Skeletons and tropicalizations

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• 作者：Walter Gubler^a ; ^{walter.gubler@mathematik.uni-regensburg.de" class="auth_mail" title="E-mail the corresponding author} ; Joseph Rabinoff^b ; ^{jrabinoff@math.gatech.edu" class="auth_mail" title="E-mail the corresponding author} ; Annette Werner^c ; ^{werner@math.uni-frankfurt.de" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Tropicalization ; Skeleton ; Berkovich space ; Analytic geometry ; Non-Archimedean geometry
• 刊名：Advances in Mathematics
• 出版年：2016
• 出版时间：14 May 2016
• 年：2016
• 卷：294
• 期：Complete
• 页码：150-215
• 全文大小：1123 K

文摘

Let K   be a complete, algebraically closed non-archimedean field with ring of integers 16000827&_mathId=si1.gif&_user=111111111&_pii=S0001870816000827&_rdoc=1&_issn=00018708&md5=7eecf324908c2076e9871e102b5ba71c" title="Click to view the MathML source">K^∘$K^{\circ }$ and let X be a K  -variety. We associate to the data of a strictly semistable 16000827&_mathId=si1.gif&_user=111111111&_pii=S0001870816000827&_rdoc=1&_issn=00018708&md5=7eecf324908c2076e9871e102b5ba71c" title="Click to view the MathML source">K^∘$K^{\circ }$-model 16000827&_mathId=si13.gif&_user=111111111&_pii=S0001870816000827&_rdoc=1&_issn=00018708&md5=a78fda7939ef6fdbfca2f06318d83932" title="Click to view the MathML source">X$\mathcal{X}$ of X plus a suitable horizontal divisor H   a skeleton 16000827&_mathId=si19.gif&_user=111111111&_pii=S0001870816000827&_rdoc=1&_issn=00018708&md5=a539970d7a23f4a16e7a99e4a62799e5" title="Click to view the MathML source">S(X,H)$S(\mathcal{X},H)$ in the analytification of X. This generalizes Berkovich's original construction by admitting unbounded faces in the directions of the components of H  . It also generalizes constructions by Tyomkin and Baker–Payne–Rabinoff from curves to higher dimensions. Every such skeleton has an integral polyhedral structure. We show that the valuation of a non-zero rational function is piecewise linear on 16000827&_mathId=si19.gif&_user=111111111&_pii=S0001870816000827&_rdoc=1&_issn=00018708&md5=a539970d7a23f4a16e7a99e4a62799e5" title="Click to view the MathML source">S(X,H)$S(\mathcal{X},H)$. For such functions we define slopes along codimension one faces and prove a slope formula expressing a balancing condition on the skeleton. Moreover, we obtain a multiplicity formula for skeletons and tropicalizations in the spirit of a well-known result by Sturmfels–Tevelev. We show a faithful tropicalization result saying roughly that every skeleton can be seen in a suitable tropicalization. We also prove a general result about existence and uniqueness of a continuous section to the tropicalization map on the locus of tropical multiplicity one.