On Severi type inequalities
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Mathematische Annalen
On Severi type inequalities Zhi Jiang1 Received: 18 April 2019 / Revised: 20 May 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We study Severi type inequalities for big line bundles on irregular varieties via cohomological rank functions. We show that these Severi type inequalities are related to some naturally defined birational invariants of the general fibers of the Albanese morphisms. As an application, we show that the volume of an irregular threefold of general type is at least 38 . We also show that the volume of a smooth projective variety X of general type and of maximal Albanese dimension is at least 2(dim X )!. Moreover, if vol(X ) = 2(dim X )!, the canonical model of X is a double cover of a principally polarized abelian variety (A, ) branched over some divisor D ∈ |2|.
1 Introduction Xiao proved an inequality among certain Chern numbers for surfaces with fibrations to curves [36], which is now called Xiao’s slope inequality. Assume that f : X → C is a relatively minimal and not locally trivial fibration from a smooth projective surface to a smooth curve. Let g be the genus of a general fiber of f and let b be the genus of C. Then Xiao showed that 2 K S/C
1 ≥4 1− (χ (O S ) − (g − 1)(b − 1)) . g
Note that χ (O S ) = χ (ω S ) is a birational invariant of S. If S is minimal surface, then 2 = K S2 − 2K S · f ∗ K C = vol(S) − 8(g − 1)(b − 1) and we can rewrite Xiao’s K S/C
Communicated by Vasudevan Srinivas. The author is partially supported by NSFC Grant Nos. 11871155 and 11731004.
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Zhi Jiang [email protected] Shanghai Center for Mathematical Sciences, Xingjiangwan Campus, Fudan University, Shanghai 200438, People’s Republic of China
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Z. Jiang
inequality as an inequality between birational invariants: vol(S) ≥
4(g − 1) (g 2 − 1)(b − 1) χ (ω S ) + . g g
This inequality played an essential role in Pardini’s proof of Severi’s inequality of surfaces [28]. Given a surface S of general type, of maximal Albanese dimension, Pardini noted that one can construct étale covers S M → S such that vol(S M ) = M 2q(S) vol(S) and χ (ω SM ) = M 2q(S) χ (ω S ) and fibrations f M : S M → P1 such that the genus of a general fiber of f M is O(M 2q(S)−2 ), for each M ∈ N. Applying Xiao’s equality for these fibrations f M , Pardini proved that vol(S) ≥ 4χ (ω S ), which is called Severi’s inequality. Severi’s inequality was extended to varieties of maximal Albanese dimensions of higher dimensions independently by Barja [2] and Zhang [38]. Let X be a smooth projective variety of general type and of maximal Albanese dimension, then vol(X ) ≥ 2(dim X )!χ (ω X ). A crucial feature in both proofs is that, by generic vanishing, one could regard χ (ω X ) as h 0 (X , ω X ⊗ Q), where Q ∈ Pic0 (X ) is a general numerically trivial line bundle, and hence χ (ω X ) ≥ 0. Note that this inequality gives a natural lower bound for vol(X ) when χ (ω X ) > 0, however, χ (ω X ) could be 0 in dimension ≥ 3 [18]. A refined Severi’s inequality of surfaces was obtained by Lu
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