this paper, we work over an algebraically closed field $K$. Basically we follow the standard notation and terminology in algebraic geometry. For example, for a vector bundle $\mathcal{E}$, $\mathbb{P}(\mathcal{E})$ denotes ${\mathop{\rm Proj}\nolimits}S(\mathcal{E})$, where $S(\mathcal{E})$ denotes the symmetric algebra of $\mathcal{E}$. The tautological line bundle $\mathcal{O}_{\mathbb{P}(\mathcal{E})}(1)$ is also denoted by $H(\mathcal{E})$. For a coherent sheaf $\mathcal{F}$ on a smooth projective variety $X$, we denote by $c_i(\mathcal{F})$ the $i$-th Chern class of $\mathcal{F}$. For a smooth projective variety $X$, denote by $D^b(X)$ the bounded derived category of the abelian category of coherent sheaves on $X$, and call $D^b(X)$ the bounded derived category of coherent sheaves on $X$ for short. We say that a vector bundle is spanned or globally generated if it is generated by global sections. For “spinor bundles”, we follow Kapranov’s convention [@MR0939472]; our spinor bundles are spanned and they are the duals of those of Ottaviani’s [@ot]. See [@MR3275418 §5 Definition 1] for a precise definition of our spinor bundles. Finally we refer to [@MR2095472] for the definition and basic properties of nef vector bundles.

Preliminaries {#Preliminaries}
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Let $X$ be an $n$-dimensional smooth projective variety over $K$, and suppose that there exists a full strong exceptional sequence $G_0,\dots,G_m$ 