I am looking for a mini-course on Gauduchon metrics, or at least a clear explanation of what it is and why it has been developed.
A Gauduchon metric is a Hermitian metric on a complex manifold with certain curvature properties. Specifically, it is a metric whose associated 2-form is equal to the sum of a Kähler form and an exact form. This condition leads to some important properties, including the existence of certain integrability conditions and a global estimate for holomorphic functions.
One of the main applications of Gauduchon metrics is in Kähler geometry, where they play an important role in the study of the Kähler cone and related topics. They are also useful in some contexts in algebraic geometry, particularly in the study of moduli spaces and stability conditions. Additionally, Gauduchon metrics have found applications in mathematical physics, including the study of mirror symmetry and string theory.
A Gauduchon metric is a type of Riemannian metric on a complex manifold that satisfies certain curvature conditions. It was introduced by Paul Gauduchon in the 1980s as a generalization of Kähler metrics, which are another important type of metric on complex manifolds.
One of the main properties of a Gauduchon metric is that it has vanishing torsion, which means that the Levi-Civita connection associated with the metric preserves the complex structure of the manifold. Another important property is that the Ricci tensor of a Gauduchon metric is a real (1,1)-form, which has important consequences for the geometry of the manifold.
Gauduchon metrics have been studied extensively in complex geometry and mathematical physics, and have applications in areas such as string theory and mirror symmetry. They provide a natural framework for studying the geometry of moduli spaces of stable bundles on complex manifolds, and are closely related to the geometry of Hodge theory and the theory of harmonic forms.
A good resource for learning more about Gauduchon metrics is the book "The Geometry of Moduli Spaces of Sheaves" by Daniel Huybrechts, which provides a detailed introduction to the subject and its applications.
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