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SUMMARY:Limit of archimedean heights and Ceresa cycles
DTSTART:20261002T130000Z
DTEND:20261002T143000Z
DTSTAMP:20261015T123300Z
UID:indico-event-17433@indico.math.cnrs.fr
CONTACT:gaiur@ihes.fr\;+33758543130
DESCRIPTION:Speakers: Souvik Goswami (IISER - Indian Institute of Science 
 Education and Research)\n\nFor a smooth\, projective and complex variety $
 X$\, and a pair of algebraic cycles $Z$ and $W$ that are homologous to zer
 o and are in complementary codimensions\, the archimedean height pairing i
 s a real number that can be interpreted as a secondary intersection pairin
 g that measures how far a certain cohomology group of the pair $(X\\setmin
 us |Z|\, |W|)$ is from being real split. This interpretation makes it suit
 able for variational study: Let $\\Delta := \\{t\\in \\mathbb{C}\;\\\,|t|<
 1\\}$\, and $\\Delta^{\\ast}=\\Delta\\setminus \\{0\\}$. Given a family $\
 \{X_t\\}_{t\\in \\Delta}$ of projective and complex varieties that are smo
 oth for $t\\in \\Delta^{\\ast}$\, while $X_0$ is a simple normal crossing 
 divisor\, and a family $\\{Z_t\, W_t\\}_{t\\in \\Delta^{\\ast}}$ of algebr
 aic cycles in complementary codimensions that are homologous to zero\, the
  function $h := \\Delta^{\\ast}\\to\\mathbb{R}$\, with $h(t)$ being the ar
 chimedean height pairing of the pair $(Z_t\, W_t)$\, is smooth. But $h$ do
 es not extend to $t=0$. R.~Hain\, G.~Pearlstein\, P.~Brosnan et al. have s
 hown that the asymptotic function $\\widetilde h(t)= h(t)+\\mu \\log|t|$ e
 xtends continuously to $t=0$\, where $\\mu\\in\\mathbb{Q}$ depends on the 
 monodromy of the variation of mixed Hodge structures associated with these
  pairings. Fixing a parameter $t$\, one can ask if $\\widetilde h(0)$ is g
 iven by the archimedean height pairing of algebraic cycles associated with
  the central fiber $X_{0}$. In a joint work with Dr. Irene Spelta\, I have
  shown that this question has an affirmative answer in case of height pair
 ings of Ceresa cycles (arxiv.org/abs/2604.01842). The purpose of this talk
  will be to explore this result and some interesting questions that still 
 remains unanswered.\n\nhttps://indico.math.cnrs.fr/event/17433/
LOCATION:Salle Marie-Louise Dubreil-Jacotin (IHP - Bâtiment Borel)
URL:https://indico.math.cnrs.fr/event/17433/
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