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$(1,1)$ forms with specified Lagrangian phase: a priori estimates and algebraic obstructions
Cambridge Journal of Mathematics ( IF 1.6 ) Pub Date : 2020-04-21
Tristan C. Collins, Adam Jacob, Shing-Tung Yau

Let $(X, \alpha)$ be a Kähler manifold of dimension $n$, and let $[\omega] \in H^{1,1} (X, \mathbb{R})$. We study the problem of specifying the Lagrangian phase of $\omega$ with respect to $\alpha$, which is described by the nonlinear elliptic equation\[\sum^{n}_{i=1} \arctan (\lambda_i) = h(x)\]where $\lambda_i$ are the eigenvalues of $\omega$ with respect to $\alpha$. When $h(x)$ is a topological constant, this equation corresponds to the deformed Hermitian–Yang–Mills (dHYM) equation, and is related by mirror symmetry to the existence of special Lagrangian submanifolds. We introduce a notion of subsolution for this equation, and prove a priori $C^{2, \beta}$ estimates when $\lvert h \rvert \gt (n-2) \frac{\pi}{2}$ and a subsolution exists. Using the method of continuity we show that the dHYM equation admits a smooth solution in the supercritical phase case, whenever a subsolution exists. Finally, we discover some Bridgeland-stability-type cohomological obstructions to the existence of solutions to the dHYM equation and we conjecture that when these obstructions vanish the dHYM equation admits a solution. We confirm this conjecture for complex surfaces.

中文翻译:

具有指定拉格朗日相位的$(1,1)$形式:先验估计和代数障碍

令$(X,\ alpha)$为维$ n $的Kähler流形,令$ [\ω\ in H ^ {1,1}(X,\ mathbb {R})$。我们研究了相对于$ \ alpha $指定$ \ omega $的拉格朗日相位的问题,该问题由非线性椭圆方程\ [\ sum ^ {n} _ {i = 1} \ arctan(\ lambda_i)描述= h(x)\],其中$ \ lambda_i $是$ \ omega $相对于$ \ alpha $的特征值。当$ h(x)$是拓扑常数时,该方程式对应于变形的Hermitian-Yang-Mills(dHYM)方程,并且通过镜像对称关系与特殊的Lagrangian子流形相关。我们为该方程式引入了子解的概念,并证明了先验$ C ^ {2,\ beta} $估计何时$ \ lvert h \ rvert \ gt(n-2)\ frac {\ pi} {2} $并且存在子解。使用连续性方法,我们表明,只要存在子解,dHYM方程就可以在超临界相情况下获得光滑解。最后,我们发现了dHYM方程解的存在性的一些Bridgeland稳定型同调障碍,并且我们推测当这些障碍消失时dHYM方程可以接受一个解。对于复杂曲面,我们证实了这一猜想。
更新日期:2020-04-21
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