複素解析幾何セミナー

過去の記録 ~11/14次回の予定今後の予定 11/15~

開催情報 月曜日 10:30~12:00 数理科学研究科棟(駒場) 128号室
担当者 平地 健吾, 高山 茂晴, 細野 元気

2018年11月19日(月)

10:30-12:00   数理科学研究科棟(駒場) 128号室
Gerard Freixas i Montplet 氏 (Centre National de la Recherche Scientifique)
BCOV invariants of Calabi-Yau varieties (ENGLISH)
[ 講演概要 ]
The BCOV invariant of Calabi-Yau threefolds was introduced by Fang-Lu-Yoshikawa, themselves inspired by physicists Bershadsky-Cecotti-Ooguri-Vafa. It is a real number, obtained from a combination of holomorphic analytic torsion, and suitably normalized so that it only depends on the complex structure of the threefold. It is conjecturaly expected to encode genus 1 Gromov-Witten invariants of a mirror Calabi-Yau threefold. In order to confirm this prediction for a remarkable example, Fang-Lu-Yoshikawa studied the asymptotic behavior for degenerating families of Calabi-Yau threefolds acquiring at most ordinary double point (ODP) singularities. Their methods rely on former results by Yoshikawa on the singularities of Quillen metrics, together with more classical arguments in the theory of degenerations of Hodge structures and Hodge metrics. In this talk I will present joint work with Dennis Eriksson (Chalmers) and Christophe Mourougane (Rennes), where we extend the construction of the BCOV invariant to any dimension and we give precise asymptotic formulas for degenerating families of Calabi-Yau manifolds. Under several hypothesis on the geometry of the singularities acquired, our general formulas drastically simplify and prove some conjectures or predictions in the literature (Liu-Xia for semi-stable minimal families in dimension 3, Klemm-Pandharipande for ODP singularities in dimension 4, etc.). For this, we slightly improve Yoshikawa's results on the singularities of Quillen metrics, and we also provide a complement to Schmid's asymptotics of Hodge metrics when the monodromy transformations are non-unipotent.