---
title: "Towards Categorical Kähler Geometry"
authors: ["Fabian Haiden", "Ludmil Katzarkov", "Maxim Kontsevich", "Pranav Pandit"]
year: 2026
venue: "arXiv:math.AG"
arxiv: "2609.00978"
doi: null
source: "https://arxiv.org/abs/2609.00978"
topics: ["curiosity", "maths-foundations"]
seed_rank: 1587
seed_batch: "curiosity-2026-09-05"
reviewed: "2026-09-05"
pool: "maths-foundations"
relevance_score: 10
lineage: bridgeland-stability
cites:
  - title: "Towards Categorical Kähler Geometry"
    url: "https://arxiv.org/abs/2609.00978"
    year: 2026
    arxiv: "2609.00978"
    doi: null
see: []
---

# Towards Categorical Kähler Geometry

## One-sentence takeaway

Kontsevich–Katzarkov–Haiden–Pandit sketch Kähler geometry inside derived noncommutative categories, refining Bridgeland stability with metrics and BPS mass.

## Why it's lovely

Why you might love this: Bridgeland’s stability space wants to be a Kähler manifold; this programme writes the metric into the category. An emerging theory of Kähler metrics in derived noncommutative geometry — metrized objects, mass measures with a BPS inequality, harmonic metrics, minimizing flows, complexified Kähler potentials — in both Archimedean and non-Archimedean flavours. Peak curiosity sparkle, developed through examples (Fukaya categories, quivers / C*-algebras, spectral networks, comonadic adjunctions of stable ∞-categories).

## Key ideas

- Refinement of Bridgeland stability conditions that encodes underlying differential-geometric structure.
- Proposed ingredients, Archimedean and non-Archimedean: metrized objects; mass measures satisfying a BPS inequality; harmonic metrics; minimizing flows; complexified Kähler potentials.
- Developed through examples rather than a single axiomatic dump: Fukaya categories, quiver representations and associated C*-algebras, spectral networks, comonadic adjunctions of stable ∞-categories.

## Caveats

Programme / outline paper: contours and proposals, not a finished dictionary of theorems. “Towards” is honest — examples and proposed elements, not a classification of categorical Kähler metrics. Read as a research programme, not as a completed correspondence with classical Kähler geometry.

## Implications for Broadside

Stability / derived-geometry sparkle. The move (put the metric on the category, not only on a moduli space of objects) is the kind of slogan Broadside likes even when there is no Anoptic ticket yet.

## Links

- arXiv abs: https://arxiv.org/abs/2609.00978
- PDF: https://arxiv.org/pdf/2609.00978.pdf