On syzygies of projective bundles over abelian varieties (Q1730875)

Many papers (a big one is [\textit{G. Pareschi}, J. Am. Math. Soc. 13, No. 3, 651--664 (2000; Zbl 0956.14035)]) studied syzygies of line bundles on abelian variety. Here the author proves a syzygy theorem fo projective bundles over abelian varieties. Let $E$ a rank $r$ vector bundle on a complex abelian variety $X$ with $\dim X =g$. Assume that $E$ has a filtration $0 =E_0\subset E_1\subset \cdots \subset E_{r-1}\subset E_r=E$ with each $E_i/E_{i-1}\in \mathrm{Pic}^0(X)$. Let $\pi :\mathbb {P}(E)\to X$ be the associated $\mathbb {P}^{r-1}$-bundle morphism and $\mathcal {O}_{\mathbb {P}(E)}(1)$ the hyperplane line bundle. The author proves that $\mathcal {O}_{\mathbb {P}(E)}(1)\otimes \pi ^\ast (A)^{\otimes p+3}$ has property $N_p$ on $\mathbb {P}(E)$ for every ample line bundle $A$ on $X$. The method of proof is an adaptation of [\textit{E. Park}, J. Pure Appl. Algebra 211, No. 1, 15--23 (2007; Zbl 1121.14044)].