Chern Numbers of Ample Vector Bundles on Toric Surfaces

Let E be an ample rank r bundle on a smooth toric projective surface, S, whose topological Euler characteristic is e(S). In this article, we prove a number of surprisingly strong lower bounds for c1(E) and c2(E). First, we show Corollary (3.2), which says that, given S and E as above, if e(S) ≥ 5, then c1(E) ≥ r2e(S). Though simple, this is much stronger than the known lower bounds over not necessarily toric surfaces. For example, see [BSS94, Lemma 2.2], where it is shown that there are many rank two ample vector bundles with (c1(E), c2(E)) = (2, 1) on products of two smooth curves, at least one of which has positive genus. We then prove an estimate, Theorem 3.6, which is quite strong for large e(S) and r. As e(S) goes to ∞ with r fixed, the leading term of this lower bound is (4r + 2)e(S) ln2(e(S)/12), while if e is fixed and r goes to ∞, the leading term of this lower bound is 3(e(S) − 4)r. For example, c1(E) ≥ 3r2e(S), for r ≤ 3 if e(S) ≥ 13, or for r ≤ 6 if e(S) ≥ 19, or for r ≤ 141 if e(S) ≥ 100. Or again, c1(E) ≥ 5r2e(S), for r ≤ 10 if e(S) ≥ 100. We include a three line Maple program in Remark 3.7 for plotting the expression for the lower bound. The strategy is to use the adjunction process to find lower bounds for c1(E). Toric geometry has two major implications for the adjunction process. First, given an ample rank r vector bundle E on a smooth toric surface S, there is the inequality − det E ·KS ≥ e(S)(rank E). Adjunction theory yields the lower bound for c1(E) given in Theorem 3.2, which implies that c1(E) > re(S) for e(S) ≥ 7. The second important fact is that h(tKS +det E) > 0 for integers t between 0 and at least rankE+ln2(e(S)/6). Adjunction theory yields the strong lower bound given in Theorem (3.6) for c1(E) when e(S) ≥ 7. Using Bogomolov’s instability theorem, we get the strong lower bound given in Theorem (3.9) for the second Chern class, c2(E), of a rank two ample vector bundle. Basically if c2(E) is less than one fourth the lower bound already derived for c1(E), then we have an unstable bundle, and Bogomolov’s instability theorem combined with the Hodge index theorem give strong enough conditions to get a contradiction. The short list of exceptions to the bound c2(E) > e(S) are classified. Even assuming E very ample on a nontoric surface, the best general result [BSS96] shows only that c2(E) ≥ 1 with equality for P2. Inequalities derived from adjunction theory usually have the form, “some inequality is true if certain projective invariants are large enough.” Typically examples exist outside the range where the adjunction theoretic method works. For rank two ample vector bundles E we use a variety of special methods, including adjunction theory and Bogomolov’s instability theorem, to enumerate the exceptions to either the inequality c1(E) ≥ 4e(S) or the inequality c2(E) ≥ e(S) holding. The exceptions are collected in Table 1.