Topological stability obstructions and geometry of Algebraic Surfaces

In this thesis we study a few topics in the field of complex differential and algebraic geometry. In the first part it suggests a new interpretation of a vector bundle on a families of algebraic varieties (or any structures) as an object in the corresponding moduli spaces. Further, we find a new simple expression for the (2, 2)-form $c_2(E)$ , in terms of $h_{ij}$ , for a vector bundle E of an arbitrary rank on the one-dimensional family of Riemann surfaces, and as a consequence show that $c_2(E) > 0$ (Bogomolov & Lukzen, 2022). Thus it gives a new way to prove the Chen-Donaldson-Sun theorem. When the bundle E inherits certain singularities (of the number N), we offer an expression for the $c_2(E)$. The next topic of the thesis is "curves on the algebraic surfaces". Brunebarbe-Klinger- Totaro theorem asserts that X has a nonzero symmetric differential if there is a finite-dimensional representation of $pi_1(X)$ with infinite image. We give a proof of the similar bound. The last topic concerns the deformation theory of surfaces of general type. We prove the Severi inequality from the different angle for a particular types of surfaces.