论文标题
变形的Hermitian-yang-mills方程和谐波多项式的水平集
The Deformed Hermitian-Yang-Mills Equation and Level Sets of Harmonic Polynomials
论文作者
论文摘要
假设$ v（x，y）：\ mathbb c \ rightarrow \ mathbb r $是整个谐波多项式，右半平面没有关键点。令$ z_1，z_2 \ in \ mathbb c $躺在$ v $的级别集上，并假设$ {\ rm re}（z_2）> {\ rm re}（z_1）\ geq0 $。我们提供了必要和充分的条件，仅取决于多项式$ v $的代数属性，因为当存在一个平稳的真实函数$ f $时，其图$ x+如果（x）$位于$ v $连接$ z_1 $的电平曲线上，将其连接到$ z_1 $。受git的启发，我们在适当的函数空间上构建了功能性的kempf-构建功能，并在且仅当存在这样的图表时，就可以从下方和正确地界定功能。作为一个应用，我们发现稳定条件等于存在于射影捆绑的遗传性hermitian-yang-mills方程的解决方案捆绑$ x_ {r，m}：= \ mathbb p（\ Mathcal o _ {\ Mathcal o _ {\ Mathbb p^M} \ Mathb p^M} \ oplus \ oplus \ Mathcal \ Mathcal o _ \ \ \ \ Mathbb p^m}（-1）^{\ oplus（r+1）}）$带有卡拉比对称性。
Suppose $v(x,y):\mathbb C\rightarrow \mathbb R$ is an entire harmonic polynomial with no critical points in the right half plane. Let $z_1, z_2\in\mathbb C$ lie on a level set of $v$ , and assume ${\rm Re}(z_2)>{\rm Re}(z_1)\geq0$. We give a necessary and sufficient condition, depending only on algebraic properties of the polynomial $v$, for when there exists a smooth real function $f$ whose graph $x+if(x)$ lies on a level curve of $v$ connecting $z_1$ to $z_2$. Inspired by GIT, we construct a Kempf-Ness functional on an appropriate function space, and prove the functional is bounded from below and proper if and only if a such a graph exists. As an application, we find a stability condition equivalent to the existence of a solution to the deformed Hermitian-Yang-Mills equation on the family of projective bundles $ X_{r,m}:=\mathbb P(\mathcal O_{\mathbb P^m}\oplus \mathcal O_{\mathbb P^m}(-1)^{\oplus (r+1)})$ with Calabi Symmetry.
