$(1,1)$ forms with specified Lagrangian phase: <i>a priori</i> estimates and algebraic obstructions

Abstract

Let (X, α) be a Kähler manifold of dimension n, and let [ω] ∈ H 1,1 (X, R). We study the problem of specifying the Lagrangian phase of ω with respect to α, which is described by the nonlinear elliptic equationwhere λi are the eigenvalues of ω with respect to α. 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,β estimates when |h| > (n − 2) π 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.

A flow of conformally balanced metrics with Kähler fixed points

“…The existence of such subsolutions provides the existence of a genuine solution, as established in [64,17] (see also [68,15,63] for extensions and generalizations).…”

The Deformed Hermitian–Yang–Mills Equation in Geometry and Physics

“…To relate this to the Bridgeland stability condition we would like to think of inequality (9) as saying that the surjection L ։ L ⊗ O V does not destabilize L, where O V is the skyscraper sheaf with support on V . Unfortunately this is not quite correct (unless T d(X) = 1), since Finally we note that if L admits a solution of the deformed Hermitian-Yang-Mills equation then by the BPS bound in Proposition 2.2 we have |Z ω,X (L)| ch(L) > 0 which is precisely the second condition required in the definition of a Bridgeland stability condition.…”

“…Conjecture 3.5 (Collins-Jacob-Yau [9]). There exists a solution to the deformed Hermitian-Yang-Mills equation in the class a with lifted angle θ ∈ (n − 2) π 2 , n π 2 ) if and only if (8) holds for all proper, irreducible analytic subvarieties V X with dim C V = p.…”

“…We end by noting some evidence for the conjecture. First of all, the conjecture holds in complex dimension 2 [21,9]. In general, we consider an asymptotic version of the conjecture.…”

Special Lagrangian and deformed Hermitian Yang–Mills on tropical manifold

“…For example, Jacob and Yau [6] considered an analogue of Lagrangian mean curvature flows for Hermitian connections. Collins, Jacob and Yau [2] gave a priori estimates and proved the existence of deformed Hermitian Yang-Mills connections under some assumptions and found some obstructions to the existence. By using a different method from [2], Pingali [9] gave similar a priori estimates for deformed Hermitian Yang-Mills connections.…”