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Jumps, folds and singularities of Kodaira moduli spaces
Author(s) -
Dunajski Maciej,
Gundry James,
Tod Paul
Publication year - 2018
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms.12154
Subject(s) - mathematics , moduli space , pure mathematics , holomorphic function , gravitational singularity , kähler manifold , twistor space , mathematical analysis , manifold (fluid mechanics) , canonical bundle , bounded function , mathematical physics , twistor theory , mechanical engineering , engineering
For any integer k we construct an explicit example of a twistor space which contains a one‐parameter family of jumping rational curves, where the normal bundle changes from O ( 1 ) ⊕ O ( 1 ) to O ( k ) ⊕ O ( 2 − k ) . For k > 3 the resulting anti‐self‐dual Ricci‐flat manifold is a Zariski cone in the space of holomorphic sections of O ( k ) . In the case k = 2 we recover the canonical example of Hitchin's folded hyper‐Kähler manifold, where the jumping lines form a three‐parameter family. We show that in this case there exist normalisable solutions to the Schrödinger equation which extend through the fold.