Logarithmic Kodaira dimension and zeros of holomorphic log-one-forms
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Mathematische Annalen
Logarithmic Kodaira dimension and zeros of holomorphic log-one-forms Chuanhao Wei1 Received: 16 November 2019 / Revised: 20 March 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract In this paper, we prove that the zero-locus of any global holomorphic log-one-form on a projective log-smooth pair (X ; D) of log-general type must be non-empty.
1 Introduction In this paper, we prove the conjecture that appeared in [19], which is a natural generalization of the work of Popa and Schnell [12]. Theorem 1 The zero-locus of any global holomorphic log-one-form on a projective log-smooth pair (X , D) of log-general type must be non-empty. Actually, we can prove a more general result, which in the D = 0 case, is a conjecture initially proposed in [8] and is proved in [12]. Theorem 2 Let (X , D) be a projective log-smooth pair, and let W ⊂ H 0 X , 1X (log D) be a linear subspace that consists of global holomorphic log-one-forms with empty zero-locus. Then the dimension of W can be at most dim X −κ (X , D), where κ stands for the logarithmic Kodaira dimension. This is a corollary to the following main theorem of this paper. Before stating the theorem, we recall the notations and some propositions about quasi-abelian varieties in the sense of Iitaka, given in [19]. See also [3,5] for more details.
Communicated by Vasudevan Srinivas.
B 1
Chuanhao Wei [email protected] Stony Brook University, Stony Brook, USA
123
C. Wei
Definition 1 T r ,d is a quasi-abelian variety (in the sense of Iitaka), if it is an extension of a d-dimensional abelian variety Ad by an algebraic torus Grm , i.e. it is a connected commutative algebraic group which has the following Chevalley decomposition 1 → Grm → T r ,d → Ad → 1. In particular, T r ,d is a principal Grm -bundle over Ad . Consider the following group homomorphism: ρ : Grm → P G L(r , C), given by
⎡ ⎢ ⎢ ρ(λ1 , . . . , λr ) = ⎢ ⎣
1
0 λ1
0
..
.
⎤ ⎥ ⎥ ⎥ ⎦
λr
Let P r ,d := T r ,d ×ρ Pr = T r ,d × Pr /Grm , which is a Pr -bundle over Ad . We can view P r ,d as a compactification of T r ,d , which naturally carries the T r ,d action on it. Denote the boundary divisor L, which is simply normal crossing (SNC). The stratification given by the intersections of components of L is T r ,d -invariant, i.e. each stratum is T r ,d -invariant. This implies that the sheaf of log-one-forms 1P r ,d (log L)
on P r ,d , L is a trivial r + d-rank vector bundle, with its global sections are given by those T r ,d -invariant log-one-forms. We say that (X , D) is a log-smooth pair, if X is a smooth variety, and D is a reduced divisor on X , with normal crossing support. Given two log smooth pairs X , D X and
Y , D Y , and a morphism f : X → Y , we say that f is a morphism of log-pairs and
denoted by f : X , D X → Y , D Y , if f −1 D Y := Supp f ∗ D Y ⊂ D X . Given a holomorphic log-one-form θ on a log smooth pair (X , D), we use Z (θ ) to denote the zero-locus of θ , as a global section of the locally free sheaf 1X (log
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