PreprintCC BY 4.0arXiv
Towards Hodge-Riemann relations for non-Archimedean analogs of valuations on convex sets
Abstract
In [8], a non-Archimedean analogue of the space of translation-invariant even valuations on convex sets was introduced. In [7], motivated by a further analogy with the classical theory, this space was equipped with two multiplicative structures, the product and the convolution. Both structures satisfy Poincare duality and the (non-mixed) hard Lefschetz theorem. In this paper, we formulate a conjecture concerning a more general mixed versions of the hard Lefschetz theorem and the Hodge-Riemann relations. We prove the non-mixed Hodge-Riemann relations in degree 1 for the product and, equivalently, in codegree 1 for the convolution.
Related
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BibTeX
@misc{alesker2026towards,
title = {{Towards Hodge-Riemann relations for non-Archimedean analogs of valuations on convex sets}},
author = {Semyon Alesker},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2606.25641},
eprint = {2606.25641},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.25641},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Towards Hodge-Riemann relations for non-Archimedean analogs of valuations on convex sets
AU - Semyon Alesker
PY - 2026
DA - 2026/11/04
PB - arXiv
DO - 10.48550/arxiv.2606.25641
UR - https://arxiv.org/abs/2606.25641
AB - In [8], a non-Archimedean analogue of the space of translation-invariant even valuations on convex sets was introduced. In [7], motivated by a further analogy with the classical theory, this space was equipped with two multiplicative structures, the product and the convolution. Both structures satisfy Poincare duality and the (non-mixed) hard Lefschetz theorem. In this paper, we formulate a conjecture concerning a more general mixed versions of the hard Lefschetz theorem and the Hodge-Riemann relations. We prove the non-mixed Hodge-Riemann relations in degree 1 for the product and, equivalently, in codegree 1 for the convolution.
ER -