Numerical bounds of canonical varieties

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Let X be a minimal, complex, projective, Gorenstein variety of dimension n. We say that X is canonical if for some (any) desingularization σ : Y −→ X, the map associated to the canonical linear series |KY | is birational. We note KX for the canonical divisor of X and ωX = OX(KX) the canonical sheaf. Let pg = h (X,ωX), q = h (X,OX). There are several known bounds for K X depending on pg, the most general one being the bound K X ≥ (n + 1)pg + dn (dn constant) given by Harris ([9]). Bounds including other invariants are known for canonical surfaces, K S ≥ 3pg + q − 7 ([12], [7]), and for surfaces and threefolds fibred over curves ([20], [25]). In this paper we prove some results for canonical surfaces and threefolds. In the case of canonical surfaces there are some known results which show that under some additional hypotheses, the bound K S ≥ 3pg + q − 7 can be considerably improved (see Remark 2.2). We give here some other special cases (Remark 2.2) for which is not sharp and prove (Theorem 2.1) that, in fact, K S = 3pg + q − 7 only if q = 0 whenever pg(S) ≥ 6. Canonical surfaces with K S = 3pg − 7 are known to exist and classified ([1]).Then we can hope that a good bound for canonical surfaces including the irregularity should be of type K S ≥ 3pg + aq − 7, a > 1. Since for q = 1 it is known ([16]) that K S ≥ 3pg, a should be 7, although unfortunately examples of low K S (with q ≥ 2) are not known. In the case of canonical threefolds we prove that K X ≥ 4pg + 6q− 32. In particular, we prove that the results of Ohno for canonical fibred threefolds are not sharp.