Tian's theorem for Grassmannian embeddings and degeneracy sets of random sections
Fuente:
arXiv
Salvato in:
| Autori principali: | , , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2025
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866910970263109632 |
|---|---|
| author | Bayraktar, Turgay Coman, Dan Liu, Bingxiao Marinescu, George |
| author_facet | Bayraktar, Turgay Coman, Dan Liu, Bingxiao Marinescu, George |
| contents | Let $(X,ω)$ be a compact Kähler manifold, $(L,h^L)$ be a positive line bundle, and $(E,h^E)$ be a Hermitian holomorphic vector bundle of rank $r$ on $X$. We prove that the pullback by the Kodaira embedding associated to $L^p\otimes E$ of the $k$-th Chern class of the dual of the universal bundle over the Grassmannian converges as $p\to\infty$ to the $k$-th power of the Chern form $c_1(L,h^L)$, for $0\leq k\leq r$. If $c_1(L,h^L)=ω$ we also determine the second term in the semiclassical expansion, which involves $c_1(E,h^E)$. As a consequence we show that the limit distribution of zeros of random sequences of holomorphic sections of high powers $L^p\otimes E$ is $c_1(L,h^L)^r$. Furthermore, we compute the expectation of the currents of integration along degeneracy sets of random holomorphic sections. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_19731 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Tian's theorem for Grassmannian embeddings and degeneracy sets of random sections Bayraktar, Turgay Coman, Dan Liu, Bingxiao Marinescu, George Complex Variables Differential Geometry Probability Primary 32L10, Secondary 32A60, 32U40, 53C55, 60D05 Let $(X,ω)$ be a compact Kähler manifold, $(L,h^L)$ be a positive line bundle, and $(E,h^E)$ be a Hermitian holomorphic vector bundle of rank $r$ on $X$. We prove that the pullback by the Kodaira embedding associated to $L^p\otimes E$ of the $k$-th Chern class of the dual of the universal bundle over the Grassmannian converges as $p\to\infty$ to the $k$-th power of the Chern form $c_1(L,h^L)$, for $0\leq k\leq r$. If $c_1(L,h^L)=ω$ we also determine the second term in the semiclassical expansion, which involves $c_1(E,h^E)$. As a consequence we show that the limit distribution of zeros of random sequences of holomorphic sections of high powers $L^p\otimes E$ is $c_1(L,h^L)^r$. Furthermore, we compute the expectation of the currents of integration along degeneracy sets of random holomorphic sections. |
| title | Tian's theorem for Grassmannian embeddings and degeneracy sets of random sections |
| topic | Complex Variables Differential Geometry Probability Primary 32L10, Secondary 32A60, 32U40, 53C55, 60D05 |
| url | https://arxiv.org/abs/2504.19731 |