Asian Journal of Mathematics

Volume 22 (2018)

Number 5

The deformation of pairs $(X, E)$ lifting from base family

Pages: 841 – 862



Kefeng Liu (School of Mathematics, Capital Normal University, Beijing, China; and Department of Mathematics, University of California at Los Angeles)

Jie Tu (Center of Mathematical Sciences, Zhejiang University, Hangzhou, China)


We study the holomorphic family of pairs $\lbrace (X_t, E_t) \rbrace$ by the calculation of Kuranishi data on $\lbrace E_t \rbrace$, where each $E_t$ is a holomorphic vector bundle over the compact complex manifold $X_t$. The splitting of holomorphic cotangent bundle over $E_0$ via any integrable connection decomposes the Kuranishi data into the horizontal part and the vertical part. The horizontal part is the Kursnishi data of base family $\lbrace X_t \rbrace$. When the vertical part is vanishing under the decomposition by a Nakano semi-positive Chern connection $\nabla$, i.e. $\lbrace (X_t, E_t) \rbrace$ is lifting from base family $\lbrace X_t \rbrace$ via $\nabla$, we get an infinitesimal extension of $\overline{\partial}$-closed bundle valued $(n, q)$-form by the recursive method.


Kuranishi data, deformation of pairs, recursive method

2010 Mathematics Subject Classification

32G05, 32G08

Received 4 March 2016

Accepted 3 November 2016

Published 9 November 2018