Monge-Ampère equation, hyperkähler structure and adapted complex structure
Fuente:
arXiv
Salvato in:
| Autore principale: | |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2024
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866910455360913408 |
|---|---|
| author | Kan, Su-Jen |
| author_facet | Kan, Su-Jen |
| contents | In the tangent bundle of $(M,g)$, it is well-known that the Monge-Ampère equation $(\partial\bar\partial \sqrtρ)^n=0$ has the asymptotic expansion $
ρ(x+iy)=\sum_{ij} g_{ij} (x) y_{i} y_{j} + O(y^4)$ near $M$. Those 4th order terms are made explicit in this article: $$ρ(x+iy)=\sum_{i}y_{i}^2-\frac 13\sum_{pqij} R_{i p j q}(0)x_p x_q y_{i}y_{j}+O(5).$$ At $M$, sectional curvatures of the Kähler metric $2i\partial\bar\partialρ$ can be computed. This has enabled us to find a family of Kähler manifolds whose tangent bundles have admitted complete hyperkähler structures whereas the adapted complex structure can only be partially defined on the tangent bundles.
In these cases, the study of the adapted complex structure is equivalent to the study of some gauge transformations on the baby Nahm's equation $\dot T_1+[T_0,T_1]=0.$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_13287 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Monge-Ampère equation, hyperkähler structure and adapted complex structure Kan, Su-Jen Differential Geometry Complex Variables 32C09, 32Q15, 53C26 In the tangent bundle of $(M,g)$, it is well-known that the Monge-Ampère equation $(\partial\bar\partial \sqrtρ)^n=0$ has the asymptotic expansion $ ρ(x+iy)=\sum_{ij} g_{ij} (x) y_{i} y_{j} + O(y^4)$ near $M$. Those 4th order terms are made explicit in this article: $$ρ(x+iy)=\sum_{i}y_{i}^2-\frac 13\sum_{pqij} R_{i p j q}(0)x_p x_q y_{i}y_{j}+O(5).$$ At $M$, sectional curvatures of the Kähler metric $2i\partial\bar\partialρ$ can be computed. This has enabled us to find a family of Kähler manifolds whose tangent bundles have admitted complete hyperkähler structures whereas the adapted complex structure can only be partially defined on the tangent bundles. In these cases, the study of the adapted complex structure is equivalent to the study of some gauge transformations on the baby Nahm's equation $\dot T_1+[T_0,T_1]=0.$ |
| title | Monge-Ampère equation, hyperkähler structure and adapted complex structure |
| topic | Differential Geometry Complex Variables 32C09, 32Q15, 53C26 |
| url | https://arxiv.org/abs/2405.13287 |