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SUMMARY:Piotr Achinger (IMPAN Warsaw)
DTSTART:20201022T160000Z
DTEND:20201022T172000Z
DTSTAMP:20260423T035457Z
UID:RAMpAGe/21
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/RAMpAGe/21/"
 >Hodge theory over $\\mathbf{C}((t))$</a>\nby Piotr Achinger (IMPAN Warsaw
 ) as part of Recent Advances in Modern p-Adic Geometry (RAMpAGe)\n\n\nAbst
 ract\nI will describe some ways in which Hodge theory makes its way into t
 he\ngeometry of rigid-analytic varieties over $\\mathbf{C}((t))$. Namely\,
  such spaces\nhave a "Betti realization"\, well-defined up to homotopy (jo
 int work\nwith Talpo)\, and their cohomology carries a mixed Hodge Structu
 re\n(Steenbrink\, Stewart-Vologodsky\, Berkovich). The notion of "projecti
 ve\nreduction" introduced by Li and studied by Hansen-Li is a good working
 \nanalog of the Kaehler condition. In this case\, Hodge symmetry holds\,\n
 even though it fails in some cases over the $p$-adic numbers (Petrov).\nMo
 reover\, there is a Riemann-Hilbert correspondence (work in\nprogress)\, w
 hich should allow us to define variations of mixed Hodge\nstructure in thi
 s context. All of these analogs rely on corresponding\nstatements regardin
 g the logarithmic special fiber of a semistable\nmodel. Open problems abou
 nd.\n
LOCATION:https://researchseminars.org/talk/RAMpAGe/21/
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