Tangent cones to positive-(1,1) De Rham currents
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Costante Bellettini
Abstract
We consider positive-(1,1) De Rham currents in arbitrary almost complex manifolds and prove the uniqueness of the tangent cone at any point where the density does not have a jump with respect to all of its values in a neighborhood. Without this assumption, counterexamples to the uniqueness of tangent cones can be produced already in ℂn, hence our result is optimal. The key idea is an implementation, for currents in an almost complex setting, of the classical blow-up of curves in algebraic or symplectic geometry. Unlike the classical approach in ℂn, we cannot rely on plurisubharmonic potentials.
Funding source: Swiss Polytechnic Federal Institute Graduate Research Fellowship
Award Identifier / Grant number: ETH-01 09-3
Funding source: Giorgio and Elena Petronio Fellowship
Funding source: National Science Foundation
Award Identifier / Grant number: DMS-1128155
The author wishes to thank Tristan Rivière, who introduced him to (1,1)-currents and stimulated him to work on the subject. Part of this paper overlaps with the Ph.D. thesis of the author, who is therefore grateful to ETH Zürich for the excellent environment. The research there was partially supported by the Swiss Polytechnic Federal Institute Graduate Research Fellowship ETH-01 09-3. The work was corrected, completed and put in its final form at Princeton University and at the Institute for Advanced Study. The author acknowledges financial support from the Giorgio and Elena Petronio Fellowship and the National Science Foundation under agreement No. DMS-1128155.
© 2015 by De Gruyter
Articles in the same Issue
- Frontmatter
- The limit of the Yang–Mills flow on semi-stable bundles
- Tangent cones to positive-(1,1) De Rham currents
- A characterization of vertex operator algebras V+ℤα: I
- Degenerate neckpinches in Ricci flow
- On the local Bump–Friedberg L-function
- A variational characterization of J-holomorphic curves
- Mahler measure and elliptic curve L-functions at s = 3
- Uniform bounds for bounded geodesic image theorems
- Linear stability of Perelman's ν-entropy on symmetric spaces of compact type
Articles in the same Issue
- Frontmatter
- The limit of the Yang–Mills flow on semi-stable bundles
- Tangent cones to positive-(1,1) De Rham currents
- A characterization of vertex operator algebras V+ℤα: I
- Degenerate neckpinches in Ricci flow
- On the local Bump–Friedberg L-function
- A variational characterization of J-holomorphic curves
- Mahler measure and elliptic curve L-functions at s = 3
- Uniform bounds for bounded geodesic image theorems
- Linear stability of Perelman's ν-entropy on symmetric spaces of compact type
