$(1,1)$ forms with specified Lagrangian phase: A priori estimates and algebraic obstructions

Tristan C. Collins Massachusetts Institute of Technology Adam Jacob University of California, Davis Shing-Tung Yau Harvard University

Geometric Analysis and Geometric Topology mathscidoc:2005.15002

Cambridge Journal of Mathematics, (2), 407-452, 2020
Let $(X,\alpha)$ be a K\"ahler 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 \[ \sum_{i=1}^{n} \arctan(\la_i)= h(x) \] where $\la_i$ 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 of the mirror. We introduce a notion of subsolution for this equation, and prove a priori $C^{2,\beta}$ estimates when $|h|>(n-2)\frac{\pi}{2}$ 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 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.
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@inproceedings{tristan2020$(1,1)$,
  title={$(1,1)$ forms with specified Lagrangian phase: A priori estimates and algebraic obstructions},
  author={Tristan C. Collins, Adam Jacob, and Shing-Tung Yau},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20200513090538317310677},
  booktitle={Cambridge Journal of Mathematics},
  number={2},
  pages={407-452},
  year={2020},
}
Tristan C. Collins, Adam Jacob, and Shing-Tung Yau. $(1,1)$ forms with specified Lagrangian phase: A priori estimates and algebraic obstructions. 2020. In Cambridge Journal of Mathematics. pp.407-452. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20200513090538317310677.
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