally generated line bundle \(\mathcal{M}\) may not be a closed embedding, but by tensoring it 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 10 For an ample line bundle \(\mathcal{L}\) defined on a projective variety \(X\) and an arbitrary line bundle \(\mathcal{M}\), for sufficiently large \(n\) the sheaf \(\mathcal{M} \otimes \mathcal{L}^{\otimes n}\) is very ample.

Proof

First, since \(\mathcal{L}\) is ample, for some suitable \(m>0\) the sheaf \(\mathcal{L}^{\otimes m}\) is very ample. On the other hand, by Proposition 4 (Serre Vanishing) we can make the higher cohomology of \(\mathcal{M}\otimes \mathcal{L}^{\otimes k}\) vanish for sufficiently large \(k\), so for such \(k\) the sheaf \(\mathcal{M}\otimes \mathcal{L}^{\otimes k}\) is globally generated. Now by Proposition 9,

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

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

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References

[Har] 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.