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SUMMARY:Hélène Esnault (FU Berlin/Harvard/Copenhagen)
DTSTART:20231204T230000Z
DTEND:20231204T235000Z
DTSTAMP:20260423T004545Z
UID:CHAT/7
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/CHAT/7/">Cod
 imension one in Algebraic and Arithmetic Geometry</a>\nby Hélène Esnault
  (FU Berlin/Harvard/Copenhagen) as part of CHAT (Career\, History And Thou
 ghts) series\n\n\nAbstract\nThe notions of $\\textit{weight}$ in complex g
 eometry and in $\\ell$-adic theory in geometry over a finite field\nhave b
 een developed by Deligne and by the Grothendieck  school. The analogy betw
 een the theories is  foundational \nand led to  predictions and theorems o
 n both sides. \nOn the complex Hodge theory side\, not only do we have the
  weight filtration\, but we  also have the Hodge filtration. \nThe analogy
  on the $\\ell$-adic side over a finite field hasn’t really been documen
 ted by Deligne. \nThinking of this gave the way to understand the Lang--Ma
 nin conjecture according to which smooth projective\n rationally connected
  varieties over a finite field possess a rational point. \n$\\url{http://p
 age.mi.fu-berlin.de/esnault/preprints/helene/62-chowgroup.pdf}$\n\nOn the 
 other hand\, we know the formulation in complex geometry of the Hodge conj
 ecture: on a smooth projective complex variety $X$\, \na sub-Hodge structu
 re of $H^{2j}(X)$  of Hodge type   $(j\,j)$ should be supported on a  codi
 mension $j$ cycle. The analog  $\\ell$-adic conjecture \nhas been formulat
 ed by Tate\, even over a number field. Grothendieck’s generalized Hodge 
 conjecture is straightforwardly formulated: \na sub-Hodge structure $H$ of
  $H^i(X)$ of Hodge type $(i-1\,1)\, (i-2\,2)\, \\ldots\, (1\,i-1)$ should 
 be supported on a codimension $1$ cycle. \nEquivalently it should die at t
 he generic point of the variety. \nThis is difficult to formulate because 
 Hodge structures are complicated to describe. But there is one instance fo
 r which we can bypass the Hodge formulation:\n$H=H^i(X)$ and $H^{0\,i}=H^i
 (X\, \\mathcal O)(=H^{i\,0}=H^0(X\, \\Omega^i))=0$. Then the conjecture de
 scends to the field of definition of $X$ and becomes purely algebraic. \nI
 t is on the one hand related to the (quite bold) motivic conjectures predi
 cting that $H^i(X\,\\mathcal O)=0$ for all $i\\neq 0$ should be equivalent
  to the triviality of the Chow group of $0$-cycles over a large field (thi
 s brings us back to the proof of the Lang--Manin conjecture). On the other
  hand\, as it is purely algebraic\, one can try to think of it in the fram
 ework of today’s $p$-adic Hodge theory.\n
LOCATION:https://researchseminars.org/talk/CHAT/7/
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