1 Introduction

In the last few decades, a number of notions of positivity have been used to understand the geometry of higher dimensional projective varieties. The cone of nef \(\mathbb {R}\)-divisors on a smooth projective variety X, denoted by \(\mathrm{Nef}^{\,1}(X)\), and the closure of the cone generated by effective \(\mathbb {R}\)-divisors on X called a pseudo-effective cone, denoted by \(\overline{\mathrm{Eff}}{}^{\,1}(X)\), are two fundamental invariants of X which play a very crucial role in this understanding.

Most of the developments in this direction have been successfully summarized in [17, 18]. In [29], Zariski proved that for any effective divisor D on a smooth projective surface X, there exist uniquely determined \(\mathbb {Q}\)-divisors P and N with \(D = P + N\) satisfying the following properties:

  1. (i)

    P is nef and N is effective.

  2. (ii)

    Either N is zero or has negative definite intersection matrix of its components.

  3. (iii)

    for every irreducible component C of N.

Fujita [9] later extended the above decomposition for pseudo-effective \(\mathbb {R}\)-divisors on surfaces. This decomposition gives a great deal of information about the linear series on a smooth surface. For example, such decomposition is used as a very powerful tool to prove the rationality of the volume of integral divisors on surfaces. Many attempts have been made to generalize this decomposition to higher dimensional varieties, for instance, the Fujita–Zariski decomposition [10] and the CKM–Zariski decomposition [28]. In fact, the rationality of the volume of a big divisor admitting a CKM–Zariski decomposition is proved similarly. However, Cutkosky [7] constructed examples of effective big divisors on higher dimensional varieties with irrational volume, which proved that a CKM–Zariski decomposition on a smooth projective variety does not exist in general. Nakayama [27] constructed a related example to show that it is impossible to find a CKM–Zariski decomposition even if one allows N and P to be \(\mathbb {R}\)-divisors in the definition of CKM–Zariski decomposition.

Definition 1.1

(Weak Zariski Decomposition) A weak Zariski decomposition for an \(\mathbb {R}\)-divisor D on a normal projective variety X consists of a normal variety \(X'\), a projective bi-rational morphism \(f :X'\! \rightarrow X\), and a numerical equivalence \(f^*D \equiv P + N\) such that \(P \in \mathrm{Nef}^{\,1}(X')\) and \(N \in {\mathrm{Eff}}{}^{\,1}(X')\).

The existence of weak Zariski decomposition also has very useful consequences. For example, in [1], it is shown that the existence of weak Zariski decomposition for adjoint divisors \(K_X + B\) of a log canonical pair (X, B) is equivalent to the existence of log minimal model for (X, B). However, in [19] an example is found of a pseudo-effective \(\mathbb {R}\)-divisor on the blow-up of \(\mathbb {P}^3\) in nine general points, which does not admit weak Zariski decomposition.

The existence of weak Zariski decomposition of pseudo-effective \(\mathbb {R}\)-divisors on projective bundles \(\mathbb {P}_X(E)\) was studied in [25, 27], when the base space X is a smooth curve and a smooth projective variety with Picard number 1 respectively. In this paper, we consider a vector bundle E of rank r on a smooth projective variety X together with the projectivization map \(\pi :\mathbb {P}_X(E) \rightarrow X\). We investigate the existence of weak Zariski decomposition for pseudo-effective \(\mathbb {R}\)-divisors on \(\mathbb {P}_X(E)\). By [19], the numerical classes of \(\mathbb {R}\)-divisors in a smooth projective variety X admitting weak Zariski decomposition form a cone in \(\overline{\mathrm{Eff}}{}^{\,1}(X)\), and hence it is enough to prove existence of such decomposition for every extremal ray of the pseudo-effective cone \(\overline{\mathrm{Eff}}{}^{\,1}(X)\) (see [25, Proposition 1]). In particular, if \({\mathrm{Eff}}{}^{\,1}(X)\) is a closed cone (e.g., when X is a Mori dream space), then weak Zariski decomposition always exists. Also, if \(\mathrm{Nef}^{\,1}(\mathbb {P}_X(E)) = \overline{\mathrm{Eff}}{}^{\,1}(\mathbb {P}_X(E))\) for some vector bundle E on a specific X, then weak Zariski decomposition always exists on \(\mathbb {P}_X(E)\) because all the extremal rays are nef generators in this case. However, pseudo-effectivity does not imply nefness in \(\mathbb {P}_X(E)\) in general.

In his paper [22], Miyaoka found that in characteristic 0, a vector bundle E on a smooth projective curve C is semistable if and only if the nef cone \(\mathrm{Nef}^{\,1}(\mathbb {P}_C(E))\) and pseudo-effective cone \(\overline{\mathrm{Eff}}{}^{\,1}(\mathbb {P}_C(E))\) coincide. However, examples are given in [3] to show that this characterization of semistability in terms of equality of nef and pseudo-effective cones is not true for vector bundles on higher dimensional projective varieties. More specifically, there are examples of semistable bundles E on any smooth projective variety X with not numerically zero, but \(\mathrm{Nef}^{\,1}(\mathbb {P}_X(E)) \subsetneq \overline{\mathrm{Eff}}{}^{\,1}(\mathbb {P}_X(E))\) (see [3, Section 5]). In Sect. 3, we prove the following result.

Theorem 1.2

Let E be a semistable vector bundle of rank r on a smooth complex projective variety X with , and \(\pi :\mathbb {P}_X(E) \rightarrow X\) be the projectivization map. Denote . Then the following are equivalent:

  1. (i)

    \(\mathrm{Nef}^{\,1}(X) = \overline{\mathrm{Eff}}{}^{\,1}(X)\).

  2. (ii)

    for every effective divisor D in \(\mathbb {P}_X(E)\).

  3. (iii)

    \(\mathrm{Nef}^{\,1}(\mathbb {P}_X(E)) = \overline{\mathrm{Eff}}{}^{\,1}(\mathbb {P}_X(E))\).

In addition, if \(\mathrm{Nef}^{\,1}(X) = \overline{\mathrm{Eff}}{}^{\,1}(X)\) is a finite polyhedron generated by nef classes \(L_1, L_2, \dots ,L_k\), then

$$\begin{aligned} \mathrm{Nef}^{\,1}(\mathbb {P}_X(E)) = \overline{\mathrm{Eff}}{}^{\,1}(\mathbb {P}_X(E)) = \bigl \{y_0\lambda _E + y_1\pi ^*L_1+\cdots +y_k\pi ^*L_k\mid y_i \in \mathbb {R}_{\geqslant 0}\bigr \}, \end{aligned}$$

In general, the nef cones \(\mathrm{Nef}^{\,1}(\mathbb {P}_X(E))\) might not be a finite polyhedron when the Picard number \(\rho (X)\) is at least 3, and in such cases these cones are not so easy to calculate. As applications of the above result, we show the equality \(\mathrm{Nef}^{\,1}(\mathbb {P}_X(E)) = \overline{\mathrm{Eff}}{}^{\,1}(\mathbb {P}_X(E))\) for semistable bundles E with on different smooth projective varieties X with high Picard numbers (see Corollaries 3.1 and 3.7). For instance, we prove the following:

Corollary 3.8

Let X be a smooth complex projective variety X with \(\overline{\mathrm{Eff}}{}^{\,1}(X) = \mathrm{Nef}^{\,1}(X)\) and \(E_1,E_2,\dots ,E_k\) be finitely many semistable vector bundles on X of ranks \(r_1,r_2,\dots ,r_k\) respectively with for all \(i \in \{1,2,\dots ,k\}\). Then

We also give several examples to illustrate Theorem 1.2. In all these cases, weak Zariski decomposition always exists for pseudo-effective divisors as mentioned in our previous remark. In Sect. 4, we prove the following:

Theorem 4.4

Let E and \(E'\) be two vector bundles of rank m and n respectively on a smooth complex projective curve C. Then weak Zariski decomposition exists for any pseudo-effective \(\mathbb {R}\)-divisors on the fibre product .

Note that where \(\pi _1 :\mathbb {P}_C(E') \rightarrow C\) is the projection map. Here \(\mathbb {P}_C(E')\) has Picard number 2 and unless both E and \(E'\) are semistable bundles on C (see Corollary 3.8). Also, is not a Mori dream space unless \(C = \mathbb {P}^1\). In case the \(C = \mathbb {P}^1\), the fibre product gives an example of a Bott tower of height 3. The proof of Theorem 4.4 makes use of some of the techniques from [25].

Like the cones of divisors, the closure of cones generated by effective k-cycles \(\overline{\mathrm{Eff}}_k(X)\) and its dual \(\mathrm{Nef}^{\,k}(X)\) are also of great importance for any projective variety X. It quite challenging to calculate these cones of higher co-dimensional cycles in higher dimensional varieties (see [5, 6, 8, 11]). Let E be an elliptic curve and \(B = E^{\otimes n}\) be the n-fold product. Then B is an abelian variety. In [8], it is shown that for any \( 2 \leqslant k \leqslant n\), there exist nef cycles of codimension k in \(\mathrm{Nef}^{\,k}(B)\) that are not pseudo-effective. Hence, the inclusion \(\mathrm{Nef}^{\,k}(X) \subseteq \overline{\mathrm{Eff}}{}^{\,k}(X)\) for higher codimensional cycles \((2 \leqslant k \leqslant n)\) does not hold in general. However, in [11], it is proved that a vector bundle E of rank r on a smooth complex projective curve C is semistable if and only if it is k-homogeneous, i.e., \(\overline{\mathrm{Eff}}{}^{\,k}(\mathbb {P}_C(E)) = \mathrm{Nef}^{\,k}(\mathbb {P}_{C}(E))\) for all \(k \in \{1,2,\dots ,r-1\}\). But there is an example of a rank 2 stable bundle E on \(\mathbb {P}^2\) (see [11, Example 3.4]) which is not 1-homogeneous, i.e., \(\overline{\mathrm{Eff}}{}^{\,1}(\mathbb {P}_{\mathbb {P}^2}(E)) \ne \mathrm{Nef}^{\,1}(\mathbb {P}_{\mathbb {P}^2}(E))\). We note that any semistable bundle E on \(\mathbb {P}^2\) with is a trivial bundle up to some twist by a line bundle, and hence can never be stable. Motivated by this observation, we prove the following result in Sect. 5.

Theorem 5.2

Let E be a semistable vector bundle of rank \(r \geqslant 2\) with on a smooth complex projective surface X of Picard number \(\rho (X) = 1\). We denote the numerical class of a fibre of the projection map \(\pi :\mathbb {P}_X(E) \rightarrow X\) by F. If \(L_X \in N^1(X)_{\mathbb {R}}\) denotes the numerical class of an ample generator of the Néron–Severi group \(N^1(X)_{\mathbb {Z}}\), then for all \( 1< k < r\),

Moreover, E is a k-homogeneous bundle on X, i.e., \(\overline{\mathrm{Eff}}{}^{\,k}(\mathbb {P}_X(E)) = \mathrm{Nef}^{\,k}(\mathbb {P}_X(E))\) for all \(1 \leqslant k < r\).

We also obtain similar results for k-homogeneous bundles on some smooth ruled surfaces (see Theorem 5.3). Note that ruled surfaces have Picard number 2.

2 Preliminaries

2.1 Nef cone and pseudo-effective cone

Throughout this article, all the algebraic varieties are assumed to be irreducible. Let X be a projective variety of dimension n and (respectively ) denote the free abelian group generated by k-dimensional (respectively k-codimensional) subvarieties on X. The Chow groups \(A_k(X)\) are defined as the quotient of modulo rational equivalence. When X is a smooth irreducible projective variety, we denote .

Two cycles are said to be numerically equivalent, denoted by \(Z_1\equiv Z_2\), if for all . The numerical groups \( N^k(X)_{\mathbb {R}}\) are defined as the quotient of modulo numerical equivalence. When X is smooth, we define for all k.

Let

be the subgroup of consisting of numerically trivial divisors. The quotient is called the Néron Severi group of X, and is denoted by \(N^1(X)_{\mathbb {Z}}\). The Néron–Severi group \(N^1(X)_{\mathbb {Z}}\) is a free abelian group of finite rank. Its rank, denoted by \(\rho (X)\), is called the Picard number of X. In particular, \(N^1(X)_{\mathbb {R}}\) is called the real Néron–Severi group and . For X smooth, the intersection product induces a perfect pairing

which implies \(N^k(X)_{\mathbb {R}} \simeq (N_k(X)_{\mathbb {R}})^\vee \) for every k satisfying \(0\leqslant k \leqslant n\). The direct sum is a graded \(\mathbb {R}\)-algebra with multiplication induced by the intersection form.

The convex cone generated by the set of all effective k-cycles in \(N_k(X)_\mathbb {R}\) is denoted by \(\mathrm{Eff}_k(X)\) and its closure \(\overline{\mathrm{Eff}}_k(X)\) is called the pseudo-effective cone of k-cycles in X. For any \(0 \leqslant k \leqslant n\), . The nef cone are defined as follows:

An irreducible curve C in X is called movable if there exists an algebraic family of irreducible curves \(\{C_t\}_{t\in T}\) such that \(C = C_{t_0}\) for some \(t_0 \in T\) and \(\bigcup _{t \in T} C_t \subset X\) is dense in X.

A class \(\gamma \in N_1(X)_{\mathbb {R}}\) is called movable if there exists a movable curve C such that \(\gamma = [C]\) in \(N_1(X)_{\mathbb {R}}\). The closure of the cone generated by movable classes in \(N_1(X)_{\mathbb {R}}\), denoted by \(\overline{\mathrm{ME}}(X)\), is called the movable cone. By [2], \(\overline{\mathrm{ME}}(X)\) is the dual cone to \(\overline{\mathrm{Eff}}{}^{\,1}(X)\). Also, always \(\mathrm{Nef}^{\,1}(X) \subseteq \overline{\mathrm{Eff}}{}^{\,1}(X)\) (see [17, Chapter 2]). We refer the reader to [17, 18] for more details about these cones.

2.2 Semistability of vector bundles

Let X be a smooth complex projective variety of dimension n with a fixed ample line bundle H on it. For a torsion-free coherent sheaf of rank r on X, the H-slope of is defined as

A torsion-free coherent sheaf on X is said to be H-semistable if for all subsheaves of . A vector bundle E on X is called H-unstable if it is not H-semistable. For every vector bundle E on X, there is a unique filtration

$$\begin{aligned} 0 = E_k \subsetneq E_{k-1} \subsetneq E_{k-2} \subsetneq \cdots \subsetneq E_{1} \subsetneq E_0 = E \end{aligned}$$

of subbundles of E, called the Harder–Narasimhan filtration of E, such that \(E_i/E_{i+1}\) is an H-semistable torsion-free sheaf for each \(i \in \{ 0,1,2,\dots ,k-1\}\) and \(\mu _H(E_{k-1}/E_{k})> \mu _H(E_{k-2}/E_{k-1})>\cdots > \mu _H(E_{0}/E_{1})\). We define and . We recall the following result from [26] or [4, Theorem 1.2].

Theorem 2.1

Let E be a vector bundle of rank r on a smooth complex projective variety X. Then the following are equivalent:

  1. (i)

    E is semistable and .

  2. (ii)

    .

  3. (iii)

    For every pair of the form \((\phi ,C)\), where C is a smooth projective curve and \(\phi :C \rightarrow X\) is a non-constant morphism, \(\phi ^*(E)\) is semistable.

Since nefness of a line bundle does not depend on the fixed polarization (i.e., fixed ample line bundle H) on X, Theorem 2.1 implies that the semistability of a vector bundle E with is independent of the fixed polarization H. We will not mention about the polarization H from now on whenever we speak of semistability of such bundles E with . We have the following lemmata as easy applications of Theorem 2.1.

Lemma 2.2

Let \(\psi :X \rightarrow Y\) be a morphism between two smooth complex projective varieties and E be a semistable bundle on Y with . Then the pullback bundle \(\psi ^*(E)\) is also semistable with .

Proof

Let \(\phi :C \rightarrow X\) be a non-constant morphism from a smooth curve C to X. If the image of \(\phi \) is contained in any fibre of \(\psi \), then the pullback bundle \(\phi ^*\psi ^*(E)\) is trivial, and hence semistable. Now let us assume that the image is not contained in any fibre of \(\phi \). As E is a semistable bundle on Y with , by Theorem 2.1 the pullback bundle under the non-constant morphism is semistable on C. Hence \(\psi ^*\!\) is a semistable bundle on X with . \(\square \)

Lemma 2.3

Let E be a semistable vector bundle of rank r on a smooth complex projective variety X with . Then for any positive integer m and any line bundle on X, is also a semistable bundle on X with

Proof

It is enough to prove that for any smooth complex projective curve C and any non-constant map \(\phi :C \rightarrow X\), the pullback bundle is a semistable bundle on C. We note that

As E is a semistable bundle on X with , by Theorem 2.1 we have that \(\phi ^*\!E\) is semistable. Hence is also semistable. \(\square \)

The projective bundle \(\mathbb {P}_X(E)\) associated to a vector bundle E over a projective variety X is defined as . We will simply write \(\mathbb {P}(E)\) whenever the base space X is clear from the context.

3 Equality of nef and pseudo-effective cones

We note that for a vector E on a smooth irreducible projective complex variety X, the equality \(\mathrm{Nef}^{\,1}(\mathbb {P}(E)) = \overline{\mathrm{Eff}}{}^{\,1}(\mathbb {P}(E))\) always implies the equality \(\mathrm{Nef}^{\,1}(X) = \overline{\mathrm{Eff}}{}^{\,1}(X)\). However, the converse is not true in general (see [3, Section 5]). In this section, we prove that \(\mathrm{Nef}^{\,1}(\mathbb {P}(E)) = \overline{\mathrm{Eff}}{}^{\,1}(\mathbb {P}(E))\) if and only if \(\mathrm{Nef}^{\,1}(X) = \overline{\mathrm{Eff}}{}^{\,1}(X)\) under the assumption that E is a semistable bundle on X with (cf. Theorem 1.2). We also give several examples, especially on abelian varieties and fibre products of projective bundles to illustrate our results.

Proof of Theorem 1.2

(i) \(\Rightarrow \) (ii) Let D be an effective divisor on \(\mathbb {P}(E)\) such that for some integer m and a line bundle . Then

which implies \(m \geqslant 0\).

Let \(\gamma = [C] \) be a movable class in \(N_1(X)_{\mathbb {R}}\). Then C belongs to an algebraic family of curves \(\{C_t\}_{t\in T}\) such that \(\bigcup _{t \in T} C_t\) covers a dense subset of X. So we can find a curve \(C_{t_1}\) in this family such that

Let \(\eta _{t_1} :\widetilde{C}_{t_1}\! \rightarrow C_{t_1}\) be the normalization of the curve \(C_{t_1}\) and we call where \(i :C_{t_1} \!\hookrightarrow X\) is the inclusion.

As E is a semistable bundle on X with , is also semistable on X and by Lemma 2.3. Therefore by Theorem 2.1 we have is also semistable on \(\widetilde{C}_{t_1}\).

Since \(\eta _{t_1}\) is a surjective map, we have

This implies that , and hence for a movable class \(\gamma \in N_1(X)_{\mathbb {R}}\).

Using the duality property of movable cone \(\overline{\mathrm{ME}}(X)\) we conclude that

(ii) \(\Rightarrow \) (iii) We will show that every effective divisor D in \(\mathbb {P}(E)\) is nef. Let for some positive integer m and a line bundle .

Now,

so that . Since E is a semistable bundle on X with , by Theorem 2.1 we have .

Hence,

Therefore, \(\mathrm{Eff}^1(\mathbb {P}(E)) \subset \mathrm{Nef}^{\,1}(\mathbb {P}(E)) \subset \overline{\mathrm{Eff}}{}^{\,1}(\mathbb {P}(E))\). Taking closure, we get \(\mathrm{Nef}^{\,1}(\mathbb {P}(E)) = \overline{\mathrm{Eff}}{}^{\,1}(\mathbb {P}(E))\).

(iii) \(\Rightarrow \) (i) We claim that \(\mathrm{Nef}^{\,1}(X) = \overline{\mathrm{Eff}}{}^{\,1}(X)\). If not then there is an effective divisor D in X which is not nef. Therefore the pullback \(\pi ^*(D)\) is also effective. Since \(\pi \) is a surjective proper morphism, \(\pi ^*(D)\) can never be nef, which contradicts that \(\mathrm{Nef}^{\,1}(\mathbb {P}(E)) = \overline{\mathrm{Eff}}{}^{\,1}(\mathbb {P}(E))\). Hence we are done.

In addition, let \(\mathrm{Nef}^{\,1}(X) = \overline{\mathrm{Eff}}{}^{\,1}(X)\) be a finite polyhedron generated by nef classes \(L_1,L_2,\dots ,L_k\). Then for any effective divisor D on \(\mathbb {P}(E)\) such that for some positive integer m and , we have . Let (say) for some non-negative real numbers \(x_i\)’s. The above calculation then shows that

This shows that

$$\begin{aligned} \mathrm{Eff}^1(\mathbb {P}(E)) \subseteq \bigl \{y_0\lambda _E + y_1\pi ^*\!L_1+\cdots +y_k\pi ^*\!L_k\mid y_i \in \mathbb {R}_{\geqslant 0}\bigr \} \subseteq \mathrm{Nef}^{\,1}(\mathbb {P}(E)). \end{aligned}$$

Therefore

$$\begin{aligned} \mathrm{Nef}^{\,1}(\mathbb {P}(E)) = \overline{\mathrm{Eff}}{}^{\,1}(\mathbb {P}(E)) = \bigl \{y_0\lambda _E + y_1\pi ^*\!L_1+\cdots +y_k\pi ^*\!L_k\mid y_i \in \mathbb {R}_{\geqslant 0}\bigr \}. \end{aligned}$$

\(\square \)

Corollary 3.1

Let X be a smooth complex projective variety of dimension n with Picard number \(\rho (X) = 1\) and E be a semistable vector bundle of rank r on X with . Then

where \(L_X\) is the numerical class of an ample generator of the Neŕon–Severi group \(N^1(X)_{\mathbb {Z}}\simeq \mathbb {Z}\).

Proof

Let D be an effective divisor on X and \(D\equiv mL_X\) for some \(m\in \mathbb {Z}\). Then . As \(L_X\) is ample, by the Nakai criterion for ampleness we have \(L_X^{n} > 0\). This shows that \(m \geqslant 0\) and \(D \equiv mL_X \in \mathrm{Nef}^{\,1}(X)\). So \(\mathrm{Nef}^{\,1}(X) = \overline{\mathrm{Eff}}{}^{\,1}(X)\) and hence the result follows from Theorem 1.2.\(\square \)

Remark 3.2

Miyaoka [22, Theorem 3.1] proved that a vector bundle E of rank r on a smooth complex projective curve C is semistable if and only if

where , f is the numerical class of a fibre of \(\pi :\mathbb {P}(E)\rightarrow C\) and is an ample generator of \(N^1(C)_{\mathbb {Z}} \simeq \mathbb {Z}\). The smooth curve C has Picard number 1 and any second Chern class vanishes on C. So Corollary 3.1 can be thought as a partial generalization of Miyaoka’s result in higher dimensions.

Example 3.3

Let X be a smooth projective variety with Picard number \(\rho (X) = 1\), and be a rank two bundle on X such that (here \(L_X\) is the ample generator for \(N^1(X)_{\mathbb {Z}}\)). Then E is semistable with . Therefore,

Example 3.4

Let X be a smooth complex projective variety with Picard number 1 and . Then for any line bundle L on X, there is a non-split extension

$$\begin{aligned} 0 \rightarrow L \rightarrow E \rightarrow L \rightarrow 0. \end{aligned}$$

In this case, E is a semistable bundle of rank 2 with . Moreover, for any positive integer r, the vector bundles of the forms and \(E^{\oplus r}\) are examples of semistable bundles of ranks \(2r+1\) and 2r respectively with . In all these cases, nef cone and pseudo-effective cones of divisors coincide.

Example 3.5

Let G be a connected algebraic group acting transitively on a complex projective variety X. Then every effective divisor on X is nef, i.e., \(\mathrm{Nef}^{\,1}(X) = \overline{\mathrm{Eff}}{}^{\,1}(X)\). This is because any irreducible curve \(C\subset X\) meets the translate gD of an effective divisor D properly for a general element \(g \in G\). Since G is connected, \(gD \equiv D\). Therefore , and hence D is nef. Examples of such homogeneous varieties X include smooth abelian varieties, flag manifolds etc. So for any semistable vector bundle E on such a homogeneous space X with , using Theorem 1.2 we have \(\mathrm{Nef}^{\,1}(\mathbb {P}(E)) = \overline{\mathrm{Eff}}{}^{\,1}(\mathbb {P}(E))\). For example, let C be a general elliptic curve and be the self product. Then X is an abelian surface and \(\mathrm{Nef}^{\,1}(X)\) is a non-polyhedral cone (see [17, Lemma 1.5.4]). Let \(p_i :X \rightarrow C\) be the projection maps. For any semistable vector bundle F of rank r on C, the pullback bundle is a semistable bundle with . Hence \(\mathrm{Nef}^{\,1}(\mathbb {P}(E)) = \overline{\mathrm{Eff}}{}^{\,1}(\mathbb {P}(E))\).

Example 3.6

The above example can be extended as follows. Let B be any smooth curve and C be an elliptic curve. Then the product of C with the Jacobian variety of B, i.e., is an abelian variety. For any semistable vector bundle F on C, the pullback bundle under the first projection \(p_1\), is a semistable bundle with . Hence \(\mathrm{Nef}^{\,1}(\mathbb {P}(E)) = \overline{\mathrm{Eff}}{}^{\,1}(\mathbb {P}(E))\).

A vector bundle E on an abelian variety X is called weakly-translation invariant (semi-homogeneous in the sense of Mukai) if for every closed point \(x \in X\), there is a line bundle \(L_x\) on X depending on x such that for all \(x\in X\), where \(T_x\) is the translation morphism given by \(x\in X\).

Corollary 3.7

Let E be a semi-homogeneous vector bundle of rank r on an abelian variety X. Then \(\mathrm{Nef}^{\,1}(\mathbb {P}(E)) = \overline{\mathrm{Eff}}{}^{\,1}(\mathbb {P}(E))\).

Proof

By a result due to Mukai, E is Gieseker semistable (see [14, Chapter 1] for definition) with respect to some polarization and it has projective Chern classes zero, i.e., if c(E) is the total Chern class, then \(c(E) = \{ 1+ c_1(E)/r\}^r\) (see [23, p. 260, Theorem 5.8], [23, p. 266, Proposition 6.13]; also see [24, p. 2]). Gieseker semistablity implies slope semistability (see [14]). So, in particular, we have E is slope semistable with . Hence the result follows.\(\square \)

Corollary 3.8

Let X be a smooth complex projective variety X with \(\overline{\mathrm{Eff}}{}^{\,1}(X) = \mathrm{Nef}^{\,1}(X)\) and \(E_1,E_2,\dots ,E_k\) be finitely many semistable vector bundles on X of ranks \(r_1,r_2,\dots ,r_k\) respectively with for all \(i \in \{1,2,\dots ,k\}\). Then

Moreover, if \(\mathrm{Nef}^{\,1}(X) = \overline{\mathrm{Eff}}{}^{\,1}(X)\) is a finite polyhedron and

is the projection map for \(r\in \{1,2,\dots ,k\}\) with \(X_0 = X\), then for each \(1 \leqslant r \leqslant k\)

where \(L_1,L_2,\dots ,L_r\) are the nef generators of the nef cone of \(X_{r-1}\) and .

Conversely, if X is a smooth curve, then implies that \(E_i\) is semistable for each i.

Proof

We will proceed by induction on k. For \(k=1\), this is precisely the statement of Theorem 1.2. Now suppose the theorem holds true for \(k-1\) many vector bundles. Consider the following fibre product diagram:

figure a

Note that

Since \(E_k\) is semistable with on X, by Lemma 2.2 its pullback \(\psi _k^*E_k\) under \(\psi _k\) is also semistable with . By induction hypothesis we have

Therefore applying Theorem 1.2 we get the result.

Conversely, if X is a curve and the equality holds, then inductively \(\mathrm{Nef}^{\,1}(\mathbb {P}(E_i)) = \overline{\mathrm{Eff}}{}^{\,1}(\mathbb {P}(E_i))\) for each \(i\in \{1,2,\dots ,k\}\). This implies that each \(E_i\) is semistable by [22, Theorem 3.1]. \(\square \)

Remark 3.9

In general, the computations of nef cones \(\mathrm{Nef}^{\,1}(\mathbb {P}_X(E))\) of any projective bundle \(\mathbb {P}_X(E)\) over an arbitrary X can be very complicated, even on smooth surfaces X (see [21]). In [15], of fibre product over a smooth curve C is calculated for any two vector bundles \(E_1\) and \(E_2\) (not necessarily semistable bundles). We note that the proof in [15] easily generalizes to fibre product of finitely many copies of projective bundles over C.

Remark 3.10

In particular, Corollary 3.8 generalizes [16, Theorem 4.1]. Moreover, Corollary 3.8 indicates that the hypothesis about semistability of E in Theorem 1.2 is necessary. This is because on a smooth complex projective curve C, take one semistable bundle \(E_1\) and another unstable bundle \(E_2\). Then . But \(\mathrm{Nef}^{\,1}(\mathbb {P}(E_1)) = \overline{\mathrm{Eff}}{}^{\,1}(\mathbb {P}(E_1))\) and where \(\pi _1 :\mathbb {P}(E_1) \rightarrow C\) is the projection map. Note that \(\pi _1^*E_2\) is unstable with (see [20]). In view of this remark, we finish this section with the following question.

Question 3.11

Let E be a vector bundle on a smooth projective variety X and \(\mathrm{Nef}^{\,1}(\mathbb {P}_X(E)) = \overline{\mathrm{Eff}}{}^{\,1}(\mathbb {P}_X(E))\). Does this imply that E is a semistable bundle with

4 Weak Zariski decomposition on fibre product of projective bundles

Let E and \(E'\) be two vector bundles of ranks m and n respectively on a smooth complex projective curve C with and . Let

$$\begin{aligned} 0 = E_{k} \subset E_{k-1} \subset \cdots \subset E_{1} \subset E_0 = E \end{aligned}$$

and

$$\begin{aligned} 0 = E'_{l} \subset E'_{l-1} \subset \cdots \subset E'_{1} \subset E'_0 = E' \end{aligned}$$

be the Harder–Narasimhan filtrations of E and \(E'\) respectively. Let us fix , , , \(\mu (Q_1) = \mu _1 = {d_1}/{r_1}\), , , , \(\mu (Q'_1) = \mu '_1 = {d'_1}/{r'_1}\).

Consider the following fibre product diagram:

figure b

Let \(f_1\) and \(f_2\) denote the numerical classes of fibres of \(\pi _1\) and \(\pi _2\) respectively, and , , \(F = \pi ^*\!f_1 = \pi '^*\!f_2\). Therefore,

Also, by [15, Theorem 3.1],

We use the above notations in the rest of this section. The intersection products in X are as follows:

figure c

Since \(\pi _1\) is a smooth map, in particular it is flat and hence \(\pi _1^*\) is an exact functor. Also we observe that for any semistable vector bundle V on C and for any ample line bundle H on \(\mathbb {P}(E')\), \(\pi _1^*(V)\) is H-semistable with and . These observations immediately imply that

$$\begin{aligned} 0 = \pi _1^*E_{k} \subset \pi _1^*E_{k-1} \subset \cdots \subset \pi _1^*E_{1} \subset \pi _1^*E_0 = \pi _1^*E \end{aligned}$$

is the unique Harder–Narasimhan filtration of \(\pi _1^*E\) with respect to any ample line bundle H on \(\mathbb {P}(E')\) with . Similar argument holds for \(\pi _2^*E'\). We fix for all \(i \in \{0,1,2,\dots ,k\}\) and for all \(i \in \{0,1,2,\dots ,l\}\). Define .

The projection map can be seen as a rational map. The indeterminacies of this rational map are resolved by blowing up . Now we consider the following commutative diagram:

figure d

By following the ideas of [11, Proposition 2.4] (see [11, Remark 2.5]), one can find a locally free sheaf on Y such that \((\widetilde{X},Y,\eta )\) is the projective bundle over Y with \(\eta \) as the projection map. Moreover, if , and , then \(\gamma = \beta ^*\xi \) and \(\eta ^*\xi _1 = \beta ^*\xi - \widetilde{E}\) where \(\widetilde{E}\) is the numerical class of the exceptional divisor of the map \(\beta \). Also, .

In the rest of this section, we give a proof of existence of weak Zariski decomposition for every pseudo-effective \(\mathbb {R}\)-divisor on . We consider the following three cases:

Case 1: We assume that \(E'\) is semistable and E is unstable with . We show that \(\overline{\mathrm{Eff}}{}^{\,1}(X)\) is a finite polyhedron and weak Zariski decomposition exists for every extremal ray of \(\overline{\mathrm{Eff}}{}^{\,1}(X)\). In particular, this proves the existence of weak Zariski decomposition for every pseudo-effective \(\mathbb {R}\)-divisor on X in this case (see [25, Proposition 1]).

Case 2: We assume that both E and \(E'\) are unstable with and . The proof of existence of weak Zariski decomposition in this case is identical to the proof in Case 1.

Case 3: In this case, we assume that at least one of them, say E is unstable with . We define a cone map

which is an isomorphism. Inductively, the Harder–Narasimhan filtration of E gives the isomorphisms

also prove that \(C^{(1)}(D)\) admits a weak Zariski decomposition if D does. Finally, we show the existence of weak Zariski decomposition for any pseudo-effective \(\mathbb {R}\)-divisors on X by using the results in Cases 1 and 2 recursively.

Lemma 4.1

Let E and \(E'\) be as above. Further assume that \(E'\) is semistable and E is unstable with . Then,

where \(\mu '\! = \mu (E')\). In this case, weak Zariski decomposition exists for pseudo-effective \(\mathbb {R}\)-divisors in .

Proof

Note that , \(\zeta - \mu 'F\!, F \in \mathrm{Nef}^{\,1}(X) \subset \overline{\mathrm{Eff}}{}^{\,1}(X)\). Hence

To prove the converse, consider an element \(\alpha = a\xi + b\zeta + cF \in \overline{\mathrm{Eff}}{}^{\,1}(X)\). Then using the intersection products in X, we get

Note that \(\alpha = a(\xi + (d_1-d)F) + b(\zeta - \mu 'F) + ( b\mu ' - a(d_1-d) + c ) F \).

This proves the equality. Note that, in this case weak Zariski decompostion exists for every extremal ray of \(\overline{\mathrm{Eff}}{}^{\,1}(X)\). Hence the result follows from [25, Proposition 1]. \(\square \)

Lemma 4.2

Let E and \(E'\) be two unstable bundles as above with and . Then

In this case also, weak Zariski decomposition exists for pseudo-effective \(\mathbb {R}\)-divisors in .

Proof

The proof is similar to the proof of Lemma 4.1.\(\square \)

Lemma 4.3

Let E and \(E'\) be two bundles as above and E is unstable with . Recall the following commutative diagram:

figure e

Then

  1. (i)

    The cone map

    defined as follows:

    induces an isomorphism of onto .

  2. (ii)

    If \(D \in \overline{\mathrm{Eff}}{}^{\,1}(Y)\) admits weak Zariski decomposition, then its image \(C^{(1)}(D) \in \overline{\mathrm{Eff}}{}^{\,1}(X)\) under the cone map also admits weak Zariski decomposition.

Proof

(i) We have where . We define \(\phi _1 :N^1(X)_{\mathbb {R}} \rightarrow N^1(Y)_{\mathbb {R}}\) by

$$\begin{aligned} \phi _1( a\xi + b\zeta + c F) = a\xi _1 + b\nu + c\rho ^*\!f_1. \end{aligned}$$

This gives an isomorphism between real vector spaces \(N^1(X)_{\mathbb {R}}\) and \(N^1(Y)_{\mathbb {R}}\).

Also we define \(U_1 :N^1(Y)_{\mathbb {R}} \rightarrow N^1(X)_{\mathbb {R}}\) as follows:

$$\begin{aligned}U_1(\alpha ) = \beta _*\eta ^*(\alpha ). \end{aligned}$$

In particular, \(U_1(a\xi _1 + b\nu + c\rho ^*\!f_1) = a\xi + b\zeta + cF\).

As \(\eta \) is flat and \(\beta \) is bi-rational, the above map \(U_1\) is well-defined. We construct an inverse for \(U_1\). Define \(D_1 :N^1(X)_{\mathbb {R}} \rightarrow N^1(Y)_{\mathbb {R}}\) as follows:

Note that . Similarly, \(D_1(\zeta ) = \nu \) and \(D_1(\pi ^*\!f_1) = \rho ^*(f_1)\). This shows that the maps \(D_1\) and \(\phi _1\) are the same maps.

Next we will show that the map \(D_1\) sends effective divisor in X to \(\overline{\mathrm{Eff}}{}^{\,1}(Y)\). For any effective divisor \(\alpha \), we can write \(\beta ^*(\alpha ) = {\alpha }' \!+ j_*\widetilde{\alpha }\) for some \(\widetilde{\alpha }\), where \(\alpha '\) is the strict transform under the map \(\beta \) and \(j :\widetilde{E} \rightarrow \widetilde{X}\) is the canonical inclusion. (Here by abuse of notation \(\widetilde{E}\) is also the support of \(\widetilde{E}\).) Since \(\delta \) is intersection of nef divisors and \(\eta _*\) maps pseudo-effective divisors to pseudo-effective divisors, it is enough to prove that . This is true because and .

(ii) As D admits weak Zariski decomposition, there is a bi-rational transformation \(\psi :Y'\! \rightarrow Y\) such that \(\psi ^*(D) = P + N\), with \( P \in \mathrm{Nef}^{\,1}(Y')\) and \(N \in \mathrm{Eff}^1(Y')\). Then we have the following commutative diagram:

figure f

We denote . Let \(D' \!= a\xi + b \zeta + c \pi ^*\!f_1\) for some \(a,b,c \in \mathbb {R}\). Then by the above construction of the map \(C^{(1)}\), we have \(D = a\xi _1 + b \nu + c\rho ^*\!f_1\).

We will show that \(\beta ^*(D') = \eta ^*(D) + a\widetilde{E}\). Recall that , and \(\beta ^*\zeta = \eta ^*\nu \). Also, \(\beta ^*\xi = \eta ^*\xi _1 + \widetilde{E}\).

Using these relations we conclude

$$\begin{aligned} \beta ^*(D')&=\beta ^*(a\xi +b\zeta +c\pi ^*\!f_1)= a\beta ^*\xi + b\beta ^*\zeta + c\beta ^*(\pi ^*\!f_1)\\&= a\eta ^*\xi _1 + a\widetilde{E} + b\eta ^*\nu + c\eta ^*(\rho ^*\!f_1)\\&=\eta ^*(a\xi _1+b\nu +c\rho ^*\!f_1) + a\widetilde{E}= \eta ^*(D) + a\widetilde{E}. \end{aligned}$$

Therefore,

The map \(\widetilde{\psi }\) is bi-rational. Hence is also bi-rational, \(\widetilde{\eta }^{\,*}(P) \in \mathrm{Nef}^{\,1}(Y' {\times }_Y \widetilde{X} )\) and \(\widetilde{\eta }^{\,*}(N) + \psi ^*(a\widetilde{E}) \in \mathrm{Eff}^1(Y' {\times }_Y \widetilde{X} )\). Thus we obtain a weak Zariski decomposition of \(D'\! = C^{(1)}(D)\).\(\square \)

Theorem 4.4

Let E and \(E'\) be two vector bundles of rank m and n respectively on a smooth complex projective curve C. Then weak Zariski decomposition exists for the fibre product .

Proof

The following three cases can occur:

Case I : If both E and \(E'\) are semistable bundles on C, then

and hence weak Zariski decomposition exists trivially.

Case II : Let exactly one of E and \(E'\) be unstable, and without loss of generality, we assume that \(E'\) is semistable. Let

$$\begin{aligned} 0 = E_{k} \subset E_{k-1} \subset \cdots \subset E_{1} \subset E_0 = E \end{aligned}$$

be the Harder–Narasimhan filtration of E. If , then by Lemma 4.1 weak Zariski decomposition exists for any pseudo-effective \(\mathbb {R}\)-divisors in . If , then the cone map as in Lemma 4.3

$$\begin{aligned} C^{(1)} :N^1(Y) \rightarrow N^1(X) \end{aligned}$$

induces an isomorphism onto \(\overline{\mathrm{Eff}}{}^{\,1}(\mathbb {P}(\pi _1^*E))\).

Recursively, we have isomorphisms of cone maps such that

$$\begin{aligned} \overline{\mathrm{Eff}}{}^{\,1}(\mathbb {P}(\pi _1^*E_{k-1})) \simeq \cdots \simeq \overline{\mathrm{Eff}}{}^{\,1}(\mathbb {P}(\pi _1^*E_{1})) \simeq \overline{\mathrm{Eff}}{}^{\,1}(\mathbb {P}(\pi _1^*E)). \end{aligned}$$

Now the subbundle \(E_{k-1}\) is a semistable vector bundle on C. Therefore weak Zariski decomposition exists on by Case I. Since cone maps preserve pseudo-effectivity, by applying Lemma 4.3 repeatedly we get that weak Zariski decomposition exists for .

Case III : In this case, both E and \(E'\) are unstable. By arguing similarly as in Case II, and by using Lemma 4.2 and Case II, we have the existence of weak Zariski decomposition for pseudo-effective \(\mathbb {R}\)-divisors in this case also.\(\square \)

5 Results about homogeneous bundles

A vector bundle E on a projective variety X is called k-homogeneous if \(\overline{\mathrm{Eff}}{}^{\,k}(\mathbb {P}(E)) = \mathrm{Nef}^{\,k}(\mathbb {P}(E))\). Let E be a vector bundle of rank \(r \geqslant 2\) on a smooth complex projective surface X and \(\pi :\mathbb {P}(E) \rightarrow X\) be the projection map. Then for each \(k \in \{2,\dots ,r-1\}\),

where . We will denote the numerical class of a fibre of \(\pi \) by F. We will continue to use the above notations in what follows. Then the intersection products on \(\mathbb {P}(E)\) are as follows:

We first prove the following useful lemma.

Lemma 5.1

If E is a semistable bundle of rank \(r \geqslant 2\) on a smooth complex projective surface X with , then \(\lambda _E^{r} = 0\).

Proof

$$\begin{aligned} \lambda _E^{r} = \biggl (\xi - \frac{1}{r}\,\pi ^*\!c_1(E)\biggr )^{\!r}\!\! = \xi ^{r} - \xi ^{r-1}\pi ^*\!c_1(E) + \frac{1}{r^2}\,\frac{r(r-1)}{2}\,\xi ^{r-2}(\pi ^*\!c_1(E))^2\!. \end{aligned}$$
(1)

Since , we have \(c^2_1(E) = \frac{2r}{r-1}c_2(E)\). Replacing this in (1) and using Grothendieck’s relation we get the result.\(\square \)

Theorem 5.2

Let E be a semistable vector bundle of rank \(r \geqslant 2\) with on a smooth complex projective surface X of Picard number \(\rho (X) = 1\). We denote the numerical class of a fibre of the projection \(\pi :\mathbb {P}(E) \rightarrow X\) by F. If \(L_X \in N^1(X)_{\mathbb {R}}\) denotes the numerical class of an ample generator of the Néron–Severi group \(N^1(X)_{\mathbb {Z}}\), then

Moreover, E is a k-homogeneous bundle on X, i.e., \(\overline{\mathrm{Eff}}{}^{\,k}(\mathbb {P}(E)) = \mathrm{Nef}^{\,k}(\mathbb {P}(E))\) for all \( 1 \leqslant k < r \).

Proof

By Theorem 2.1, \(\lambda _E\) is nef. So for any \(k\in \{2,\dots ,r-1\}\), \(\lambda _E^k\), \(\lambda _E^{k-1}\pi ^*L_X\) and \(\lambda _E^{k -2}F\) are intersections of nef divisors and hence they are pseudo-effective. Also \(\lambda _E^r = 0\).

Now if \(a\lambda _E^k + b \lambda _E^{k-1}\pi ^*L_X + c\lambda _E^{k-2}F \in \overline{\mathrm{Eff}}{}^{\,k}(\mathbb {P}(E))\), then

This implies \( b \geqslant 0\) (since \(L_X\) being ample, \(L_X^2 \geqslant 0\)),

This proves that for all \( 1< k < r\),

So for any \(1< k < r\),

Any element with \(x,y,z \in \mathbb {R}\) is in \(\mathrm{Nef}^{\,k}(\mathbb {P}(E))\) if and only if

Therefore, \(\overline{\mathrm{Eff}}{}^{\,k}(\mathbb {P}(E)) = \mathrm{Nef}^{\,k}(\mathbb {P}(E))\) for all \(k \in \{1,2,\dots ,r-1\}\).\(\square \)

Theorem 5.3

Let \(\rho :X = \mathbb {P}_C(G) \rightarrow C\) be a ruled surface defined by a semistable rank 2 bundle G of slope \(\mu \) on a smooth complex projective curve C and . Assume that E is a semistable bundle of rank \(r \geqslant 2\) on X with . Denote the numerical classes of a fibre of \(\rho \) and a fibre of the projection map \(\pi :\mathbb {P}(E) \rightarrow X\) by f and F respectively. Then

for all \(1< k < r\). Also, E is a k-homogeneous bundle on X, i.e., \(\overline{\mathrm{Eff}}{}^{\,k}(\mathbb {P}(E)) = \mathrm{Nef}^{\,k}(\mathbb {P}(E))\) for all \(k \in \{1,2,\dots ,r-1\}\).

Proof

By Theorem 2.1, \(\lambda _E\) is nef. Also

For any \(k\in \{2,\dots ,r-1\}\), \(\lambda _E^kslant\), \(\lambda _E^{k-1}(\pi ^*\eta - \mu \pi ^*\!f)\), \(\lambda _E^{k-1}\pi ^*\!f\) and \(\lambda _E^{k -2}F\) are intersections of nef divisors and hence they are pseudo-effective.

Now if \(a\lambda _E^k + b \lambda _E^{k-1}\pi ^*\eta + c\lambda _E^{k-1}\pi ^*\!f + d\lambda _E^{k-2}F \in \overline{\mathrm{Eff}}{}^{\,k}(\mathbb {P}(E))\), then using the intersection products as above we have

This implies \( b\eta ^2 - \mu b\eta f + c \eta f = 2b\mu - b\mu + c = b\mu + c \geqslant 0\). Hence

This proves that for all \( 1< k < r\). A similar argument as in Theorem 5.2 proves that \(\overline{\mathrm{Eff}}{}^{\,k}(\mathbb {P}(E)) = \mathrm{Nef}^{\,k}(\mathbb {P}(E))\) for all \( 0< k < r\).\(\square \)

Remark 5.4

Let E be a semistable bundle of rank r on a smooth projective complex variety of dimension n and . Then E is projectively flat. So if c(E) is the total Chern class of E, then \(c(E) = \bigl ( 1 + \frac{c_1(E)}{r}\bigr )^r\). In particular, \(c_i(E) = \left( {\begin{array}{c}r\\ i\end{array}}\right) \bigl (\frac{c_1(E)}{r}\bigr )^i\) for all i. This implies that \(\lambda _E^r = \bigl (\xi - \frac{1}{r}\pi ^*c_1(E)\bigr )^{r} = \sum _i(-1)^i\left( {\begin{array}{c}r\\ i\end{array}}\right) \xi ^{r-i}\pi ^*\bigl (\frac{c_1(E)}{r}\bigr )^i = 0\) by applying Grothendieck relation. So similar approaches can be taken to calculate \(\overline{\mathrm{Eff}}{}^{\,k}(\mathbb {P}(E)), \mathrm{Nef}^{\,k}(\mathbb {P}(E))\) over higher dimensional smooth varieties X. However the computations of these cones will be complicated in general.