On two complex Hessian equations and convergence of corresponding flows

Abstract

This thesis consists of three parts. Two important complex Hessian equations are studied on certain compact K\"ahler manifolds from different perspectives. The first one is the {\em J-equation} introduced independently by S.K. Donaldson and X.X. Chen from different point of view. The second one is the {\em deformed Hermitian Yang Mills (dHYM) equation} which has connection to the mirror symmetry in string theory.

There is a notion of (global) {\em slopes} for both equations. It is known that they admit smooth solutions with the global slopes if and only if certain Nakai-Moishezon (NM) type criterion holds. In the first part, our aim is to find some appropriate {\em singular} solutions of the equations when the NM-type criterion fails--this is the so-called {\em unstable case}. An algebro-geometric characterization of the slopes is formulated -- which we call the {\em minimal J-slope} for the J-equation and the {\em maximal dHYM-slope} for the dHYM equation. There is a natural {\em weak} (i.e. {\em singular}) version of the equations replacing the standard wedge product with a more generalized product, called the {\em non-pluripolar product}. We settle the existence and uniqueness problem for the singular J and dHYM equation on compact K\"ahler surfaces. More precisely, for the J-equation we show that there exists a unique closed $(1,1)$-{\em K\"ahler current} solving the singular J-equation on a compact K\"ahler surface with the minimal J-slope. Analogous result is established for the singular dHYM equation on compact K\"ahler surfaces with the maximal dHYM-slope. Furthermore, we conjecture analogous existence and uniqueness result for higher dimensions.

In the second part, we study the convergence behaviour of the {\em J-flow}, which is the parabolic version of the J-equation, on certain generalized projective bundles using the {\em Calabi Symmetry} in the J-unstable case. An {\em invariant version} of the minimal J-slope is introduced for these bundles. Furthermore, we show that the flow converges to some unique limit in the weak sense of currents, and the limiting current solves the singular J-equation with the invariant minimal J-slope. This result resolves the invariant version of our conjecture for the J-equation on these examples with symmetry.

In the third part, we study the convergence behaviour of a flow, called the {\em cotangent flow}, for the dHYM equation in the dHYM-unstable case on the blowup of $\mathbb{C}\mathbb{P}^2$ or $\mathbb{C}\mathbb{P}^3$. Analogous to our results in the second part, we show that the flow converges to some unique limit, and the limiting current solves the singular dHYM equation with the (invariant) maximal dHYM-slope.