Upper Level Sets of Lelong Numbers on Hirzebruch Surfaces

• Published in: The Journal of geometric analysis 2023-08, Vol.33 (8), Article 250
• Main Authors: Kişisel, Ali Ulaş Özgür, Yazici, Ozcan
• Format: Article
• Language: English
• Subjects: Abstract Harmonic Analysis; Convex and Discrete Geometry; Differential Geometry; Dynamical Systems and Ergodic Theory; Fourier Analysis; Geometry; Global Analysis and Analysis on Manifolds; Homology; Mathematics; Mathematics and Statistics
• ISSN: 1050-6926; 1559-002X
• DOI: 10.1007/s12220-023-01312-y

Let F a denote the Hirzebruch surfaces and T α , α ′ ( F a ) denotes the set of positive, closed (1, 1)-currents on F a whose cohomology class is α F + α ′ H where F and H generates the Picard group of F a . E β + ( T ) denotes the upper level sets of Lelong numbers ν ( T , x ) of T ∈ T α , α ′ ( F a ) . When a = 0 , ( F a = P 1 × P 1 ), for any current T ∈ T α , α ′ ( P 1 × P 1 ) , we show that E ( α + α ′ ) / 3 + ( T ) is contained in a curve of total degree 2, possibly except 1 point. For any current T ∈ T α , α ′ ( F a ) , we show that E β + ( T ) is contained in either in a curve of bidegree (0, 1) or in a + 1 curves of bidegree (1, 0) where β ≥ ( α + ( a + 1 ) α ′ ) / ( a + 2 ) .