Effective Hilbert-Samual Inequality And Slope Inequality On Arithmetic Surfaces

In the study of number theory, it is always a very popular way trying to view the problems over number fields as analogues of them over function fields, which is the starting point of Arakelov geometry.In the theory of Arakelov geometry, one of the most important problems is to find the relationship between some fundamental invariants of an arithmetic line bundle L on an arithmetic variety X, for example, the self-intersection number Ldimx, effective sections h0(L) and the character Xsup.The arithmetic Hilbert-Samuel formula was first set up by Gillet-Soule [GS2] and S. Zhang [Zh4] to indicate the asymptotic relationship between the self-intersection of L(?)n and Xsup(L(?)n) for ample L and n sufficiently large. Then this result was generalized to the arithmetic nef line bundles and big line bundles by Moriwaki [Mo2] and X. Yuan [Yul]. Moreover, the arithmetic Fujita approximation was studied by H. Chen [Ch2] and X. Yuan [Yu2], which reflects the asymptotic relationship of big line bundles and ample line bundles. In this paper, we consider the effective comparison between L2, h0(L) and Xsup(L) when X is an arithmetic surface. Namely, we give the upper bounds of h0(L) and Xsup(L) in terms of L2, which can be viewed as an effective arithmetic Hilbert-Samuel formula on arithmetic surfaces. It is also a generalization of the classical Noether inequality on algebraic surfaces.Positivity is also an important topic in Arakelov geometry. The positivity of ω2of the Arakelov canonical bundle of an arithmetic surface was proved by Ullmo [Ul] and S. Zhang [Zh2], which serves as a key step in the proof of the Bogomolov conjecture. As an application of our effective result, we give different lower bounds of ωX2of an arithmetic surface X in terms of h0(ωX), Xsup(ωX) and the Faltings height XFal(ωX).Our result is an arithmetic version of the slope inequality on fibered surfaces [CH, Xil]. In particular, as a corollary, we recover a result of Bost [Bol] with explicit error terms.