中央大学基幹理工学部　幾何・トポロジーセミナー

Boris Botvinnik (University of Oregon)

タイトル：Generalized positive scalar curvature on spin^c manifolds

アブストラクト：This talk is based on joint work with J. Rosenberg and P. Piazza. Let (M, L) be a compact non-spin spin^c manifold. Fix a Riemannian metric g on M and a connection A on L, and let D_L be the associated spin Dirac operator. Let R_g^tw := R_g + 2ic(Ω) be the twisted scalar curvature (which takes values in the endomorphisms of the spinor bundle), where R_g is the scalar curvature of g and 2ic(Ω) comes from the curvature 2-form of the connection A. Then the Lichnerowicz–Schrödinger formula for the square of the Dirac operator takes the form D_L² = ∇^*∇ + ¼R_g^tw. It turns out that a closed non-spin simply connected spin^c manifold of dimension n ≥ 5 admits a pair (g, A) such that R_g^tw > 0 if and only if the index (M, L) := ind D_L vanishes in KU_n. We introduce a scalar-valued generalized scalar curvature R_g^gen := R_g − 2|Ω|_op where |Ω|_op is the pointwise operator norm of Clifford multiplication c(Ω) acting on spinors. We show that the positivity condition on the operator R_g^tw is equivalent to the positivity of the scalar function R_g^gen. We prove a corresponding trichotomy theorem concerning the curvature R_g^gen and study its implications. We also show that the space of pairs (g, A) with R_g^gen > 0 has non-trivial topology, and we address a conjecture concerning the non-triviality of the "index difference" map. Finally, we address the spin^c version of the Gromov–Lawson–Rosenberg conjecture.

2026_Botvinnik.pdf