Big Line Bundles over Arithmetic Varieties

Abstract

We prove a Hilbert-Samuel type result of arithmetic big line bundles in Arakelov geometry, which is an analogue of a classical theorem of Siu. An application of this result gives equidistribution of small points over algebraic dynamical systems, following the work of Szpiro-Ullmo-Zhang. We also generalize Chambert-Loir's non-archimedean equidistribution.

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• The Dynamical Andre-Oort Conjecture: Unicritical Polynomials
• Differentiability of relative volumes over an arbitrary non-Archimedean field
• Effective bound of linear series on arithmetic surfaces
• An arithmetic Hilbert-Samuel theorem for singular hermitian line bundles and cusp forms
• Roth's Theorem over arithmetic function fields
• Arakelov geometry, heights, equidistribution, and the Bogomolov conjecture
• Geometric Bogomolov conjecture in arbitrary characteristics
• Algebraic zeros divisors on the projective line having small diagonals and small heights and their application to adelic dynamics
• Impossible intersections in a Weierstrass family of elliptic curves
• A Bertini-type theorem for free arithmetic linear series
• Simultaneously preperiodic points for a family of polynomials in positive characteristic
• Quantitative equidistribution of Galois orbits of small points in the N-dimensional torus
• Space spanned by characteristic exponents
• Geometry of PCF parameters in spaces of quadratic polynomials
• Slope boundedness and Equidistribution theorem
• Higher dimensional essential minima and equidistribution of cycles
• Monge-Ampère measures for toric metrics on abelian varieties