文摘
Let K be a complete, algebraically closed non-archimedean field with ring of integers class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816000827&_mathId=si1.gif&_user=111111111&_pii=S0001870816000827&_rdoc=1&_issn=00018708&md5=7eecf324908c2076e9871e102b5ba71c" title="Click to view the MathML source">K∘class="mathContainer hidden">class="mathCode"><math altimg="si1.gif" overflow="scroll">K∘math> and let X be a K -variety. We associate to the data of a strictly semistable class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816000827&_mathId=si1.gif&_user=111111111&_pii=S0001870816000827&_rdoc=1&_issn=00018708&md5=7eecf324908c2076e9871e102b5ba71c" title="Click to view the MathML source">K∘class="mathContainer hidden">class="mathCode"><math altimg="si1.gif" overflow="scroll">K∘math>-model class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816000827&_mathId=si13.gif&_user=111111111&_pii=S0001870816000827&_rdoc=1&_issn=00018708&md5=a78fda7939ef6fdbfca2f06318d83932" title="Click to view the MathML source">Xclass="mathContainer hidden">class="mathCode"><math altimg="si13.gif" overflow="scroll">mathvariant="script">Xmath> of X plus a suitable horizontal divisor H a skeleton class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816000827&_mathId=si19.gif&_user=111111111&_pii=S0001870816000827&_rdoc=1&_issn=00018708&md5=a539970d7a23f4a16e7a99e4a62799e5" title="Click to view the MathML source">S(X,H)class="mathContainer hidden">class="mathCode"><math altimg="si19.gif" overflow="scroll">S(mathvariant="script">X,H)math> 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 class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816000827&_mathId=si19.gif&_user=111111111&_pii=S0001870816000827&_rdoc=1&_issn=00018708&md5=a539970d7a23f4a16e7a99e4a62799e5" title="Click to view the MathML source">S(X,H)class="mathContainer hidden">class="mathCode"><math altimg="si19.gif" overflow="scroll">S(mathvariant="script">X,H)math>. 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.