On the vanishing of weight one Koszul cohomology of abelian varieties (Q2801722)

If \(L\) is an ample line bundle on a complex abelian variety \(X\), then \textit{G. Pareschi} [J. Am. Math. Soc. 13, No. 3, 651--664 (2000; Zbl 0956.14035)], proved that \(L^a\) has property \((N_p)\) for \(a \geq p+3\), thereby establishing a conjecture of Lazarsfeld.NEWLINENEWLINEIn the article under review, the authors establish two complementary results which pertain to twisted Koszul cohomology groups of the form \(K_{p,1}(X,B;L^a)\); here \(B\) is a line bundle on \(X\) with the property that the line bundle \(bL-B\) is ample for some integer \(b \geq 1\).NEWLINENEWLINETo state the authors' first result, let \(L\) be an ample line bundle on an abelian variety \(X\) of dimension \(n\geq 3\), fix an integer \(a \geq 2\) and set \(r_a := h^0(X,L^a)-1\). The authors prove that if \(B\) is a line bundle on \(X\) with the property that \(bL-B\) is ample for some \(b \geq 1\), then NEWLINE\[NEWLINEK_{p,1}(X,B;L^a) = 0,NEWLINE\]NEWLINE for \(p \leq r_a - n\), provided that \( p \geq r_a - a(n-1)+b(1-\frac{1}{a})\) and \(a \geq b\).NEWLINENEWLINEThe authors' second result applies to globally generated line bundles \(L\) on polarized abelian varieties \((X,H)\) of dimension \(n \geq 3\) such that \(L-2H\) is ample. To state this result, let \(r := h^0(X,L) -1\) and let \(B\) be a line bundle on \(X\) such that \(L-B\) is ample. The authors prove that NEWLINE\[NEWLINEK_{p,1}(X,B;L) = 0,NEWLINE\]NEWLINE for \(p \leq r-n\), if either \(p > r-2n+2\) or \(p = r-2n+2\) and \(-B\) is nef.NEWLINENEWLINEAn important aspect to the proof of these results is Lemma 2.1 of the paper under review. This lemma applies to smooth complex projective varieties of dimension at least \(2\) and gives a sufficient condition for twisted Koszul cohomology groups of weight \(1\) \(p\)-syzygies to vanish. This condition is phrased in terms of the surjectivity of suitably defined multiplication maps.