On the Hodge Extension of Local Kähler Forms
Abstract
Let X be a Stein manifold, and let K={h≤0} be a compact sublevel set of a strictly plurisubharmonic function h defined near K. It is known that K is meromorphically convex in X if and only if ddch extends to a Hodge form on X. We show that if H2(X,ℤ) is the direct sum of a free abelian group and a torsion group, for example if it is finitely generated, then any extension of ddch to a Kähler form on X can be modified, after replacing h by a positive multiple, into a Hodge form. Thus meromorphic convexity of K is equivalent to the extendability of ddch as a Kähler form. We also show that the topological assumption cannot be dropped: we construct a Stein manifold X of dimension $6$ with H2(X,ℤ)≅ℚ, a sublevel set K as above that is not meromorphically convex although ddch extends to a Kähler form, and a totally real torus in K that is isotropic for a Kähler form but is not meromorphically convex. This shows that the Hodge condition in Guedj's characterization of meromorphically convex totally real submanifolds cannot be weakened to a Kähler condition, and gives counterexamples to two criteria stated by Cieliebak and Eliashberg.
Related
- Matsumura's extension problem for pluricanonical forms in Kähler families I: the smooth and essentially Moishezon cases
- On the Hodge conjecture for hyper-Kähler varieties of generalized Kummer type of dimension divisible by four
- Discriminants of Hermitian forms, maximal degenerations and Kontsevich's tropical approach to the Hodge conjecture
- The Heat Kernel of the Hodge Laplacian on an Invariant Subspace of 1-forms on the Hyperbolic Space ℍn
- Holomorphic 1-forms without zeros on Kähler manifolds
BibTeX
@misc{shafikov2026hodge,
title = {{On the Hodge Extension of Local Kähler Forms}},
author = {Rasul Shafikov},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.38055},
eprint = {2610.38055},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.38055},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On the Hodge Extension of Local Kähler Forms
AU - Rasul Shafikov
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.38055
UR - https://arxiv.org/abs/2610.38055
AB - Let X be a Stein manifold, and let K={h≤0} be a compact sublevel set of a strictly plurisubharmonic function h defined near K. It is known that K is meromorphically convex in X if and only if dd^ch extends to a Hodge form on X. We show that if H_2(X,ℤ) is the direct sum of a free abelian group and a torsion group, for example if it is finitely generated, then any extension of dd^ch to a Kähler form on X can be modified, after replacing h by a positive multiple, into a Hodge form. Thus meromorphic convexity of K is equivalent to the extendability of dd^ch as a Kähler form. We also show that the topological assumption cannot be dropped: we construct a Stein manifold X of dimension $6$ with H_2(X,ℤ)≅ℚ, a sublevel set K as above that is not meromorphically convex although dd^ch extends to a Kähler form, and a totally real torus in K that is isotropic for a Kähler form but is not meromorphically convex. This shows that the Hodge condition in Guedj's characterization of meromorphically convex totally real submanifolds cannot be weakened to a Kähler condition, and gives counterexamples to two criteria stated by Cieliebak and Eliashberg.
ER -