# Bounding singular surfaces via Chern numbers

## TL;DR

This paper establishes bounds on the minimal model program steps for singular surfaces using Chern numbers and demonstrates boundedness of certain classes of pairs of general type with specific singularities and Chern number constraints.

## Contribution

It introduces bounds on the minimal model program for singular surfaces based on discrepancies and Chern numbers, and proves boundedness of classes of pairs with specified invariants.

## Key findings

- Bound on minimal model program steps in terms of discrepancies and Chern numbers.
- Boundedness of classes of pairs of general type with specified invariants.
- Establishment of a bounded family of surfaces with controlled singularities and Chern numbers.

## Abstract

We prove the existence of a bound on the number of steps of the minimal model program for singular surfaces in terms of discrepancies and top Chern numbers. As an application, we prove that given $R\in\mathbb{R}$ and $\epsilon\in (0,1)$, the class $\mathcal{F}(R,\epsilon)$ of $2$-dimensional pairs $(X,D)$ of general type with $\epsilon$-klt singularities, $D$ with standard coefficients, and $4c_2(X,D)-c_1^2(X,D)\leq R$, forms a bounded family.