\(J\)-equation on holomorphic vector bundles. (English) Zbl 1536.32007
Motivated by the article [R. Dervan et al. “Z-critical connections and Bridgeland stability conditions”, Preprint, arXiv:2012.10426], the author introduces a version of the \(J\)-equation on holomorphic vector bundles over compact Kähler manifolds and investigates some fundamental properties as well as examples of solutions. In particular, he provides an algebraic condition called (asymptotic) \(J\)-stability in terms of subbundles on compact Kähler surfaces, and a numerical criterion on vortex bundles via dimensional reduction. Also, he discusses an application for the vector bundle version of the deformed Hermitian-Yang-Mills equation in the small volume regime. The author explains that there is a moment map/GIT framework associated to the \(J\)-equation, i.e., the \(J\)-equation can be viewed as the zero of the moment map on a suitable infinite-dimensional symplectic manifold. He studies some existence and non-existence results for the \(J\)-equation and considers an example on projective spaces which gives the first non-trivial example for the \(J\)-equation.
This work consists of the following basic parts:
1. Introduction. This section is an introduction to the subject and states the results.
2. Preliminaries. In this section, the author sums up some properties on the space of connections and hermitian metrics.
3. Positivity concepts of hermitian metrics on vector bundles. Here, the author gives the definition of \(J\)-positivity which is crucial to assure the ellipticity and also defines, in a particular case, the notion of \(J\)-Griffiths positivity as a weaker version of the \(J\)-positivity.
4. Moment map interpretation. In this section, the author gives the moment map interpretation for the \(J\)-equation. Geometrically, the set of \(J\)-positive connections is characterized as a subset of the space of integrable connections where the symplectic form is non-degenerate.
5. \(J\)-stability for rank-2 vector bundles over compact Kähler surfaces. Here, the author shows that the existence of a \(J\)-Griffiths positive solution implies the \(J\)-(semi)stability for rank-\(2\) vector bundles over compact Kähler surfaces.
6. Examples. The author discusses three examples, namely projective spaces, sufficiently smooth vector bundles, and vortex bundles.
7. The deformed Hermitian-Yang-Mills equation. Here, the author constructs dHYM-positive solutions on simple vector bundles in the small volume regime assuming the existence of a \(J\)-positive solution.
This work consists of the following basic parts:
1. Introduction. This section is an introduction to the subject and states the results.
2. Preliminaries. In this section, the author sums up some properties on the space of connections and hermitian metrics.
3. Positivity concepts of hermitian metrics on vector bundles. Here, the author gives the definition of \(J\)-positivity which is crucial to assure the ellipticity and also defines, in a particular case, the notion of \(J\)-Griffiths positivity as a weaker version of the \(J\)-positivity.
4. Moment map interpretation. In this section, the author gives the moment map interpretation for the \(J\)-equation. Geometrically, the set of \(J\)-positive connections is characterized as a subset of the space of integrable connections where the symplectic form is non-degenerate.
5. \(J\)-stability for rank-2 vector bundles over compact Kähler surfaces. Here, the author shows that the existence of a \(J\)-Griffiths positive solution implies the \(J\)-(semi)stability for rank-\(2\) vector bundles over compact Kähler surfaces.
6. Examples. The author discusses three examples, namely projective spaces, sufficiently smooth vector bundles, and vortex bundles.
7. The deformed Hermitian-Yang-Mills equation. Here, the author constructs dHYM-positive solutions on simple vector bundles in the small volume regime assuming the existence of a \(J\)-positive solution.
Reviewer: Ahmed Lesfari (El Jadida)
MSC:
| 32J27 | Compact Kähler manifolds: generalizations, classification |
| 32L05 | Holomorphic bundles and generalizations |
| 32W50 | Other partial differential equations of complex analysis in several variables |
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