Limit of archimedean heights and Ceresa cycles

par Souvik Goswami (IISER - Indian Institute of Science Education and Research)

vendredi 2 oct. 2026, 15:00 → 16:30 Europe/Paris

Salle M-L Dubreil-Jacotin, Bâtiment Borel, 4ème étage (IHP)

For a smooth, projective and complex variety $X$, and a pair of algebraic cycles $Z$ and $W$ that are homologous to zero and are in complementary codimensions, the archimedean height pairing is a real number that can be interpreted as a secondary intersection pairing that measures how far a certain cohomology group of the pair $(X\setminus |Z|, |W|)$ is from being real split. This interpretation makes it suitable 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 smooth 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 algebraic cycles in complementary codimensions that are homologous to zero, the function $h := \Delta^{\ast}\to\mathbb{R}$, with $h(t)$ being the archimedean height pairing of the pair $(Z_t, W_t)$, is smooth. But $h$ does not extend to $t=0$. R.~Hain, G.~Pearlstein, P.~Brosnan et al. have shown that the asymptotic function $\widetilde h(t)= h(t)+\mu \log|t|$ extends 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 given 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 pairings 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.