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# Cambridge Journal of Mathematics

## Volume 8 (2020)

### Number 2

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

Pages: 407 – 452

DOI: https://dx.doi.org/10.4310/CJM.2020.v8.n2.a4

#### Authors

#### Abstract

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.

#### 2010 Mathematics Subject Classification

14J32, 53C07

Tristan C. Collins was supported in part by NSF grant DMS-1506652, NSF grant DMS-1810924 and an Alfred P. Sloan fellowship.

Adam Jacob was supported in part by NSF grant DMS-1204155.

Received 5 August 2019

Published 21 April 2020