LEADER 04171nam 22006015 450 001 996418269703316 005 20200706091649.0 010 $a981-15-1728-2 024 7 $a10.1007/978-981-15-1728-0 035 $a(CKB)4100000010121888 035 $a(DE-He213)978-981-15-1728-0 035 $a(MiAaPQ)EBC6112039 035 $a(PPN)242842720 035 $a(EXLCZ)994100000010121888 100 $a20200129d2020 u| 0 101 0 $aeng 135 $aurnn|008mamaa 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aArakelov Geometry over Adelic Curves$b[electronic resource] /$fby Huayi Chen, Atsushi Moriwaki 205 $a1st ed. 2020. 210 1$aSingapore :$cSpringer Singapore :$cImprint: Springer,$d2020. 215 $a1 online resource (XVIII, 452 p.) 225 1 $aLecture Notes in Mathematics,$x0075-8434 ;$v2258 311 $a981-15-1727-4 327 $aIntroduction -- Metrized vector bundles: local theory -- Local metrics -- Adelic curves -- Vector bundles on adelic curves: global theory -- Slopes of tensor product -- Adelic line bundles on arithmetic varieties -- Nakai-Moishezon?s criterion -- Reminders on measure theory. 330 $aThe purpose of this book is to build the fundament of an Arakelov theory over adelic curves in order to provide a unified framework for research on arithmetic geometry in several directions. By adelic curve is meant a field equipped with a family of absolute values parametrized by a measure space, such that the logarithmic absolute value of each non-zero element of the field is an integrable function on the measure space. In the literature, such construction has been discussed in various settings which are apparently transversal to each other. The authors first formalize the notion of adelic curves and discuss in a systematic way its algebraic covers, which are important in the study of height theory of algebraic points beyond Weil?Lang?s height theory. They then establish a theory of adelic vector bundles on adelic curves, which considerably generalizes the classic geometry of vector bundles or that of Hermitian vector bundles over an arithmetic curve. They focus on an analogue of the slope theory in the setting of adelic curves and in particular estimate the minimal slope of tensor product adelic vector bundles. Finally, by using the adelic vector bundles as a tool, a birational Arakelov geometry for projective variety over an adelic curve is developed. As an application, a vast generalization of Nakai?Moishezon?s criterion of positivity is proven in clarifying the arguments of geometric nature from several fundamental results in the classic geometry of numbers. Assuming basic knowledge of algebraic geometry and algebraic number theory, the book is almost self-contained. It is suitable for researchers in arithmetic geometry as well as graduate students focusing on these topics for their doctoral theses. 410 0$aLecture Notes in Mathematics,$x0075-8434 ;$v2258 606 $aAlgebraic geometry 606 $aCommutative algebra 606 $aCommutative rings 606 $aFunctional analysis 606 $aAlgebraic Geometry$3https://scigraph.springernature.com/ontologies/product-market-codes/M11019 606 $aCommutative Rings and Algebras$3https://scigraph.springernature.com/ontologies/product-market-codes/M11043 606 $aFunctional Analysis$3https://scigraph.springernature.com/ontologies/product-market-codes/M12066 615 0$aAlgebraic geometry. 615 0$aCommutative algebra. 615 0$aCommutative rings. 615 0$aFunctional analysis. 615 14$aAlgebraic Geometry. 615 24$aCommutative Rings and Algebras. 615 24$aFunctional Analysis. 676 $a516.35 700 $aChen$b Huayi$4aut$4http://id.loc.gov/vocabulary/relators/aut$0791279 702 $aMoriwaki$b Atsushi$4aut$4http://id.loc.gov/vocabulary/relators/aut 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a996418269703316 996 $aArakelov Geometry over Adelic Curves$92379878 997 $aUNISA