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• W1770393878 abstract "Let <inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper X><mml:semantics><mml:mi>X</mml:mi><mml:annotation encoding=application/x-tex>X</mml:annotation></mml:semantics></mml:math></inline-formula>be a smooth proper variety over the quotient field of a Henselian discrete valuation ring with algebraically closed residue field of characteristic <inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=p><mml:semantics><mml:mi>p</mml:mi><mml:annotation encoding=application/x-tex>p</mml:annotation></mml:semantics></mml:math></inline-formula>. We show that for any coherent sheaf <inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper E><mml:semantics><mml:mi>E</mml:mi><mml:annotation encoding=application/x-tex>E</mml:annotation></mml:semantics></mml:math></inline-formula>on <inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper X><mml:semantics><mml:mi>X</mml:mi><mml:annotation encoding=application/x-tex>X</mml:annotation></mml:semantics></mml:math></inline-formula>, the index of <inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper X><mml:semantics><mml:mi>X</mml:mi><mml:annotation encoding=application/x-tex>X</mml:annotation></mml:semantics></mml:math></inline-formula>divides the Euler–Poincaré characteristic<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=chi left-parenthesis upper X comma upper E right-parenthesis><mml:semantics><mml:mrow><mml:mi>χ<!-- χ --></mml:mi><mml:mo stretchy=false>(</mml:mo><mml:mi>X</mml:mi><mml:mo>,</mml:mo><mml:mi>E</mml:mi><mml:mo stretchy=false>)</mml:mo></mml:mrow><mml:annotation encoding=application/x-tex>chi (X,E)</mml:annotation></mml:semantics></mml:math></inline-formula>if<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=p equals 0><mml:semantics><mml:mrow><mml:mi>p</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:annotation encoding=application/x-tex>p=0</mml:annotation></mml:semantics></mml:math></inline-formula>or<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=p greater-than dimension left-parenthesis upper X right-parenthesis plus 1><mml:semantics><mml:mrow><mml:mi>p</mml:mi><mml:mo>></mml:mo><mml:mi>dim</mml:mi><mml:mo>⁡<!-- ⁡ --></mml:mo><mml:mo stretchy=false>(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=false>)</mml:mo><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:annotation encoding=application/x-tex>p>dim (X)+1</mml:annotation></mml:semantics></mml:math></inline-formula>. If<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=0 greater-than p less-than-or-equal-to dimension left-parenthesis upper X right-parenthesis plus 1><mml:semantics><mml:mrow><mml:mn>0</mml:mn><mml:mo>></mml:mo><mml:mi>p</mml:mi><mml:mo>≤<!-- ≤ --></mml:mo><mml:mi>dim</mml:mi><mml:mo>⁡<!-- ⁡ --></mml:mo><mml:mo stretchy=false>(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=false>)</mml:mo><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:annotation encoding=application/x-tex>0>pleq dim (X)+1</mml:annotation></mml:semantics></mml:math></inline-formula>, the prime-to-<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=p><mml:semantics><mml:mi>p</mml:mi><mml:annotation encoding=application/x-tex>p</mml:annotation></mml:semantics></mml:math></inline-formula>part of the index of <inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper X><mml:semantics><mml:mi>X</mml:mi><mml:annotation encoding=application/x-tex>X</mml:annotation></mml:semantics></mml:math></inline-formula>divides<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=chi left-parenthesis upper X comma upper E right-parenthesis><mml:semantics><mml:mrow><mml:mi>χ<!-- χ --></mml:mi><mml:mo stretchy=false>(</mml:mo><mml:mi>X</mml:mi><mml:mo>,</mml:mo><mml:mi>E</mml:mi><mml:mo stretchy=false>)</mml:mo></mml:mrow><mml:annotation encoding=application/x-tex>chi (X,E)</mml:annotation></mml:semantics></mml:math></inline-formula>. Combining this with the Hattori–Stong theorem yields an analogous result concerning the divisibility of the cobordism class of <inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper X><mml:semantics><mml:mi>X</mml:mi><mml:annotation encoding=application/x-tex>X</mml:annotation></mml:semantics></mml:math></inline-formula>by the index of <inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper X><mml:semantics><mml:mi>X</mml:mi><mml:annotation encoding=application/x-tex>X</mml:annotation></mml:semantics></mml:math></inline-formula>. As a corollary, rationally connected varieties over the maximal unramified extension of a<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=p><mml:semantics><mml:mi>p</mml:mi><mml:annotation encoding=application/x-tex>p</mml:annotation></mml:semantics></mml:math></inline-formula>-adic field possess a zero-cycle of<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=p><mml:semantics><mml:mi>p</mml:mi><mml:annotation encoding=application/x-tex>p</mml:annotation></mml:semantics></mml:math></inline-formula>-power degree (a zero-cycle of degree <inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=1><mml:semantics><mml:mn>1</mml:mn><mml:annotation encoding=application/x-tex>1</mml:annotation></mml:semantics></mml:math></inline-formula>if<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=p greater-than dimension left-parenthesis upper X right-parenthesis plus 1><mml:semantics><mml:mrow><mml:mi>p</mml:mi><mml:mo>></mml:mo><mml:mi>dim</mml:mi><mml:mo>⁡<!-- ⁡ --></mml:mo><mml:mo stretchy=false>(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=false>)</mml:mo><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:annotation encoding=application/x-tex>p>dim (X)+1</mml:annotation></mml:semantics></mml:math></inline-formula>). When<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=p equals 0><mml:semantics><mml:mrow><mml:mi>p</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:annotation encoding=application/x-tex>p=0</mml:annotation></mml:semantics></mml:math></inline-formula>, such statements also have implications for the possible multiplicities of singular fibers in degenerations of complex projective varieties." @default.
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• W1770393878 date "2015-06-18" @default.
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• W1770393878 title "Index of varieties over Henselian fields and Euler characteristic of coherent sheaves" @default.
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