# The correspondence formula of Dolbeault complex on pair deformation

## TL;DR

This paper establishes a correspondence formula for the Dolbeault complex in holomorphic families of pairs, linking the complex structures and vector bundles across small deformations of the underlying manifolds.

## Contribution

It introduces a new formula that relates Dolbeault complexes of deformed pairs to the original, enhancing understanding of complex structure variations.

## Key findings

- Derived a correspondence between Dolbeault complexes under small deformations
- Established a formula connecting complex structures and vector bundles across deformations
- Provided a framework for analyzing holomorphic family variations

## Abstract

Given a holomorphic family of pairs $\{(X_t,E_t)\}$, where each $E_t$ is holomorphic vector bundle over compact complex manifold $X_t$. For small enough $t$, we get a correspondence between the Dolbeault complex of $E_t$-valued $(p,q)$-forms on $X_t$ and the one of $E_0$-valued $(p,q)$-forms on $X_0$.