Ordinary and calibrated differential operators Application to curvilinear webs
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| Formato: | Preprint |
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2025
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| _version_ | 1866917917636952064 |
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| author | Lehmann, Daniel |
| author_facet | Lehmann, Daniel |
| contents | We study the space of the solutions $s$ of any system of partial differential equations $ D(j^ks)=0 $ defined by a linear and homogeneous differential operator $ D:J^kE\to F $ of any order $k\geq 1$, which is ``ordinary" (i.e. which is generic in some sense among all $D$'s), $E$ and $F$ being vector bundles over a $n$-dimensional manifold $V$, and $D$ being assumed to be surjective at any point of $V$. In some range of the ranks $p$ and $q$ of these bundles ($p < q\leq np$ in the case $k=1$), we first give an upper-bound $π(n,k,p,q)$ for the dimension of the space ${\mathcal S}_m$ of the germs of solutions at a generic point $m$ of the ambiant manifold. If these ranks satisfy moreover to some condition of integrality (in the case $k=1$, $\frac{p(n-1)}{q-p}$ must be an integer), and we then say that $D$ is ``calibrated", we build a vector bundle $\mathcal E$ of rank $π(n,k,p,q)$ on $V$, provided with a tautological connection $\nabla$, whose curvature is an obstruction for the dimension of ${\mathcal S}_m$ to reach its maximal value. We also prove a ``theorem of concentration'' : relatively to some convenient trivialization of $\mathcal E$, some coefficients of this curvature vanish systematically.
As an example, we provide, for any curvilinear $d$-web on $V$, a differential operator $D$ of order one, which is always ordinary and calibrated, and for which ${\mathcal S}_m$ is the space of germs of abelian relations ([L]). Thus, we recover the Damiano's upper-bound ([D1]) for the rank of such a web, and we can define in the most general case the ``curvature'' of such a web, already known for $n=2$ (see [BB] if $d=3$, and [Pa],[H1],[Pi1] for arbitrary $d$), obstruction for this rank to be maximum. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_06641 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Ordinary and calibrated differential operators Application to curvilinear webs Lehmann, Daniel Differential Geometry 53A60 We study the space of the solutions $s$ of any system of partial differential equations $ D(j^ks)=0 $ defined by a linear and homogeneous differential operator $ D:J^kE\to F $ of any order $k\geq 1$, which is ``ordinary" (i.e. which is generic in some sense among all $D$'s), $E$ and $F$ being vector bundles over a $n$-dimensional manifold $V$, and $D$ being assumed to be surjective at any point of $V$. In some range of the ranks $p$ and $q$ of these bundles ($p < q\leq np$ in the case $k=1$), we first give an upper-bound $π(n,k,p,q)$ for the dimension of the space ${\mathcal S}_m$ of the germs of solutions at a generic point $m$ of the ambiant manifold. If these ranks satisfy moreover to some condition of integrality (in the case $k=1$, $\frac{p(n-1)}{q-p}$ must be an integer), and we then say that $D$ is ``calibrated", we build a vector bundle $\mathcal E$ of rank $π(n,k,p,q)$ on $V$, provided with a tautological connection $\nabla$, whose curvature is an obstruction for the dimension of ${\mathcal S}_m$ to reach its maximal value. We also prove a ``theorem of concentration'' : relatively to some convenient trivialization of $\mathcal E$, some coefficients of this curvature vanish systematically. As an example, we provide, for any curvilinear $d$-web on $V$, a differential operator $D$ of order one, which is always ordinary and calibrated, and for which ${\mathcal S}_m$ is the space of germs of abelian relations ([L]). Thus, we recover the Damiano's upper-bound ([D1]) for the rank of such a web, and we can define in the most general case the ``curvature'' of such a web, already known for $n=2$ (see [BB] if $d=3$, and [Pa],[H1],[Pi1] for arbitrary $d$), obstruction for this rank to be maximum. |
| title | Ordinary and calibrated differential operators Application to curvilinear webs |
| topic | Differential Geometry 53A60 |
| url | https://arxiv.org/abs/2502.06641 |