On the solvability of general inverse σ_k-equations

In this talk, first, I will introduce general inverse σ_k equations in Kähler geometry. Some classical examples are the complex Monge–Ampère equation, the J-equation, the complex Hessian equation, and the deformed Hermitian–Yang–Mills equation. Second, by introducing some new real algebraic geometry techniques, we can consider more complicated general inverse σ_k equations. Last, analytically, we study the solvability of these complicated general inverse σ_k equations.