arXiv subjects

Thomas Franzinetti

Publications and source records attributed to Thomas Franzinetti.

3 recordsLinked to original sources

arXiv2026

Quantizing Geodesics in K\"ahler and Sasaki Geometry

The space of K\"ahler potentials can be quantized through the classical Fubini-Study map, relating infinite-dimensional geometric structures to finite-dimensional symmetric spaces. We prove (exactly) when the Fubini-Study image of a geodesic line in the space of positive definite Hermitian matrices gives rise to a quasi-geodesic in the space of K\"ahler potentials. Furthermore, we introduce a quantization procedure for geodesics between potentials on normal K\"ahler varieties and show how this construction extends to the Sasaki setting.

math.DG↗

arXiv2021

Isometries of the Space of Sasaki Potentials

Given any two Kähler manifolds $X_1$ and $X_2$, L. Lempert recently proved that if their spaces of Kähler potentials are isometric with respect to the Mabuchi metric, then $X_1$ and $X_2$ must be diffeomorphic. We prove that this is no longer the case for Sasaki manifolds. Then, considering regular Sasaki manifolds $M_1$ and $M_2$, we prove that if the spaces of potentials are isometric, then $M_1$ and $M_2$ must have, among others, the same universal covering space. Finally, getting rid of the regularity assumption on $M_1$ and $M_2$, we investigate the consequences of the existence of affine Mabuchi isometries: this leads to a family of Sasaki isospectral structures.

math.DG↗

arXiv2021

Curvature of the Completion of the Space of Sasaki Potentials

Given a compact Sasaki manifold, we endow the space of the Sasaki potentials with an analogue of Mabuchi metric. We show that its metric completion is negatively curved in the sense of Alexandrov.

math.DG↗