1 Introduction

Horrocks’ famous splitting theorem [3] on \({\mathbb P}^n\) says that a vector bundle \({\mathcal F}\) on \({\mathbb P}^n\) splits into a direct sum of line bundles

$$\begin{aligned} {\mathcal F}\cong \bigoplus _j {\mathcal O}(k_j) \end{aligned}$$

if and only if \({\mathcal F}\) has no intermediate cohomology, i.e., if

$$\begin{aligned} H^i({\mathbb P}^n,{\mathcal F}(k))=0 \quad \forall \, k \in {\mathbb Z}\, \hbox { and } \forall i \hbox { with }\, 0< i <n. \end{aligned}$$

In this note we prove a similar criterion for Segre–Veronese varieties

$$\begin{aligned} {\mathbb P}^{n_1}\times \cdots \times {\mathbb P}^{n_t} \hookrightarrow {\mathbb P}^N \end{aligned}$$

embedded by the complete linear system of a very ample line bundle \({\mathcal O}(H)={\mathcal O}(d_1,\ldots ,d_t)\), so \(N= (\prod _{j=1}^t {n_j+d_j \atopwithdelims ()n_j}) -1\).

Theorem 0.1

Let \({\mathcal O}(H) = {\mathcal O}(d_1,\ldots ,d_t)\) be a very ample line bundle on a product of projective spaces \({\mathbb P}={\mathbb P}^{n_1}\times \cdots \times {\mathbb P}^{n_t}\) of dimension \(m=n_1 +\cdots +n_t\) with \(t \ge 2\) factors. A torsion free sheaf \({\mathcal F}\) on \({\mathbb P}\) splits into a direct sum \({\mathcal F}\cong \bigoplus _j {\mathcal O}(k_jH)\) if and only if

$$\begin{aligned} \forall \quad i \in \{1,\ldots ,m-1\} \quad H^i({\mathbb P},{\mathcal F}(a_1,\ldots ,a_t))=0\ \end{aligned}$$

for all twists with \({\mathcal O}(a_1,\ldots ,a_t)\) such that the cohomology groups \(H^i({\mathbb P},{\mathcal O}(kH)\otimes {\mathcal O}(a_1,\ldots ,a_t))\) vanish for all \(i \in \{1,\ldots ,m-1\} \) and all \(k \in {\mathbb Z}\).

We can rephrase the theorem as follows: If a torsion free sheaf \({\mathcal F}\) on a product \({{\mathbb P}^{n_1} \times \cdots \times {\mathbb P}^{n_t}}\) has no intermediate cohomology in the range where the sheaves \({\mathcal O}(kH)\) have no intermediate cohomology, then it is a direct sum of these sheaves.

Example 0.2

For \({\mathbb P}={\mathbb P}^{n_1}\times {\mathbb P}^{n_2}\), the line bundle \({\mathcal O}(a_1,a_2)\) has a nonzero cohomology group

$$\begin{aligned} \begin{matrix} H^{n_1}({\mathcal O}(a_1,a_2))\not =0, &{} H^0({\mathcal O}(a_1,a_2))\not =0, \\ H^{m}({\mathcal O}(a_1,a_2))\not =0 , &{} \;H^{n_2}({\mathcal O}(a_1,a_2))\not =0, \end{matrix} \end{aligned}$$

for \(a=(a_1,a_2)\) in the range

$$\begin{aligned} \begin{matrix} \{a_1< -n_1, a_2 \ge 0 \}, &{} \{ a_1\ge 0, a_2\ge 0\} \\ \{ a_1< -n_1, a_2< -n_2 \} , &{} \{ a_1 \ge 0, a_2 < -n_2 \} \end{matrix} \end{aligned}$$

respectively and is zero otherwise.

In particular for \({\mathbb P}^{2}\times {\mathbb P}^{3}\) and the area \(\{-5 \le a_1 \le 1,-5 \le a_2 \le 2 \}\), nonzero cohomology and nonzero intermediate cohomology occur in the shaded regions

Thus for \({\mathcal O}(H)={\mathcal O}(4,2)\), the assumption of the theorem in this case is that the intermediate cohomology occurs only in a range as indicated in the area \(\{ -8 \le a_1 \le 12, -4\le a_2 \le 6 \}\) by the shaded region below:

Remark 0.3

For any coherent sheaf \({\mathcal F}\) on \({\mathbb P}\) the condition

$$\begin{aligned} H^i({\mathbb P},{\mathcal F}(kH))=0 \mathrm{for all } i\in \{1,\ldots ,m-1\} \mathrm{and all } k \in {\mathbb Z}\end{aligned}$$

implies that \({\mathcal F}\) is locally free unless \({\mathcal F}\) has a zero dimensional subsheaf. The condition \({\mathcal F}\) torsion free in Theorem 0.1 is only used to exclude such torsion subsheaves.

2 Preliminaries and notation

The Tate resolutions of a sheaf on products of projective spaces is a generalization of the Tate resolution on \({\mathbb P}^n\) [2]. We recall from [1] the basic notation.

Let \({\mathbb P}={{\mathbb P}^{n_1} \times \cdots \times {\mathbb P}^{n_t}}= {\mathbb P}(W_{1})\times \cdots \times {\mathbb P}(W_{t})\) be a product of t projective spaces over an arbitrary field K. Set \(V_{i} = W_{i}^{*}\) and \(V = \bigoplus _{i}V_{i}\). Let E be the \({\mathbb Z}^{t}\)-graded exterior algebra on V, where elements of \(V_{i}\subset E\) have degree \((0,\dots ,0, -1, 0,\dots ,0)\) with \(-1\) in the i-th place.

For a sheaf \({\mathcal F}\) on \({\mathbb P}\) the Tate resolution \(\mathbf{T}({\mathcal F})\) is a minimal exact complex of graded E-modules with terms

$$\begin{aligned} \mathbf{T}({\mathcal F})^d = \bigoplus _{a\in {\mathbb Z}^t} {{\,\mathrm{Hom}\,}}_K(E,H^{d-|a|}({\mathbb P}, {\mathcal F}(a))), \end{aligned}$$

where the cohomology group \(H^{d-|a|}({\mathbb P},{\mathcal F}(a))\) is regarded as a vector space concentrated in degree a, and \(|a|= \sum _{j=1}^t a_j\) denotes the total degree.

Since \(\omega _E = {{\,\mathrm{Hom}\,}}_K(E,K)\) is the free E-module of rank 1 with socle in degree 0 and hence generator in degree \((n_1+1,\ldots ,n_t+1)\), the differential of the complex \(\mathbf{T}({\mathcal F})\) is given by a matrix with entries in E. More precisely, the component \({{\,\mathrm{Hom}\,}}_K(E,H^{d-|a|}({\mathbb P}, {\mathcal F}(a))) \rightarrow {{\,\mathrm{Hom}\,}}_K(E,H^{d+1-|b|}({\mathbb P}, {\mathcal F}(b)))\) is given by a \(h^{d+1-|b|}({\mathbb P}, {\mathcal F}(b)) \times h^{d-|a|}({\mathbb P}, {\mathcal F}(a))\)-matrix with entries in

$$\begin{aligned} \Lambda ^{b-a}V := \Lambda ^{b_1-a_1} V_1 \otimes \cdots \otimes \Lambda ^{b_t-a_t} V_t. \end{aligned}$$

In particular, if \(b_j< a_j\) for some j, then the corresponding block is zero. Moreover, all blocks corresponding to cases with \(a=b\) are also zero, since \(\mathbf{T}({\mathcal F})\) is a minimal complex.

The complex \(\mathbf{T}({\mathcal F})\) has various exact free subquotient complexes: For \(c\in {\mathbb Z}^t\) a degree and \(I, J, K \subset \{1,\ldots , t \}\) disjoint subsets we have the subquotient complex \(T_c(I,J,K)\) with terms

$$\begin{aligned} {T_c(I,J,K)} ^d= \sum _{\begin{array}{c} a \in {\mathbb Z}^t \\ a_i < c_i \mathrm{for } i \in I \\ a_i = c_i { \mathrm for } i \in J \\ a_i \ge c_i \mathrm{for } i \in K \end{array}} {{\,\mathrm{Hom}\,}}_K(E,H^{d-|a|}({\mathbb P},{\mathcal F}(a))) \end{aligned}$$

By [1, Theorem 3.3 and Corollary 3.5] these complexes are exact as long as \(I \cup J \cup K \subsetneq \{1,\ldots ,t \}\). The complexes \({T_c({\emptyset },J,{\emptyset })}\) can be used to compute the direct image complex of \({\mathcal F}(c)\) along a partial projection \(\pi _J :{\mathbb P}\rightarrow \prod _{j \notin J } {\mathbb P}^{n_j}\) [1, Corollary 0.3 and Proposition 3.6].

Lemma 1.1

Let \({\mathcal F}\) be a coherent sheaf on a product of projective spaces \({\mathbb P}= {\mathbb P}^{n_1} \times \ldots \times {\mathbb P}^{n_t}\) and let \(a=(a_1,a_2,\ldots , a_t)=(a',a_t) \in {\mathbb Z}^t= {\mathbb Z}^{t-1} \times {\mathbb Z}^{}\) and \(n\in {\mathbb Z}\). If

$$\begin{aligned} H^n({\mathcal F}(a',a_t))=H^{n-1}({\mathcal F}(a',a_t+1))=\ldots =H^{n-n_t}({\mathcal F}(a',a_t+n_t))=0 \end{aligned}$$

then \(H^n({\mathcal F}(a',a_t-1))=0\) as well. A similar statement holds for the cohomology along the j-th strand \(T_a({\emptyset },{\{1,\ldots ,t\}{\setminus } \{j\}},{\emptyset })\).

Proof

We consider the strand \(T_a({\emptyset },\{1,\ldots ,t-1\},{\emptyset })\) of \(\mathbf{T}({\mathcal F})\). The differential starting at the summand \({{\,\mathrm{Hom}\,}}_K(E,H^n({\mathcal F}(a',a_t-1)) \subset T_a({\emptyset },\{1,\ldots ,t-1\},{\emptyset })\) maps in the strand to the summands

$$\begin{aligned} {{\,\mathrm{Hom}\,}}_K(E, H^n({\mathcal F}(a',a_t)))\oplus \ldots \oplus {{\,\mathrm{Hom}\,}}_K(E,H^{n-n_t}({\mathcal F}(a',a_t+n_t))). \end{aligned}$$

By assumption the target is zero. Since of \(T_a({\emptyset },\{1,\ldots ,t-1\},{\emptyset })\) is minimal and exact, the source is zero as well.

The proof of the our main theorem uses the corner complexes \(T_{\Rsh c}({\mathcal F})\) which are defined as the cone of a map of complexes

$$\begin{aligned} \varphi _{c}: T_c(\{1,\ldots ,t\},\emptyset ,\emptyset )[-t] \rightarrow T_c(\emptyset ,\emptyset ,\{1,\ldots ,t\}) \end{aligned}$$

from the last quadrant complex to the first quadrant complex. The map \(\varphi _{c}\) is the composition of t maps

$$\begin{aligned} T_c(\{1,\ldots ,k\},\emptyset ,\{k+1,\ldots ,t\})[-k] \rightarrow T_c(\{1,\ldots ,k-1\},\emptyset ,\{k,\ldots ,t\})[-k+1] \end{aligned}$$

each of which is obtained from the differential of \(\mathbf{T}({\mathcal F})\) by taking the terms with source in one quadrant and target in the next quadrant. The corner complexes are exact as well by [1, Theorem 4.3 and Corollary 4.5].

If we follow a path from the last quadrant to the first quadrant using a different order of the elements in the set \(\{1,\ldots ,t \}\), we obtain an isomorphic complex. Indeed, all of these corner complexes are exact and their differentials

$$\begin{aligned} T_{\Rsh c}({\mathcal F})^d \rightarrow T_{\Rsh c}({\mathcal F})^{d+1} \end{aligned}$$

coincide for sufficiently large cohomological degree d, since those differentials involve only terms from the first quadrant \(T_c(\emptyset ,\emptyset ,\{1,\ldots ,t\})\).

3 Proof of the main result

We use the partial order \(a \ge b\) on \({\mathbb Z}^t\) defined by \(a_j \ge b_j \mathrm{for } j=1,\ldots , t\) and write \(a > b\) if \(a \ge b\) and \(a \not =b\).

Let \({\mathcal F}\) be a coherent sheaf on \({\mathbb P}={{\mathbb P}^{n_1} \times \cdots \times {\mathbb P}^{n_t}}\). If \(H^m({\mathbb P},{\mathcal F}(a)) \not =0\) then \(H^m({\mathbb P},{\mathcal F}(b)) \not =0\) for all \(b \le a\) as we see from applying \(H^m\) to the surjection

$$\begin{aligned} H^0({\mathbb P},{\mathcal O}(a-b)) \otimes {\mathcal F}(b) \rightarrow {\mathcal F}(a). \end{aligned}$$

An extremal \(H^m\)-position of \({\mathcal F}\) is a degree \(a\in {\mathbb Z}^t\) such that \(H^m({\mathbb P},{\mathcal F}(a)) \not = 0\) but \(H^m({\mathbb P},{\mathcal F}(c))=0\) for all \(c > a\).

Proposition 2.1

Let \({\mathcal F}\) be a torsion free sheaf on \({{\mathbb P}^{n_1} \times \cdots \times {\mathbb P}^{n_t}}\) satisfying the assumption of Theorem 0.1 with respect to \({\mathcal O}(H) = {\mathcal O}(d_1,\ldots ,d_t)\). There exists an extremal \(H^m\)-position for \({\mathcal F}\) of the form

$$\begin{aligned} (a_1,\ldots ,a_t)=(kd_1-n_1-1,\ldots ,kd_t-n_t-1) \end{aligned}$$

for some \(k \in {\mathbb Z}\).

Note that \({\mathcal O}(-n_1-1,\ldots ,-n_t-1) \cong \omega _{\mathbb P}\) is the canonical sheaf on \({\mathbb P}\).

Proof

Since \({\mathcal F}\) is nonzero and torsion free, we have \(H^m({\mathcal F}(kH)\otimes \omega _{\mathbb P}) \not =0\) for \(k\ll 0\) and \(H^m({\mathcal F}(kH) \otimes \omega _{\mathbb P}) =0\) for \(k \gg 0\). Let k be the maximum such that \(H^m({\mathcal F}(kH)\otimes \omega _{\mathbb P}) \not =0\). We claim that \((kd_1-n_1-1,\ldots ,kd_t-n_t-1)\) is an extremal \(H^m\)-position. Suppose it is not. Then there exists a maximal a in the range

$$\begin{aligned} (kd_1-n_1-1, \ldots ,kd_t-n_t-1) < a \le ((k+1)d_1-n_1-1, \ldots ,(k+1)d_t-n_t-1) \end{aligned}$$

such that \(H^m({\mathcal F}(a)) \not =0\). At least for one i we have \(kd_i-n_i-1 < a_i\). Then for any \(j \not =i\) we consider \(J=\{1,\ldots ,t\}{\setminus } \{j\}\) and look at the j-th strand \({T_a({\emptyset },J,{\emptyset })}\) through a. Lemma 1.1 implies \(a_j=(k+1)d_j-n_j-1\): If \(a_j<(k+1)d_j-n_j-1\), then we cannot reach the intermediate cohomology range of \({\mathcal F}\) after at most \(n_j+1\) steps along this strand, contradicting the maximality of a. Starting with \(kd_j-n_j-1 < a_j=(k+1)d_j-n_j-1\) and interchanging the role of i and j in the argument above, we deduce \(a_i=(k+1)d_i-n_i-1\) for all i. This is a contradiction to the maximality of k.

Proposition 2.2

Let \({\mathcal F}\) be a torsion free sheaf on \({{\mathbb P}^{n_1} \times \cdots \times {\mathbb P}^{n_t}}\) satisfying the assumption of Theorem 0.1 with respect to \({\mathcal O}(H) = {\mathcal O}(d_1,\ldots ,d_t)\). If

$$\begin{aligned} (kd_1-n_1-1,\ldots ,kd_t-n_t-1) \end{aligned}$$

is an extremal \(H^m\)-position for \({\mathcal F}\), then

$$\begin{aligned} {\mathcal F}\cong {\mathcal O}(kH) \oplus {\mathcal F}'. \end{aligned}$$

Proof

We consider the corner complex \(T_{\Rsh c}({\mathcal F})\) for \(c=(kd_1-n_1,\ldots ,kd_t-n_t)\). The first part of the corner map

$$\begin{aligned} T_c(\{1,\ldots ,t\},\emptyset ,\emptyset )[-t] \rightarrow T_c(\{1,\ldots ,t-1\},\emptyset ,\{t\})[-t+1] \end{aligned}$$

with source \({{\,\mathrm{Hom}\,}}_K(E,H^m({\mathcal F}(kH) \otimes \omega _{\mathbb P})\) is a map

$$\begin{aligned} {{\,\mathrm{Hom}\,}}_K(E,H^m({\mathcal F}(kH) \otimes \omega _{\mathbb P})) \rightarrow {{\,\mathrm{Hom}\,}}_K(E,H^{m-n_t}({\mathcal F}\otimes \omega _{\mathbb P}\otimes {\mathcal O}(0,\ldots ,0,n_t+1))) \end{aligned}$$

given by a matrix with entries in \(\Lambda ^{n_t+1} V_t\). Indeed, \(H^m(({\mathcal F}\otimes \omega _{\mathbb P}\otimes {\mathcal O}(0,\ldots ,0,1))=0\) holds since \((kd_1-n_1-1,\ldots ,kd_t-n_t-1)\) is extremal. Since the map follows the strand \(T_c(\emptyset ,\{1,\ldots ,t-1\},\emptyset )\), the group \(H^{m-n_t}({\mathcal F}\otimes \omega _{\mathbb P}\otimes {\mathcal O}(0,\ldots ,0,n_t+1)\) is the first possible non-zero intermediate cohomology group by assumption. Composed with the second part of the corner complex

$$\begin{aligned} T_c(\{1,\ldots ,t-1\},\emptyset ,\{t\})[-t+1] \rightarrow T_c(\{1,\ldots ,t-2\},\emptyset ,\{t-1,t\})[-t+2] \end{aligned}$$

the image is in

$$\begin{aligned} {{\,\mathrm{Hom}\,}}_K(E,H^{m-n_t-n_{t-1}}({\mathcal F}\otimes \omega _{\mathbb P}\otimes {\mathcal O}(0,\ldots ,0,n_{t-1}+1,n_t+1))) \end{aligned}$$

since \(\Lambda ^{n_t+2} V_t=0\) and other possible intermediate cohomology groups in the strand \(T_c(\emptyset ,\{1,\ldots ,t-2\}, \emptyset )\) vanish by assumption. Repeating these arguments, we conclude that the corner map with source \({{\,\mathrm{Hom}\,}}_K(E,H^m({\mathcal F}(kH) \otimes \omega _{\mathbb P})\) has an image only in \({{\,\mathrm{Hom}\,}}_K(E,H^0({\mathcal F}(kH))\). It is given by an

$$\begin{aligned} h^0({\mathcal F}(kH)) \times h^m({\mathcal F}(kH) \otimes \omega _{\mathbb P})\mathrm{-matrix} \end{aligned}$$

with entries in the one-dimensional space

$$\begin{aligned} \Lambda ^{m+t} V =\Lambda ^{n_1+1} V_1 \otimes \cdots \otimes \Lambda ^{n_t+1} V_t. \end{aligned}$$

Consider the submatrix of the differential in the corner complex with target equal to the summand \({{\,\mathrm{Hom}\,}}_K(E,H^0({\mathcal F}(kH))\). The only other subspaces in the source which have this target come from \(H^0\)-groups:

$$\begin{aligned} {{\,\mathrm{Hom}\,}}_K(E,H^0({\mathcal F}(kH)(-1,0,\ldots ,0)), \ldots , {{\,\mathrm{Hom}\,}}_K(E,H^0({\mathcal F}(kH)(0,\ldots ,0,-1)). \end{aligned}$$

Thus this differential is given by an

$$\begin{aligned} h^0({\mathcal F}(kH)) \times [(h^m({\mathcal F}(kH)\otimes \omega _{\mathbb P}) + h^0({\mathcal F}(kH)\otimes {\mathcal B}) ) ]\mathrm{-matrix} \end{aligned}$$

with \({\mathcal B}= {\mathcal O}(-1,0,\ldots ,0) \oplus \ldots \oplus {\mathcal O}(0,\ldots ,0,-1)\). Note that \(h^0({\mathcal F}(kH)) \ge h^m({\mathcal F}(kH)\otimes \omega _{\mathbb P})\), because otherwise a generator of \({{\,\mathrm{Hom}\,}}_K(E,H^m({\mathcal F}(kH) \otimes \omega _{\mathbb P})\) would map to zero which is impossible because \(T_{\Rsh c}({\mathcal F})\) is exact and minimal. Thus in a suitable basis the matrix has shape

with \(v \in \Lambda ^{m+t} V\) a fixed basis element and \( \ell _{ij} \in V_1 \cup \cdots \cup V_t\).

We claim now that \(\ell _{1j}\) is a K-linear combination of \(\ell _{r+1j}, \ldots , \ell _{sj}\). Indeed if not, we could multiply the j-th column by a decomposable element \(w \in \Lambda ^{m+t-1} V\) which annihilates \(\ell _{r+1j}, \ldots , \ell _{sj}\) but does not annihilate \(\ell _{1j}\), so that \(\ell _{1j}w =v\). This would give us a column

for possibly zero scalars \(\lambda _2, \ldots , \lambda _r\), and the first column would be an E-linear combination of columns 2 to j. This is impossible since no generator can map to zero in \(T_{\Rsh c}({\mathcal F})\).

Let \(r_1-r\) denote the dimension of the linear span of \(\ell _{r+1j},\ldots ,\ell _{sj}\). Then after row operations we may assume that \(\varphi \) has the shape

with \(\ell _{r+1j},\ldots \ell _{r_1j}\) K-linearly independent.

Next we note that the columns of the matrix

are in the E-column span of \(\varphi \). Arguing as before, we see that \(\ell _{1j+1}\) is a linear combination of \(\ell _{r_1+1j+1},\ldots ,\ell _{sj+1}\), and repeating the arguments, we find that \(\varphi \) can be transformed by row operations into a matrix of type

We conclude that \(T_{\Rsh c}({\mathcal O}(kH))\) is a direct summand of the complex \(T_{\Rsh c}({\mathcal F})\), and

$$\begin{aligned} {\mathcal F}\cong {\mathcal O}(kH)\oplus {\mathcal F}', \end{aligned}$$

since we can recover \({\mathcal F}\) from its corner complex with the Beilinson functor \(\mathbf{U}\) applied to \(T_{\Rsh c}({\mathcal F})(a)[|a|]\) for a suitable \(a \in {\mathbb Z}^t\) by [1, Theorem 0.1]. Indeed \(\mathbf{U}(T_{\Rsh c}({\mathcal F})(a)[|a|])\) and \(\mathbf{U}(\mathbf{T}({\mathcal F})(a)[|a|])\) coincide for \( a\gg 0\).

Proof of Theorem. Let \({\mathcal F}\) be a torsion free sheaf on \({{\mathbb P}^{n_1} \times \cdots \times {\mathbb P}^{n_t}}\) with no intermediate cohomology where the sheaves \({\mathcal O}(kH)\) for \({\mathcal O}(H)={\mathcal O}(d_1,\ldots ,d_t)\) have no intermediate cohomology. By Proposition 2.1 there is an extremal \(H^m\)-position of \({\mathcal F}\) of the form

$$\begin{aligned} (k_1d_1-n_1-1,\ldots ,k_1d_t-n_t-1) \end{aligned}$$

and by Proposition 2.2 we get a summand

$$\begin{aligned} {\mathcal F}\cong {\mathcal O}(k_1H) \oplus {\mathcal F}'. \end{aligned}$$

If \({{\,\mathrm{rank}\,}}{\mathcal F}=1\), we are done: \({\mathcal F}'=0\) since \({\mathcal F}\) is torsion free. Otherwise we can argue by induction on the rank since \({\mathcal F}'\) satisfies the assumption of the Theorem again. \(\square \)