论文标题
Hitchin-CSCK系统
The Hitchin-cscK system
论文作者
论文摘要
我们提出了一个无限的二维超智勒的减少，该减少范围扩展了Fujiki和Donaldson的经典瞬间图图片，用于Kähler指标的标态曲率。我们的方法是基于由于Biquard和Gauduchon引起的HyperKähler指标的明确构建。该结构是通过如何从Hermitian Yang-Mills方程来得出Hitchin方程的动机，并产生一个矩图方程的系统，该系统通过在CSCK方程中添加“ higgs field”来修改恒定标态曲率Kähler（CSCK）条件。在复杂曲线的特殊情况下，我们恢复了唐纳森的先前结果，而对于高维流形的方程式系统尚未研究。在某些特殊情况下，我们研究了系统解决方案的存在。在Riemann表面上，我们将唐纳森方程式的存在结果扩展到了我们的系统。然后，我们研究了不接受CSCK指标的一类规则表面上的矩图方程的解决方案的存在，表明将合适的HIGGS术语添加到CSCK方程中可以稳定歧管。最后，我们利用对标量曲率的Abreu公式类似于Abreu的公式来研究对Abelian和曲曲面表面方程的方程系统。
We present an infinite-dimensional hyperkähler reduction that extends the classical moment map picture of Fujiki and Donaldson for the scalar curvature of Kähler metrics. We base our approach on an explicit construction of hyperkähler metrics due to Biquard and Gauduchon. The construction is motivated by how one can derive Hitchin's equations for harmonic bundles from the Hermitian Yang-Mills equation, and yields a system of moment map equations which modifies the constant scalar curvature Kähler (cscK) condition by adding a "Higgs field" to the cscK equation. In the special case of complex curves, we recover previous results of Donaldson, while for higher-dimensional manifolds the system of equations has not yet been studied. We study the existence of solutions to the system in some special cases. On a Riemann surface, we extend an existence result for Donaldson's equation to our system. We then study the existence of solutions to the moment map equations on a class of ruled surfaces which do not admit cscK metrics, showing that adding a suitable Higgs term to the cscK equation can stabilize the manifold. Lastly, we study the system of equations on abelian and toric surfaces, taking advantage of a description of the system in symplectic coordinates analogous to Abreu's formula for the scalar curvature.
