A Note on the Effective Non-vanishing Conjecture

We give a reduction of the irregular case for the effective non-vanishing conjecture by virtue of the Fourier-Mukai transform. As a consequence, we reprove that the effective non-vanishing conjecture holds on algebraic surfaces. In this note we consider the following so-called effective non-vanishing conjecture, which has been put forward by Ambro and Kawamata [Am99, Ka00]. Conjecture 1 (ENn). Let X be a proper normal variety of dimension n, B an effective R-divisor on X such that the pair (X,B) is Kawamata log terminal, and D a Cartier divisor on X. Assume that D is nef and that D − (KX + B) is nef and big. Then H(X,D) 6= 0. This conjecture is closely related to the minimal model program and plays an important role in the classification theory of Fano varieties. For a detailed introduction to this conjecture, we refer the reader to [Xie06]. By the Kawamata-Viehweg vanishing theorem, we have H (X,D) = 0 for any positive integer i. Thus H(X,D) 6= 0 is equivalent to χ(X,D) 6= 0. Under the same assumptions as in Conjecture 1, the Kawamata-Shokurov non-vanishing theorem says that H(X,mD) 6= 0 for all m ≫ 0. Thus the effective non-vanishing conjecture is an improvement of the non-vanishing theorem in some sense. Note that EN1 is trivial by the Riemann-Roch theorem, and that EN2 was settled by Kawamata [Ka00, Theorem 3.1] by virtue of the logarithmic semipositivity theorem. For n ≥ 3, only a few results are known. For instance, ENn holds trivially for toric varieties [Mu02], EN3(X, 0) holds for all canonical projective minimal threefolds X [Ka00, Proposition 4.1], and EN3(X, 0) also holds for almost all of canonical projective threefolds X with −KX nef [Xie05, Corollary 4.5]. In this note, we shall prove that, in the irregular case, the effective non-vanishing conjecture can be reduced to lower-dimensional cases by means of the Fourier-Mukai transform. As consequences, EN2 is reproved after Kawamata, and ENn holds for all varieties which are birational to an abelian variety. Throughout this note, we work over the complex number field C. For the definition of Kawamata log terminal (KLT, for short) and the other notions, we refer the reader to [KMM87, KM98]. For irregular varieties, the study of the Albanese map provides enough information to understand their birational structure. Therefore, through the Albanese map, we can utilize the Fourier-Mukai transform to give a reduction of the effective non-vanishing conjecture for irregular varieties. This idea was first used in [CH02]. First of all, we need the following lemma which follows easily from [Mu81, Theorem 2.2]. Lemma 2. Let A be an abelian variety, F a coherent sheaf on A. Assume that H (A,F ⊗ P ) = 0 for all P ∈ Pic(A) and all i. Then F = 0.