In this paper we construct indecomposable vector bundles associated to monads on multiprojective spaces. Specifically we establish the existence of monads on \(\mathbf P\)2n+1 × \(\mathbf P\)2n+1 × ⋯ × \(\mathbf P\)2n+1 and on \(\mathbf P\)a₁ × ⋯ × \(\mathbf P\)aₙ. We prove stability of the kernel bundle which is a dual of a generalized Schwarzenberger bundle associated to the monads on X = \(\mathbf P\)2n+1 × \(\mathbf P\)2n+1 × ⋯ × \(\mathbf P\)2n+1 and prove that the cohomology vector bundle which is simple, a generalization of special instanton bundles. We also prove stability of the kernel bundle and that the cohomology vector bundle associated to the monad on \(\mathbf P\)a₁ × ⋯ × \(\mathbf P\)aₙ is simple. Lastly, we construct explicitly the morphisms that establish the existence of monads on \(\mathbf P\)1 × ⋯ × \(\mathbf P\)1.
The existence of indecomposable low rank vector bundles on algebraic varieties in comparison with the ambient space has been a fertile area in algebraic geometry for the last 45 years. Regardless it remains intriguing, fascinating and exciting to construct new examples of indecomposable low rank vector bundles. Some of the remarkable works in this regard are: the famous Horrocks-Mumford bundle of rank 2 over \(\mathbf P^4\) [1], the Horrocks vector bundle of rank 3 on \(\mathbf P^5\) [2] the Tango bundles [3] of rank \(n-1\) on \(\mathbf P^n\) for \(n\geq3\) and the rank 2 vector bundle on \(\mathbf P^5\) in characteristic 2 by Tango [4] are all obtained as cohomologies of certain monads.
This (monads) is one of the techniques used to construct these vector bundles. They appear in many contexts within algebraic geometry. and were first introduced by Horrocks [5] where he proved that all vector bundles \(E\) on \(\mathbf P^3\) could be obtained as the cohomology bundle of a given monad. In vector bundle construction via monads on a given algebraic variety, the first task is to show the existence of monads. Fløystad [6] gave a theorem on the existence of monads over projective spaces. Costa and Miro-Roig [7] extended these results to smooth quadric hypersurfaces of dimension at least 3. Marchesi, Marques and Soares [8] generalized Fløystad’s theorem to a larger set of varieties. Maingi [10] proved the existence of monads on \(\mathbf P^n\times\mathbf P^m\), \(\mathbf P^{2n+1}\times\mathbf P^{2n+1}\), \(\mathbf P^{a_1}\times\mathbf P^{a_1}\times\mathbf P^{a_2}\times\mathbf P^{a_2}\times\cdots\times\mathbf P^{a_n}\times\mathbf P^{a_n}\) and on \(\mathbf P^n\times\mathbf P^n\times\mathbf P^m\times\mathbf P^m\) respectively and proved simplicity of the cohomology bundles associated.
A natural and efficient technique to construct monads and hence more examples of vector bundles is to vary the ambient variety and choose a different polarisation. In Section three of the paper, we first generalize the work of Maingi [10] by construction of monads on \(\mathbf P^{2n+1}\times\cdots\times\mathbf P^{2n+1}\) for a rank \(\beta-\alpha-\gamma\). We then prove stability of the kernel bundle which is a generalization of the dual of Schwarzenberger (steiner) bundles. Next we prove simplicity of the cohomolgy vector bundle. Specifically we establish the existence of monads \[0 \longrightarrow \mathcal{O}_X(-1, \cdots, -1)^{\oplus \alpha} \xrightarrow{f} \mathcal{O}_X^{\oplus \beta} \xrightarrow{g} \mathcal{O}_X(1, \cdots, 1)^{\oplus \gamma} \longrightarrow 0,\] on \(X=\mathbf P^{2n+1}\times\cdots\times\mathbf P^{2n+1}\). We shall call the monad above Type I in this paper.
Next, in Section four we establish the existence of monads on \(X=\mathbf P^{a_1}\times\cdots\times\mathbf P^{a_n}\) for the polarisation \(\mathscr{L}=\mathcal O_X(\alpha_1,\cdots,\alpha_t)\). This is a generalization of the results of Maingi [9, Theorem 3.2.] where he gave a conditional variant theorem for the existence of a monad on \(\mathbf P^{n}\times\mathbf P^{m}\).
Specifically we establish the existence of monads \[0 \longrightarrow \mathcal{O}_X(-\alpha_1, \cdots, -\alpha_t)^{\oplus \alpha} \xrightarrow{f} \mathcal{O}_X^{\oplus \beta} \xrightarrow{g} \mathcal{O}_X(\alpha_1, \cdots, \alpha_t)^{\oplus \gamma} \longrightarrow 0,\] on \(X=\mathbf P^{a_1}\times\cdots\times\mathbf P^{a_n}\) which we shall call monad Type II. We then prove stability of the kernel bundle \(\ker g\) and finally prove that the cohomology vector bundle, \(E=\ker g/\mathop{\mathrm{im}}f\) is simple.
Lastly, in Section five we construct the morphisms that establish the existence of monads \[M_{\bullet} : 0 \longrightarrow \mathcal{O}_X(-1, \cdots, -1)^{\oplus k} \xrightarrow{\overline{A}} \mathcal{O}_X^{\oplus 2n \oplus 2k} \xrightarrow{\overline{B}} \mathcal{O}_X(1, \cdots, 1)^{\oplus k} \longrightarrow 0,\] on \(\mathbf P^1\times\cdots\times\mathbf P^1\) which are matrices whose entries are multidegree monomials.
In this work we give generalizations for previous results by several authors. To be specific we build upon results by Maingi [9– 11] therefore the definitions, notation, the methods applied are quite similar and the trend follows the paper by Ancona and Ottaviani [13]. In this section we define and give notation in order to set up for the main results. Most of the definitions are from chapter two of the book by Okonek, Schneider and Spindler [14].
Definition 1. Let \(X\) be a nonsingular projective variety.
A monad on \(X\) is a complex of vector bundles: \[0 \to M_0 \xrightarrow{\alpha} M_1 \xrightarrow{\beta} M_2 \to 0,\] which is exact at \(M_0\) and at \(M_2\) i.e. \(\alpha\) is injective and \(\beta\) surjective.
A monad as defined above has a display diagram of short exact
sequences as shown below: 
The kernel of the map \(\beta\), \(F=\ker\beta\) and the cokernel of \(\alpha\), \(\mathop{\mathrm{coker}}\alpha\) for the given monad are also vector bundles and the vector bundle \(E = \ker(\beta)/\mathop{\mathrm{im}}(\alpha)\) and is called the cohomology bundle of the monad.
Definition 2. Let \(X\) be a nonsingular projective variety, let \(\mathscr{L}\) be a very ample line sheaf, and \(V,W,U\) be finite dimensional \(k\)-vector spaces. A linear monad on \(X\) is a complex of sheaves, \[M_{\bullet} : 0 \longrightarrow V \otimes \mathcal{L}^{-1} \xrightarrow{A} W \otimes \mathcal{O}_X \xrightarrow{B} U \otimes \mathcal{L} \longrightarrow 0,\] where \(A\in \mathop{\mathrm{Hom}}(V,W)\otimes H^0 \mathscr{L}\) is injective and \(B\in \mathop{\mathrm{Hom}}(W,U)\otimes H^0 \mathscr{L}\) is surjective.
The existence of the monad \(M_\bullet\) is equivalent to: \(A\) and \(B\) being of maximal rank and \(BA\) being the zero matrix.
Definition 3. Let \(X\) be a non-singular irreducible projective variety of dimension \(d\) and let \(\mathscr{L}\) be an ample line bundle on \(X\). For a torsion-free sheaf \(F\) on \(X\) we define
the degree of \(F\) relative to \(\mathscr{L}\) as \(\deg_{\mathscr{L}}F:= c_1(F)\cdot \mathscr{L}^{d-1}\), where \(c_1(F)\) is the first Chern class of \(F\),
the slope of \(F\) as \(\mu_{\mathscr{L}}(F):= \frac{\deg_{\mathscr{L}}F}{rk(F)}\).
Suppose that the Picard group Pic\((X) \simeq \mathbb Z^l\) where \(l\geq2\) is an integer then \(X\) is a polycyclic variety. Given a divisor \(B\) on \(X\) we define \(\delta_{\mathscr{L}}(B):= \deg_{\mathscr{L}}\mathcal O_{X}(B)\). Then one has the following stability criterion [15, Theorem 3]:
Theorem 1 (Generalized Hoppe criterion). Let \(G\rightarrow X\) be a holomorphic vector bundle of rank \(r\geq2\) over a polycyclic variety \(X\) equiped with a polarisation \(\mathscr{L}\) if \[H^0(X,(\wedge^sG)\otimes\mathcal O_X(B))=0,\] for all \(B\in\mathop{\mathrm{Pic}}(X)\) and \(s\in\{1,\ldots,r-1\}\) such that \(\displaystyle{\delta_{\mathscr{L}}(B)<-s\mu_{\mathscr{L}}(G)}\) then \(G\) is stable and if \(\displaystyle{\delta_{\mathscr{L}}(B)\leq-s\mu_{\mathscr{L}}(G)}\) then \(G\) is semi-stable.
Conversely if then \(G\) is (semi-)stable then \[H^0(X,G\otimes\mathcal O_X(B))=0,\] for all \(B\in\mathop{\mathrm{Pic}}(X)\) and all \(s\in\{1,\ldots,r-1\}\) such that \(\left(\delta_{\mathscr{L}}(B)\leq\right)\) \(\delta_{\mathscr{L}}(B)<-s\mu_{\mathscr{L}}(G)\).
Notation 1. Suppose the ambient space is \(X=\mathbf P^{a_1}\times\cdots\times\mathbf P^{a_n}\) then \(\mathop{\mathrm{Pic}}(X) \simeq \mathbb Z^{n}\).
We shall denote by \(g_i\) for \(i=1\cdots,n\) the generators of the Picard group of \(X\), \(\mathop{\mathrm{Pic}}(X)\).
Denote by \(\mathcal O_X(g_1, \cdots,g_{n}):= {p_1}^*\mathcal O_{\mathbf P^{a_1}}(g_1)\otimes\cdots\otimes {p_{n}}^*\mathcal O_{\mathbf P^{a_n}}(g_n)\), where \(p_i\) for \(i=1,\cdots,n\) are natural projections from \(X\) onto \(\mathbf P^{a_i}\).
For any line bundle \(\mathscr{L} = \mathcal O_X(g_1, g_2,\cdots,g_{n})\) on \(X\) and a vector bundle \(E\), we write \(E(g_1, g_2,\cdots,g_{n}) = E\otimes\mathcal O_X(g_1, g_2,\cdots,g_{n})\) and \((g_1, g_2,\cdots,g_{n}):= g_1[h_1\times\mathbf P^{a_1}]+\cdots+g_n[\mathbf P^{a_n}\times h_{n}]\) representing its corresponding divisor.
The normalization of \(E\) on \(X\) with respect to \(\mathscr{L}\) is defined as follows:
Set \(d=\deg_{\mathscr{L}}(\mathcal O_X(1,0,\cdots,0))\), since \(\deg_{\mathscr{L}}(E(-k_E,0,\cdots,0))=\deg_{\mathscr{L}}(E)-nk\cdot \mathop{\mathrm{rank}}(E)\) there is a unique integer \(k_E:=\lceil\mu_\mathscr{L}(E)/d\rceil\) such that \(1 – d.\mathop{\mathrm{rank}}(E)\leq \deg_\mathscr{L}(E(-k_E,0,\cdots,0))\leq0\). The twisted bundle \(E_{{\mathscr{L}}-norm}:= E(-k_E,0,\cdots,0)\) is called the \(\mathscr{L}\)-normalization of \(E\).
Lastly, the linear functional \(\delta_{\mathscr{L}}\) on \(\mathbb{Z}^{n}\) is defined as \(\delta_{\mathscr{L}}(p_1,p_2,\cdots,p_{n}):= \deg_{\mathscr{L}}\mathcal O_{X}(p_1,p_2,\cdots,p_{n})\).
For the \(q^{-th}\) cohomology group we use the notation \(H^q(\mathscr{F})\) in place of \(H^q(X,\mathscr{F})\), for the sake of brevity.
The following proposition is actually a corollary of Theorem 1 above, a special case of the generalized Hoppe criterion on stability.
Proposition 1. Let \(X\) be a polycyclic variety with Picard number \(n\), let \(\mathscr{L}\) be an ample line bundle and let E be a rank \(r>1\) holomorphic vector bundle over \(X\). If \(H^0(X,(\bigwedge^q E)_{{\mathscr{L}}-norm}(p_1,\cdots,p_{n})) = 0\) for \(1\leq q \leq r-1\) and every \((p_1,\cdots,p_{n})\in \mathbb{Z}^{n}\) such that \(\delta_{\mathscr{L}}\leq0\) then E is \(\mathscr{L}\)-stable.
Proposition 2. Let \( 0 \to E \to F \to G \to 0 \) be an exact sequence of vector bundles. Then we have the following exact sequence involving exterior and symmetric powers
\[ 0 \longrightarrow \bigwedge^q E \longrightarrow \bigwedge^q F \longrightarrow \bigwedge^{q-1} F \otimes G \longrightarrow \cdots \longrightarrow F \otimes S^{q-1}G \longrightarrow S^q G \longrightarrow 0. \]
Theorem 2 (Künneth formula). Let \(X\) and \(Y\) be projective varieties over a field \(k\). Let \(\mathscr{F}\) and \(\mathscr{G}\) be coherent sheaves on \(X\) and \(Y\) respectively. Let \(\mathscr{F}\boxtimes\mathscr{G}\) denote \(p_1^*(\mathscr{F})\otimes p_2^*(\mathscr{G})\) then \(\displaystyle{H^m(X\times Y,\mathscr{F}\boxtimes\mathscr{G}) \cong \bigoplus_{p+q=m} H^p(X,\mathscr{F})\otimes H^q(Y,\mathscr{G})}\).
Lemma 1. Let \(X=\mathbf P^{a_1}\times\cdots\times\mathbf P^{a_n}\) then \[\displaystyle{H^t(X,\mathcal O_X(p_1,\cdots,p_{n}) )\cong \bigoplus_{\sum\limits_{q_i=1}^t} H^{q_1}(\mathbf P^{a_1},\mathcal O_{\mathbf P^{a_1}}(p_1))\otimes H^{q_2}(\mathbf P^{a_2},\mathcal O_{\mathbf P^{a_2}}(p_2))\otimes\cdots\otimes H^{q_{n}}(\mathbf P^{a_n},\mathcal O_{\mathbf P^{a_n}}(p_{n}))}.\]
Theorem 3 ([16], Theorem 4.1). Let \(n\geq1\) be an integer and \(d\) be an integer. We denote by \(S_d\) the space of homogeneous polynomials of degree \(d\) in \(n+1\) variables (conventionally if \(d<0\) then \(S_d=0\)). Then the following statements are true:
\(H^0(\mathbf P^n,\mathcal O_{\mathbf P^n}(d))=S_d\) for all \(d\).
\(H^i(\mathbf P^n,\mathcal O_{\mathbf P^n}(d))=0\) for \(1<i<n\) and for all \(d\).
\(H^n(\mathbf P^n,\mathcal O_{\mathbf P^n}(d))\cong H^0(\mathbf P^n,\mathcal O_{\mathbf P^n}(-d-n-1))\).
Lemma 2. If \(\displaystyle{\sum\limits_{i=1}^np_i>}0\) then \(h^p(X,\mathcal O_X (-p_1,\cdots,-p_{n})^{\oplus k}) = 0\) where \(X=\mathbf P^{a_1}\times\cdots\times\mathbf P^{a_n}\) and for \(0\leq p< \dim(X) -1\), for \(k\) a positive integer.
Lemma 3. Let \(A\) and \(B\) be vector bundles canonically pulled back from \(A'\) on \(\mathbf P^n\) and \(B'\) on \(\mathbf P^m\) then \[\displaystyle{H^q\left(\bigwedge^s(A\otimes B)\right)= \sum\limits_{k_1+\cdots+k_s=q}\left\{\bigoplus_{i=1}^{s}\left(\sum\limits_{j=0}^s\sum\limits_{m=0}^{k_i}H^m(\wedge^j(A))\otimes(H^{k_i-m}(\wedge^{s-j}(B)))\right) \right\}}.\]
Proof. The proof follows from the following standard identities:
\[\displaystyle{H^q(A_1\oplus\cdots\oplus A_s) = \sum\limits_{k_1+\cdots+k_s=q}\left\{\bigoplus_{i=1}^{s}H^k_i(A_i)\right\}}.\]
\[H^q(A \otimes B) = \sum_{m=0}^{q} H^m(A) \otimes H^{q – m}(B).\]
\[\displaystyle{\wedge^s(A\otimes B)=\sum\limits_{j=0}^s\wedge^j(A)\otimes\wedge^{s-j}(B)}.\]
◻
Lemma 4 ([6], Main theorem). Let \(k\geq1\). There exists monads on \(\mathbf P^k\) whose maps are matrices of linear forms, \[ 0 \longrightarrow \mathcal{O}_{\mathbb{P}^k}(-1)^{\oplus a} \xrightarrow{A} \mathcal{O}_{\mathbb{P}^k}^{\oplus b} \xrightarrow{B} \mathcal{O}_{\mathbb{P}^k}(1)^{\oplus c} \longrightarrow 0, \] if and only if at least one of the following is fulfilled;
\((1)\) \(b\geq2c+k-1\) and \(b\geq a+c\),
\((2)\) \(b\geq a+c+k\).
Lemma 5 ([10], Theorem 3.9). Let \(n\) and \(k\) be positive integers and \(A\) and \(B\) be morphisms of linear forms as in \[B :=\left( \begin{array}{cccc|cccccccc} x_0\cdots & x_n & & &y_0 \cdots & y_n\\ &\ddots&\ddots &&\ddots&\ddots\\ && x_0\cdots x_n & & & y_0 \cdots & y_n \end{array} \right),\] and \[A :=\left( \begin{array}{cccccccc} -y_0\cdots & -y_n \\ &\ddots &\ddots\\ &&-y_0 \cdots & -y_n\\ \hline x_0 \cdots & x_n \\ &\ddots &\ddots\\ && x_0\cdots & x_n\\ \end{array} \right),\] then there exists a linear monad of the form \[ 0 \longrightarrow \mathcal{O}_{\mathbb{P}^{2n+1}}(-1)^{\oplus k} \xrightarrow{A} \mathcal{O}_{\mathbb{P}^{2n+1}}^{\oplus (2n + 2k)} \xrightarrow{B} \mathcal{O}_{\mathbb{P}^{2n+1}}(1)^{\oplus k} \longrightarrow 0. \]
Lemma 6 ([10], Theorem 3.2). Let \(X = \mathbf P^n\times\mathbf P^m\) and let \(\mathscr{L} = \mathcal O_X(\rho,\sigma)\) be an ample line bundle on \(X\). Denote by \(N = h^0(\mathcal O_X(\rho,\sigma)) – 1\). Let \(\alpha,\beta, \gamma\) be positive integers such that at least one of the following conditions holds
\((1)\) \(\beta\geq 2\gamma + N -1\), and \(\beta\geq \alpha + \gamma\),
\((2)\) \(\beta\geq \alpha + \gamma + N\).
Then, there exists a linear monad on \(X\) of the form \[ 0 \longrightarrow \mathcal{O}_X(-\rho, -\sigma)^{\oplus \alpha} \xrightarrow{A} \mathcal{O}_X^{\oplus \beta} \xrightarrow{B} \mathcal{O}_X(\rho, \sigma)^{\oplus \gamma} \longrightarrow 0. \]
Definition 4. Let \(X\) be a projective variety. A sheaf \(S\) on \(X\) is a steiner bundle if has short exact sequence of the form \[ 0 \longrightarrow \mathcal{O}_X(-1)^{\oplus a} \longrightarrow \mathcal{O}_X^{\oplus b} \longrightarrow S \longrightarrow 0. \]
They were first defined by Dolgachev and Kapranov [17].
Definition 5. [18] Let \(k\geq0\) the exact sequence of sheaves on \(\mathbf P^{2n+1}\) \[ 0 \longrightarrow \mathcal{O}_X(-1)^{\oplus k} \xrightarrow{\phi} \mathcal{O}_X^{\oplus (2n+2k)} \longrightarrow S \longrightarrow 0, \] where \(\phi\) is given by the matrix \[\left[ \begin{array}{cccc|cccccccc} x_0\cdots & x_n & & &y_0 \cdots & y_n\\ &\ddots&\ddots &&\ddots&\ddots\\ && x_0\cdots x_n & & & y_0 \cdots & y_n \end{array} \right],\] defines a \(2n+k-\)bundle \(S\) on \(\mathbf P^{2n+1}\) called a (generalized) Schwarzenberger bundle.
The display of the monad in Lemma 5 is
A special instanton bundle on \(\mathbf P^{2n+1}\) of quantum number \(k\) is defined by the exact sequence \[ 0 \longrightarrow \mathcal{O}_{\mathbb{P}^{2n+1}}(-1)^{\oplus k} \longrightarrow S^* := \ker(B) \longrightarrow E \longrightarrow 0, \] which is exactly the way Spindler and Trautmann remarkably described [18] where \(S\) is a Schwarzenberger bundle of rank \(2n+k\) which is defined by the short exact sequence \[ 0 \longrightarrow \mathcal{O}_{\mathbb{P}^{2n+1}}(-1)^{\oplus k} \xrightarrow{A} \mathcal{O}_{\mathbb{P}^{2n+1}}^{\oplus (2n+2k)} \longrightarrow S := \operatorname{coker}(A) \longrightarrow 0, \] and they were proved by Ancona and Ottaviani [13], Theorem 2.2 to be stable and in Theorem 2.8 they proved that \(E\) is simple. Independently, Bohnhorst and Spindler [19] proved the stability of rank \(n\) Schwarzenberger bundles on \(\mathbf P^n\).
In the next section we are going to establish the existence of monads on a more general space namely \(\mathbf P^{2n+1}\times\mathbf P^{2n+1}\times\cdots\times\mathbf P^{2n+1}\) and prove stability of the kernel bundle \(T\) and simplicity of the cohomology vector bundle \(E\).
The goal of this section is to construct monads over a multiprojective space of \(m\) copies of \(\mathbf P^{2n+1}\). More specifically we generalize the results of Maingi [10] by varying the ambient space. We rely on methods similar to those used in [11]. The kernel bundle \(T\) is a more generalized version of the dual of a Schwarzenberger vector bundle and we prove that it is stable and consequently we prove that the cohomology vector bundle \(E\) associated to the monad on \(X\) is simple. The vector bundle \(E\) is a generalized version of an instanton bundle.
Theorem 4. Let \(X=\mathbf P^{2n+1}\times\cdots\times\mathbf P^{2n+1}\) and \(\mathscr{L} = \mathcal O_X(1,\cdots,1)\) an ample line bundle. Denote by \(N = h^0(\mathcal O_X(1,\cdots,1)) – 1\). Then there exists a linear monad \(M_\bullet\) on \(X\) of the form \[ M_\bullet : 0 \longrightarrow \mathcal{O}_X(-1, \cdots, -1)^{\oplus \alpha} \xrightarrow{f} \mathcal{O}_X^{\oplus \beta} \xrightarrow{g} \mathcal{O}_X(1, \cdots, 1)^{\oplus \gamma} \longrightarrow 0, \] if atleast one of the following is satified
\(\beta\geq 2\gamma + N -1\), and \(\beta\geq \alpha + \gamma\),
\(\beta\geq \alpha + \gamma + N\), where \(\alpha,\beta, \gamma\) be positive integers.
Proof. For the ample line bundle \(\mathscr{L} = \mathcal O_X(1,\ldots,1)\) we have the Segre embedding \[ i^* : X = \mathbb{P}^{2n+1} \times \cdots \times \mathbb{P}^{2n+1} \hookrightarrow \mathbb{P}(H^0(X, \mathcal{O}_X(1, \ldots, 1))) \cong \mathbb{P}^N := \mathbb{P}^{(2n+2)^m – 1}, \] such that \(i^*(\mathcal O_X(1))\simeq \mathscr{L}\).
Suppose that one of the conditions of Lemma 4 is satified and we have \(a=\alpha\), \(b=\beta\), \(c=\gamma\) and \(k=2n+1\) thus there exists a linear monad \[ 0 \longrightarrow \mathcal{O}_{\mathbb{P}^{2n+1}}(-1)^{\oplus \alpha} \xrightarrow{\bar{A}} \mathcal{O}_{\mathbb{P}^{2n+1}}^{\oplus \beta} \xrightarrow{\bar{B}} \mathcal{O}_{\mathbb{P}^{2n+1}}(1)^{\oplus \gamma} \longrightarrow 0, \] on \(\mathbf P^{2n+1}\) whose morphisms are matrices \(A\) and \(B\) with entries monomials of degree one where \[\begin{aligned} A\in\mathop{\mathrm{Hom}}(\mathcal O_{\mathbf P^{2n+1}}(-1)^{\oplus\alpha},\mathcal O_{\mathbf P^{2n+1}}^{\oplus\beta})\cong H^0(\mathbf P^{2n+1},\mathcal O_{\mathbf P^{2n+1}}(1)^{\oplus\alpha\beta}), \\ B\in\mathop{\mathrm{Hom}}(\mathcal O_{\mathbf P^{2n+1}}^{\oplus\beta},\mathcal O_{\mathbf P^{2n+1}}(1)^{\oplus\gamma})\cong H^0(\mathbf P^{2n+1},\mathcal O_{\mathbf P^{2n+1}}(1)^{\oplus\beta\gamma}). \end{aligned}\]
Thus, \(A\) and \(B\) induce a monad on \(X\), \[ 0 \longrightarrow \mathcal{L}^{-1 \, \oplus \alpha} \xrightarrow{A} \mathcal{O}_X^{\oplus \beta} \xrightarrow{B} \mathcal{L}^{\oplus \gamma} \longrightarrow 0, \] where whose morphisms are matrices \(\bar{A}\) and \(\bar{B}\) with entries multidegree monomials such that
\[\bar{A}\in\mathop{\mathrm{Hom}}(\mathcal O_{X}(-1,\ldots,-1)^{\oplus\alpha},\mathcal O^{\oplus\beta}_X),\] and \[\bar{B}\in\mathop{\mathrm{Hom}}(\mathcal O^{\oplus\beta}_X , \mathcal O_{X}(1,\ldots,1)^{\oplus\gamma}).\] ◻
The kernel bundle \(T\) of the above monad is a generalization of the dual of Schwarzenberger vector bundles [19] which we now proceed to prove that it is stable.
Lemma 7. Let \(T\) be a vector bundle on \(X=\mathbf P^{2n+1}\times\cdots\times\mathbf P^{2n+1}\) defined by the sequence \[ 0 \longrightarrow T \longrightarrow \mathcal{O}_X^{\oplus \beta} \longrightarrow \mathcal{O}_X(1, \cdots, 1)^{\oplus \gamma} \longrightarrow 0, \] then \(T\) is stable.
Proof. We show that \(H^0(X,\bigwedge^q T(-p_1,\cdots,-p_m))=0\) for all \(\displaystyle{\sum\limits_i^m p_i>0}\) and \(1\leq q\leq \mathop{\mathrm{rank}}(T)\).
Consider the ample line bundle \(\mathscr{L} = \mathcal O_X(1,\cdots,1) = \mathcal O(L)\).
Its class in \[\mathop{\mathrm{Pic}}(X)= \langle [h_1\times\mathbf P^{2n+1}],\cdots,[\mathbf P^{2n+1}\times h_m]\rangle,\] corresponds to the class
\[\displaystyle{\sum\limits_{i=1}^m1\cdot[h_i\times\mathbf P^{2n+1}]},\] where \(h_i\), \(i=1,\cdots,n\) are hyperplanes of \(\mathbf P^{2n+1}\) with the intersection product induced by \(h_i^{2n+1} = 1\) and \(h_i^{2n+2}=0\).
Now from the display diagram of the monad we get \[\begin{aligned} \begin{split} c_1(T) & = c_1(\mathcal O_X^{\beta}) – c_1(\mathcal O_X(1,\cdots,1)^{\oplus\gamma}) \\ & = \beta(0,\cdots,0) – \gamma(1,\cdots,1)\\ & = (-\gamma,\cdots,-\gamma). \end{split} \end{aligned}\]
Now \(L^{(2n+1)m}>0\) hence , the degree of \(T\) is: \[\begin{aligned} \begin{split} \deg_{\mathscr{L}}T & = -\gamma([h_1\times\mathbf P^{2n+1}]+\cdots+[\mathbf P^{2n+1}\times h_m])\cdot (\sum\limits_{i=1}^m1\cdot[h_i\times\mathbf P^{2n+1}])^{m(2n+1)-1}\\ & = -\gamma L^{m(2n+1)}< 0. \end{split} \end{aligned}\]
Since \(\deg_{\mathscr{L}}T<0\), then \((\bigwedge^q T)_{\mathscr{L}-norm} = (\bigwedge^q T)\) and it suffices by Proposition 1, to prove that \(h^0(\bigwedge^q T(-p_1,\cdots,-p_m)) = 0\) with \(\displaystyle{\sum\limits_{i=1}^mp_i\geq0}\) and for all \(1\leq q\leq \mathop{\mathrm{rank}}(T)-1\). Next we twist the exact sequence \[ 0 \longrightarrow T \longrightarrow \mathcal{O}_X^{\oplus \beta} \longrightarrow \mathcal{O}_X(1, \cdots, 1)^{\oplus \gamma} \longrightarrow 0, \] by \(\mathcal O_X(-p_1,\cdots,-p_m)\) we get, \[0\longrightarrow T(-p_1,\cdots,-p_m)\longrightarrow\mathcal O_X(-p_1,\cdots,-p_m)^{\oplus\beta}\longrightarrow\mathcal O_X(1-p_1,\cdots,1-p_m)^{\oplus\gamma}\longrightarrow 0,\] and taking the exterior powers of the sequence by Proposition 2 we get \[0\longrightarrow\bigwedge^q T(-p_1,\cdots,-p_m) \longrightarrow\bigwedge^q (\mathcal O_X(-p_1,\cdots,-p_m)^{\oplus\beta})\longrightarrow\bigwedge^{q-1}(\mathcal O_X(1-2p_1,\cdots,1-2p_m)^{\oplus\beta+\gamma})\cdots.\]
Taking cohomology we have the injection: \[0\longrightarrow H^0(X,\bigwedge^{q}T(-p_1,\cdots,-p_m))\hookrightarrow H^0(X,\bigwedge^{q}(\mathcal O_X(-p_1,\cdots,-p_m)^{\oplus\beta})).\]
Set \(\mathscr{G}=\mathcal O_X(-p_1,\cdots,-p_m)^{\beta} = \mathcal O_X(-p_1,\cdots,-p_2)\otimes\mathcal O_X^{\oplus\beta}\) and using Lemma 2 \(H^0(X,\bigwedge^{q}\mathscr{G})\) expands into \(\displaystyle{H^0(X,\sum\limits_{j=0}^{q}\wedge^j\mathcal O_X(-p_1,\cdots,-p_2)\otimes\mathcal O_X^{\oplus\beta})}\) and since \(\displaystyle{\sum\limits_i^m p_i>0}\) by Lemma 3 then \[h^0(X,\bigwedge^{q}(\mathcal O_X(-p_1,\cdots,-p_m)^{\oplus\beta}))=h^0(X,\bigwedge^{q}T(-p_1,\cdots,-p_m))= 0,\] i.e. \(h^0(\bigwedge^{q}T(-p_1,\cdots,-p_m))=0\) and thus \(T\) is stable. ◻
Theorem 5. Let \(X={\mathbf P^{2n+1}}\times\cdots\times{\mathbf P^{2n+1}}\), then the cohomology vector bundle \(E\) associated to the monad \[ 0 \longrightarrow \mathcal{O}_X(-1, \cdots, -1)^{\oplus \alpha} \xrightarrow{A} \mathcal{O}_X^{\oplus \beta} \xrightarrow{B} \mathcal{O}_X(1, \cdots, 1)^{\oplus \gamma} \longrightarrow 0, \] of rank \(\beta-\alpha-\gamma\) is simple.
Proof. The display of the monad is
Since \(E\) is simple if its only endomorphisms are the homotheties then we need to prove that \(\mathop{\mathrm{Hom}}(E,E)=k\) which is equivalent to \(h^0(E\otimes E^*)\).
The first step is to take the dual short exact sequence \[ 0 \longrightarrow \mathcal{O}_X(-1, \cdots, -1)^{\oplus \alpha} \longrightarrow T \longrightarrow E \longrightarrow 0, \]
to get
\[ 0 \longrightarrow E^* \longrightarrow T^* \longrightarrow \mathcal{O}_X(1, \cdots, 1)^{\oplus \alpha} \longrightarrow 0. \]
Tensoring by \( E \) we get
\[ 0 \longrightarrow E \otimes E^* \longrightarrow E \otimes T^* \longrightarrow E(1, \cdots, 1)^{\oplus \alpha} \longrightarrow 0. \]
Now taking cohomology gives:
\[ 0 \longrightarrow H^0(X, E \otimes E^*) \longrightarrow H^0(X, E \otimes T^*) \longrightarrow H^0(E(1, \cdots, 1)^{\oplus \alpha}) \longrightarrow \cdots, \] which implies that \[\label{eq1} h^0(X,E\otimes E^*) \leq h^0(X,E\otimes T^*). \tag{1}\]Now we dualize the short exact sequence
\[ 0 \longrightarrow T \longrightarrow \mathcal{O}_X^{\oplus \beta} \longrightarrow \mathcal{O}_X(1, \cdots, 1)^{\oplus \gamma} \longrightarrow 0, \]
to get
\[ 0 \longrightarrow \mathcal{O}_X(-1, \cdots, -1)^{\oplus \gamma} \longrightarrow \mathcal{O}_X^{\oplus \beta} \longrightarrow T^* \longrightarrow 0. \]
Now twisting by \( \mathcal{O}_X(-1, \cdots, -1) \) and taking cohomology we get
\[ 0 \longrightarrow H^0(X, \mathcal{O}_X(-2, \cdots, -2)^{\oplus \gamma}) \longrightarrow H^0(X, \mathcal{O}_X(-1, \cdots, -1)^{\oplus \beta}) \longrightarrow H^0(X, T^*(-1, \cdots, -1)) \longrightarrow \] \[ \longrightarrow H^1(X, \mathcal{O}_X(-2, \cdots, -2)^{\oplus \gamma}) \longrightarrow H^1(X, \mathcal{O}_X(-1, \cdots, -1)^{\oplus \beta}) \longrightarrow H^1(X, T^*(-1, \cdots, -1)) \longrightarrow \] \[ \longrightarrow H^2(X, \mathcal{O}_X(-2, \cdots, -2)^{\oplus \gamma}) \longrightarrow H^2(X, \mathcal{O}_X(-1, \cdots, -1)^{\oplus \beta}) \longrightarrow H^2(X, T^*(-1, \cdots, -1)) \longrightarrow \cdots \]
from which we deduce \(H^0(X,T^*(-1,\cdots,-1)) = 0\) and \(H^1(X,T^*(-1,\cdots,-1)) = 0\) from Lemmas 1, 2 and Theorem 3.Lastly, tensor the short exact sequence
\[ 0 \longrightarrow \mathcal{O}(-1, \cdots, -1)^{\oplus \alpha} \longrightarrow T \longrightarrow E \longrightarrow 0, \]
by \( T^* \) to get
\[ 0 \longrightarrow T^*(-1, \cdots, -1)^{\oplus \alpha} \longrightarrow T \otimes T^* \longrightarrow E \otimes T^* \longrightarrow 0, \]
and taking cohomology we have
\[ 0 \longrightarrow H^0(X, T^*(-1, \cdots, -1)^{\oplus \alpha}) \longrightarrow H^0(X, T \otimes T^*) \longrightarrow H^0(X, E \otimes T^*) \longrightarrow \] \[ \longrightarrow H^1(X, T^*(-1, \cdots, -1)^{\oplus \alpha}) \longrightarrow \cdots \]
But \( H^1(X, T^*(-1, \cdots, -1)^{\oplus \alpha}) = 0 \) for \( \alpha > 1 \) from above. So we have
\[ 0 \longrightarrow H^0(X, T^*(-1, \cdots, -1)^{\oplus \alpha}) \longrightarrow H^0(X, T \otimes T^*) \longrightarrow H^0(X, E \otimes T^*) \longrightarrow 0. \]
This implies that \[\label{eq2} h^0(X,T\otimes T^*) \leq h^0(X,E\otimes T^*). \tag{2}\]
Since \(T\) is stable then it follows that it is simple which implies \(h^0(X,T\otimes T^*)=1\).
From (1) and now (2) and putting these together, we have \[1\leq h^0(X,E\otimes E^*) \leq h^0(X,E\otimes T^*) = h^0(X,T\otimes T^*) = 1.\] We have \(h^0(X,E\otimes E^*) = 1\) and therefore \(E\) is simple. ◻
The goal of this section is to construct monads over a multiprojectivespace \(\mathbf P^{a_1}\times\cdots\times\mathbf P^{a_n}\). More specifically we generalize the results of Maingi [10] by varying the ambient space and the polarisation \(\mathscr{L}\). We prove that the kernel bundle \(F\) is stable and thereafter we prove that the cohomology vector bundle \(E\) associated to the monad on \(X\) is simple.
Theorem 6. Let \(X = \mathbf P^{a_1}\cdots\times\mathbf P^{a_n}\) and \(\mathscr{L} = \mathcal O_X(\alpha_1,\cdots,\alpha_t)\) an ample line bundle. Denote by \(N = h^0(\mathcal O_X(\alpha_1,\cdots,\alpha_t)) – 1\). Then there exists a linear monad \(M_\bullet\) on \(X\) of the form \[ M_\bullet : 0 \longrightarrow \mathcal{O}_X(-\alpha_1, \cdots, -\alpha_t)^{\oplus \alpha} \xrightarrow{f} \mathcal{O}_X^{\oplus \beta} \xrightarrow{g} \mathcal{O}_X(\alpha_1, \cdots, \alpha_t)^{\oplus \gamma} \longrightarrow 0, \] if atleast one of the following is satified
\(\beta\geq 2\gamma + N -1\), and \(\beta\geq \alpha + \gamma\),
\(\beta\geq \alpha + \gamma + N\), where \(\alpha,\beta, \gamma\) be positive integers.
Proof. For the ample line bundle \(\mathscr{L} = \mathcal O_X(\alpha_1,\ldots,\alpha_t)\) we have the Segre embedding \[ i^* : X = \mathbb{P}^{a_1} \times \cdots \times \mathbb{P}^{a_n} \hookrightarrow \mathbb{P}\left(H^0(X, \mathcal{O}_X(\alpha_1, \ldots, \alpha_t))\right) \cong \mathbb{P}^N, \] such that \(i^*(\mathcal O_X(1))\simeq \mathscr{L}\) and where \(\displaystyle{N=\left({{a_1+\alpha_1}\choose\alpha_1}{{a_2+\alpha_2}\choose\alpha_2}\cdots{{a_n+\alpha_t}\choose\alpha_t}\right)-1}\).
Suppose that one of the conditions of Lemma 4 is satified thus there exists a linear monad \[ 0 \longrightarrow \mathcal{O}_{\mathbb{P}^N}(-1)^{\oplus \alpha} \xrightarrow{A} \mathcal{O}_{\mathbb{P}^N}^{\oplus \beta} \xrightarrow{B} \mathcal{O}_{\mathbb{P}^N}(1)^{\oplus \gamma} \longrightarrow 0. \] on \(\mathbf P^{N}\) whose morphisms are matrices \(A\) and \(B\) with entries monomials of degree one where \[\begin{aligned} A\in\mathop{\mathrm{Hom}}(\mathcal O_{\mathbf P^{N}}(-1)^{\oplus\alpha},\mathcal O_{\mathbf P^{N}}^{\oplus\beta})\cong H^0(\mathbf P^{N},\mathcal O_{\mathbf P^{N}}(1)^{\oplus\alpha\beta}), \\ B\in\mathop{\mathrm{Hom}}(\mathcal O_{\mathbf P^{N}}^{\oplus\beta},\mathcal O_{\mathbf P^{N}}(1)^{\oplus\gamma})\cong H^0(\mathbf P^{N},\mathcal O_{\mathbf P^{N}}(1)^{\oplus\beta\gamma}). \end{aligned}\]
Thus, \(A\) and \(B\) induce a monad on \(X\), \[ 0 \longrightarrow \mathcal{L}^{-1 \, \oplus \alpha} \xrightarrow{\bar{A}} \mathcal{O}_X^{\oplus \beta} \xrightarrow{\bar{B}} \mathcal{L}^{\oplus \gamma} \longrightarrow 0, \] where whose morphisms are matrices \(\bar{A}\) and \(\bar{B}\) with entries multidegree monomials such that \[\bar{A}\in\mathop{\mathrm{Hom}}(\mathcal O_{X}(-\alpha_1,\ldots,-\alpha_t)^{\oplus\alpha},\mathcal O^{\oplus\beta}_X),\] and \[\bar{B}\in\mathop{\mathrm{Hom}}(\mathcal O^{\oplus\beta}_X , \mathcal O_{X}(\alpha_1,\ldots,\alpha_t)^{\oplus\gamma}).\] ◻
Theorem 7. Let \(F\) be a vector bundle on \(X =\mathbf P^{a_1}\times\cdots\times\mathbf P^{a_n}\) defined by the short exact sequence \[ 0 \longrightarrow F \longrightarrow \mathcal{O}_X^{\oplus \beta} \xrightarrow{g} \mathcal{O}_X(\alpha_1, \cdots, \alpha_t)^{\oplus \gamma} \longrightarrow 0, \] then \(F\) is stable for an ample line bundle \(\mathscr{L} = \mathcal O_X(\alpha_1,\cdots,\alpha_t)\).
Proof. We are going to show that \(H^0(X,\bigwedge^q F(-p_1,\cdots,-p_{n}))=0\) for all \(\displaystyle{\sum\limits_{i=1}^{n}p_i>0}\) and \(1\leq q\leq \mathop{\mathrm{rank}}(F)-1\).
Consider the ample line bundle \(\mathscr{L} = \mathcal O_X(\alpha_1,\cdots,\alpha_t) = \mathcal O(L)\). Its class in \(\mathop{\mathrm{Pic}}(X)= \langle [h_i\times\mathbf P^{a_i}],i=1,\ldots,n]\rangle\) corresponds to \(\displaystyle{\sum\limits_{i=1}^{n}1.[h_i\times\mathbf P^{a_i}]}\) where each \(h_i\) is a hyperplane in \(\mathbf P^{a_i}\) with intersection product induced by \(h_i^{a_i} =1\) and \(h_i^{a_i+1}=0\) for \(i=1,\ldots,n\).
From the display of the monad we get
\[c_1(F) = c_1(\mathcal O_X^{\oplus\beta}) – c_1(\mathcal O_X(\alpha_1,\cdots,\alpha_t)^{\oplus\gamma}) =(-\gamma\alpha_1,\cdots,-\gamma\alpha_t).\]
Since \(L^{a_1+\cdots+a_n}>0\), the degree of \(F\) is \(\deg_{\mathscr{L}}F = c_1(T)\cdot\mathscr{L}^{d-1}\) that is \[\begin{aligned} \deg_{\mathscr{L}}F=-\gamma{n}{\sum\limits_{i=1}^t\alpha_i}([h_1\times\mathbf P^{a_1}]+\cdots+[\mathbf P^{a_n}\times h_{n}])\left(\displaystyle{\sum\limits_{i=1}^n1\cdot[h_i\times\mathbf P^{a_i}]}\right)^{{\sum\limits_{i=1}^{n}a_i}-1} =-\gamma{n}{\sum\limits_{i=1}^t\alpha_i}L^{(a_1+\cdots+a_n)}< 0. \end{aligned}\]
Since \(\deg_{\mathscr{L}}F<0\), then \((\bigwedge^q F)_{\mathscr{L}-norm} = (\bigwedge^q F)\) and it suffices by the generalized Hoppe Criterion (Proposition 1), to prove that \(h^0(\bigwedge^q F(-p_1,-p_2,\cdots,-p_{n})) = 0\) with \(\displaystyle{\sum\limits_{i=1}^{n}p_i>0}\) and for all \(1\leq q\leq \mathop{\mathrm{rank}}(F)-1\).
Next consider the exact sequence
\[ 0 \longrightarrow F \longrightarrow \mathcal{O}_X^{\oplus \beta} \xrightarrow{g} \mathcal{O}_X(\alpha_1, \cdots, \alpha_t)^{\oplus \gamma} \longrightarrow 0, \]
on twisting it by \( \mathcal{O}_X(-p_1, \cdots, -p_n) \) one gets,
\[ 0 \longrightarrow F(-p_1, \cdots, -p_n) \longrightarrow \mathcal{O}_X(-p_1, \cdots, -p_n)^{\oplus \beta} \xrightarrow{g} \mathcal{O}_X(\alpha_1 – p_1, \cdots, \alpha_t – p_n)^{\oplus \gamma} \longrightarrow 0, \]
and taking the exterior powers of the sequence by Proposition 2 one gets
\[ 0 \longrightarrow \bigwedge^q F(-p_1, \cdots, -p_n) \longrightarrow \bigwedge^q(\mathcal{O}_X(-p_1, \cdots, -p_n)^{\oplus \beta}) \longrightarrow \bigwedge^{q-1}(\mathcal{O}_X(\alpha_1 – 2p_1, \cdots, \alpha_t – 2p_n)^{\oplus \gamma}) \longrightarrow \cdots \]
Taking cohomology we have the injection:
\[ 0 \longrightarrow H^0(X, \bigwedge^q F(-p_1, \cdots, -p_n)) \hookrightarrow H^0(X, \bigwedge^q(\mathcal{O}_X(-p_1, \cdots, -p_n)^{\oplus \beta})). \]
From here \(h^0(X,\bigwedge^{q}F(-p_1,\cdots,-p_n)) = 0\) is proved in the same way as Lemma 7 the last part and thus \(F\) is stable. ◻
Theorem 8. Let \(X =\mathbf P^{a_1}\times\cdots\times\mathbf P^{a_n}\), then the cohomology vector bundle \(E\) associated to the monad \[ 0 \longrightarrow \mathcal{O}_X(-\alpha_1, \cdots, -\alpha_t)^{\oplus \alpha} \xrightarrow{f} \mathcal{O}_X^{\oplus \beta} \xrightarrow{g} \mathcal{O}_X(\alpha_1, \cdots, \alpha_t)^{\oplus \gamma} \longrightarrow 0, \] of rank \(\beta-\alpha-\gamma\) is simple.
Proof. The display of the monad is
Since \(T\) is stable from Theorem 7, we prove that the cohomology vector bundle \(E\) with rank \(2n\) is simple.
On taking the dual of the short exact sequence on the first row of the display diagram and tensoring by \(E\) we obtain
\[ 0 \longrightarrow E \otimes E^* \longrightarrow E \otimes F^* \longrightarrow E(t, \cdots, t)^{\oplus \alpha} \longrightarrow 0. \]
Now taking cohomology gives:
\[ 0 \longrightarrow H^0(X, E \otimes E^*) \longrightarrow H^0(X, E \otimes F^*) \longrightarrow H^0(E(\alpha_1, \cdots, \alpha_t)^{\oplus \alpha}) \longrightarrow \cdots, \]
which implies that\[\label{eq3} h^0(X,E\otimes E^*) \leq h^0(X,E\otimes F^*). \tag{3}\]
Dualize the short exact sequence on the first column of the display diagram to get \[ 0 \longrightarrow \mathcal{O}_X(-\alpha_1, \cdots, -\alpha_t)^{\oplus \gamma} \longrightarrow \mathcal{O}_X^{\oplus \beta} \longrightarrow F^* \longrightarrow 0. \]
Now twisting the short exact sequence above by \( \mathcal{O}_X(-\alpha_1, \cdots, -\alpha_t) \) one obtains the short exact sequence
\[ 0 \longrightarrow \mathcal{O}_X(-2\alpha_1, \cdots, -2\alpha_t)^{\oplus \gamma} \longrightarrow \mathcal{O}_X(-\alpha_1, \cdots, -\alpha_t)^{\oplus \beta} \longrightarrow F^*(-\alpha_1, \cdots, -\alpha_t) \longrightarrow 0. \]
Next on taking cohomology one gets
\[ 0 \longrightarrow H^0(\mathcal{O}_X(-2\alpha_1, \cdots, -2\alpha_t)^{\oplus \gamma}) \longrightarrow H^0(\mathcal{O}_X(-\alpha_1, \cdots, -\alpha_t)^{\oplus \beta}) \longrightarrow H^0(F^*(-\alpha_1, \cdots, -\alpha_t)) \longrightarrow \] \[ \longrightarrow H^1(\mathcal{O}_X(-2\alpha_1, \cdots, -2\alpha_t)^{\oplus \gamma}) \longrightarrow H^1(\mathcal{O}_X(-\alpha_1, \cdots, -\alpha_t)^{\oplus \beta}) \longrightarrow H^1(F^*(-\alpha_1, \cdots, -\alpha_t)) \longrightarrow \] \[ \longrightarrow H^2(\mathcal{O}_X(-2\alpha_1, \cdots, -2\alpha_t)^{\oplus \gamma}) \longrightarrow H^2(\mathcal{O}_X(-\alpha_1, \cdots, -\alpha_t)^{\oplus \beta}) \longrightarrow H^2(F^*(-\alpha_1, \cdots, -\alpha_t)) \longrightarrow \cdots, \]
from which we deduce \(H^0(X,F^*(-\alpha_1,\cdots,-\alpha_t)) = 0\) and \(H^1(X,F^*(-\alpha_1,\cdots,-\alpha_t)) = 0\) from Lemmas 1, 2 and Theorem 3.Lastly, tensor the short exact sequence
\[ 0 \longrightarrow \mathcal{O}(-\alpha_1, \cdots, -\alpha_t)^{\oplus k} \longrightarrow F \longrightarrow E \longrightarrow 0, \]
by \( F^* \) to get
\[ 0 \longrightarrow F^*(-\alpha_1, \cdots, -\alpha_t)^k \longrightarrow F \otimes F^* \longrightarrow E \otimes F^* \longrightarrow 0, \]
and taking cohomology we have
\[ 0 \longrightarrow H^0(X, F^*(-\alpha_1, \cdots, -\alpha_t)^k) \longrightarrow H^0(X, F \otimes F^*) \longrightarrow H^0(X, E \otimes F^*) \longrightarrow \] \[ \longrightarrow H^1(X, F^*(-\alpha_1, \cdots, -\alpha_t)^k) \longrightarrow \cdots \]
But since \(H^0(X,F^*(-\alpha_1,\cdots,-\alpha_t)) = H^1(X,F^*(-\alpha_1,\cdots,-\alpha_t)) = 0\) from above then it follows \(H^1(X,F^*(-\alpha_1,\cdots,-\alpha_t)^k)=0\) for \(k>1\), so we have \[ 0 \longrightarrow H^0\left(X, F^*(-\alpha_1, \cdots, -\alpha_t)^k\right) \longrightarrow H^0(X, F \otimes F^*) \longrightarrow H^0(X, E \otimes F^*) \longrightarrow 0. \]
This implies that \[\label{eq4} h^0(X,F\otimes F^*) \leq h^0(X,E\otimes F^*). \tag{4}\]
Since \(F\) is stable then it is simple implying \(h^0(X,F\otimes F^*)=1\).
From (3) and (4) and putting these together we have;
\[1\leq h^0(X,E\otimes E^*) \leq h^0(X,E\otimes F^*) = h^0(X,F\otimes F^*) = 1.\]
We have \(h^0(X,E\otimes E^*) = 1\) and therefore \(E\) is simple. ◻
Let \(X\) be a nonsingular projective variety. A monad \[ 0 \longrightarrow M_0 \xrightarrow{\alpha} M_1 \xrightarrow{\beta} M_2 \longrightarrow 0 \] on \(X\) exists if one can give the morphisms \(\alpha\) and \(\beta\). In this section we establish the existence of monads on \(\mathbf P^1\times\cdots\times\mathbf P^1\) by providing an explicit contruction of the morphisms derived from the matrices used by Fløystad [6] and Ancona and Ottaviani [13].
Construction 1. Let \(\psi : X = \mathbf P^1\times\cdots\times\mathbf P^1\longrightarrow \mathbf P^{N=2n+1}\) be the Segre embedding which is defined as follows:
\[[\alpha_{10}:\alpha_{11}][\alpha_{20}:\alpha_{21}]:\ldots:[\alpha_{m0}:\alpha_{m1}]\hookrightarrow [x_0:x_1:\cdots:x_n:y_0:y_2:\ldots:y_n].\]
First note that since we are taking \(m\) copies of \(\mathbf P^1\) then we have
\[N=2^m-1=2^m-2+1=2(2^{m-1}-1)+1=2n+1,\] i.e., \(N=2n+1\) where \(m\) and \(n\) are positive integers such that \(n=2^{m-1}-1\).
Thus from Lemma 5, there exists a linear monad \[ 0 \longrightarrow \mathcal{O}_{\mathbb{P}^{2n+1}}(-1)^{\oplus k} \xrightarrow{A} \mathcal{O}_{\mathbb{P}^{2n+1}}^{\oplus (2n+2k)} \xrightarrow{B} \mathcal{O}_{\mathbb{P}^{2n+1}}(1)^{\oplus k} \longrightarrow 0, \] whose morphisms \(A\) and \(B\) that establish the monad are as given in Lemma 5.
We induce a monad on \(X=\mathbf P^1\times\cdots\times\mathbf P^1\) \[ M_\bullet : 0 \longrightarrow \mathcal{O}_X(-1, \cdots, -1)^{\oplus k} \xrightarrow{\bar{A}} \mathcal{O}_X^{\oplus (2n+2k)} \xrightarrow{\bar{B}} \mathcal{O}_X(1, \cdots, 1)^{\oplus k} \longrightarrow 0, \] by giving the morphisms \(\overline{A}\) and \(\overline{B}\) with \(\overline{B}\cdot\overline{A}=0\) and \(\overline{A}\) and \(\overline{B}\) are of maximal rank.
From \(A\) and \(B\) whose entries are \(x_0,\cdots,x_n,y_0,\cdots,y_n\) the homogeneous coordinates on \(\mathbf P^{2n+1}\) we give the correspondence for the the Segre embedding using the following table:
\[\begin{array}{|c|c|} \hline homog. coord. ~~on ~~\mathbf P^{2n+1} & representation ~homog. coord.~~ on ~~X\\ \hline x_0 & a_{0000\cdots0000} \\\hline x_1 & a_{0000\cdots0001} \\\hline x_2 & a_{0000\cdots0010} \\\hline x_3 & a_{0000\cdots0011} \\\hline x_4 & a_{0000\cdots0100} \\\hline \vdots&\vdots\\\hline x_{n-1} & a_{0111\cdots1110} \\\hline x_n & a_{0111\cdots1111} \\\hline y_0 & a_{1000\cdots0000} \\\hline y_1 & a_{1000\cdots0001} \\\hline y_2 & a_{1000\cdots0010} \\\hline y_3 & a_{1000\cdots0011} \\\hline y_4 & a_{1000\cdots0100} \\\hline \vdots&\vdots\\\hline y_{n-1} & a_{1111\cdots1110} \\\hline y_n & a_{1111\cdots1111} \\ \hline \end{array}\] where \(a_{iiii\cdots iiii}\) for \(i\) is \(0\) or \(1\) are monomials of multidegree \((1,\ldots,1)\), i.e.,
\[\begin{array}{|c|c|} \hline representation ~~homog. coord. ~~on ~~\mathbf P^{2n+1} & homog. coord.~~ on ~~X\\ \hline a_{0000\cdots0000} & \alpha_{10}\alpha_{20}\alpha_{30}\alpha_{40}\cdots\alpha_{(m-3)0}\alpha_{(m-2)0}\alpha_{(m-1)0}\alpha_{m0} \\\hline a_{0000\cdots0001} & \alpha_{10}\alpha_{20}\alpha_{30}\alpha_{40}\cdots\alpha_{(m-3)0}\alpha_{(m-2)0}\alpha_{(m-1)0}\alpha_{m1} \\\hline a_{0000\cdots0010} & \alpha_{10}\alpha_{20}\alpha_{30}\alpha_{40}\cdots\alpha_{(m-3)0}\alpha_{(m-2)0}\alpha_{(m-1)1}\alpha_{m0} \\\hline a_{0000\cdots0011} & \alpha_{10}\alpha_{20}\alpha_{30}\alpha_{40}\cdots\alpha_{(m-3)0}\alpha_{(m-2)0}\alpha_{(m-1)1}\alpha_{m1} \\\hline a_{0000\cdots0100} & \alpha_{10}\alpha_{20}\alpha_{30}\alpha_{40}\cdots\alpha_{(m-3)0}\alpha_{(m-2)1}\alpha_{(m-1)0}\alpha_{m0} \\\hline \vdots&\vdots\\\hline a_{0111\cdots1110} & \alpha_{10}\alpha_{21}\alpha_{31}\alpha_{41}\cdots\alpha_{(m-3)1}\alpha_{(m-2)1}\alpha_{(m-1)1}\alpha_{m0} \\\hline a_{0111\cdots1111} & \alpha_{10}\alpha_{21}\alpha_{31}\alpha_{41}\cdots\alpha_{(m-3)1}\alpha_{(m-2)1}\alpha_{(m-1)1}\alpha_{m1} \\\hline a_{1000\cdots0000} & \alpha_{11}\alpha_{20}\alpha_{30}\alpha_{40}\cdots\alpha_{(m-3)0}\alpha_{(m-2)0}\alpha_{(m-1)0}\alpha_{m0} \\\hline a_{1000\cdots0001} & \alpha_{11}\alpha_{20}\alpha_{30}\alpha_{40}\cdots\alpha_{(m-3)0}\alpha_{(m-2)0}\alpha_{(m-1)0}\alpha_{m1} \\\hline a_{1000\cdots0010} & \alpha_{11}\alpha_{20}\alpha_{30}\alpha_{40}\cdots\alpha_{(m-3)0}\alpha_{(m-2)0}\alpha_{(m-1)1}\alpha_{m0} \\\hline a_{1000\cdots0011} & \alpha_{11}\alpha_{20}\alpha_{30}\alpha_{40}\cdots\alpha_{(m-3)0}\alpha_{(m-2)0}\alpha_{(m-1)1}\alpha_{m1} \\\hline a_{1000\cdots0100} & \alpha_{11}\alpha_{20}\alpha_{30}\alpha_{40}\cdots\alpha_{(m-3)0}\alpha_{(m-2)1}\alpha_{(m-1)0}\alpha_{m0} \\\hline \vdots&\vdots\\\hline a_{1111\cdots1110} & \alpha_{11}\alpha_{21}\alpha_{31}\alpha_{41}\cdots\alpha_{(m-3)1}\alpha_{(m-2)1}\alpha_{(m-1)1}\alpha_{m0} \\\hline a_{1111\cdots1111} & \alpha_{11}\alpha_{21}\alpha_{31}\alpha_{41}\cdots\alpha_{(m-3)1}\alpha_{(m-2)1}\alpha_{(m-1)1}\alpha_{m1} \\ \hline \end{array}\]
Specifically we define \(\overline{A}\) and \(\overline{B}\) as follows \[\overline{B} :=\left[ \begin{array}{ccc|cccccc} a_{0000\cdots0000}~~~\cdots & a_{0111\cdots1111} & &a_{1000\cdots0000} ~~~\cdots & a_{1111\cdots1111}\\ \ddots&\ddots &&\ddots&\ddots\\ & a_{0000\cdots0000}~~~ \cdots~~~ a_{0111\cdots1111} && & a_{1000\cdots0000}~~~ \cdots & a_{1111\cdots1111} \end{array} \right],\] and \[\overline{A} :=\left[ \begin{array}{cccccccc} -a_{1000\cdots0000}\cdots & -a_{1111\cdots1111} \\ &\ddots &\ddots\\ &&-a_{1000\cdots0000} \cdots & -a_{1111\cdots1111}\\ \hline a_{0000\cdots0000}\cdots & a_{0111\cdots1111} \\ &\ddots &\ddots\\ && a_{0000\cdots0000}\cdots & a_{0111\cdots1111}\\ \end{array} \right].\]
We note that
\(\overline{B}\cdot \overline{A} = 0\), and
The matrices \(\overline{B}\) and \(\overline{A}\) have maximal rank.
Hence we get the desired monad, \[ M_\bullet : 0 \longrightarrow \mathcal{O}_X(-1, \cdots, -1)^{\oplus k} \xrightarrow{\bar{A}} \mathcal{O}_X^{\oplus (2n + 2k)} \xrightarrow{\bar{B}} \mathcal{O}_X(1, \cdots, 1)^{\oplus k} \longrightarrow 0. \]
I wish to express sincere thanks to the Department of Mathematics and Actuarial Science, Catholic University of Eastern Africa and the Department of Mathematics, University of Nairobi, for providing a conducive enviroment to be able to carry out research despite the overwhelming duties in teaching and community service. I am also extremely grateful to Melissa, my wife who always encourages me to keep on and lastly to Amelia, Jerome and Wachuka who are always around when I am working on my research at home.
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