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

Tristan C. Collins (Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Mass., U.S.A.)

Adam Jacob (Department of Mathematics, University of California at Davis)

Shing-Tung Yau (Department of Mathematics, Harvard University, Cambridge, Massachusetts, U.S.A.)

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

Received 5 August 2019

Published 21 April 2020