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- Title
Plurisubharmonic functions in calibrated geometry and q-convexity.
- Authors
Verbitsky, Misha
- Abstract
Let (M,ω) be a Kähler manifold. An integrable function φ on M is called ωq-plurisubharmonic if the current ddcφ ^ ωq-1 is positive. We prove that φ is ωq-pluri-subharmonic if and only if φ is subharmonic on all q-dimensional complex subvarieties. We prove that ωq -plurisubharmonic function is q-convex, and admits a local approximation by smooth, ωq-plurisubharmonic functions. For any closed subvariety Z subset M, dimℂ Z ≤ q - 1, there exists a strictly ωq-plurisubharmonic function in a neighbourhood of Z (this result is known for q-convex functions). This theorem is used to give a new proof of Sibony's lemma on integrability of positive closed (p, p)-forms which are integrable outside of a complex subvariety of codimension ≥ p + 1.
- Subjects
PLURISUBHARMONIC functions; CONVEX functions; SUBHARMONIC functions; APPROXIMATION theory; MANIFOLDS (Mathematics)
- Publication
Mathematische Zeitschrift, 2010, Vol 264, Issue 4, p939
- ISSN
0025-5874
- Publication type
Article
- DOI
10.1007/s00209-009-0498-7