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.

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 $X_0$.

Equations

$$\pi :\mathcal{X}\to \Delta$$

$$F=(\sigma ,\pi ):\mathcal{X}\to X_0\times \Delta ,X_t:={\pi }^{-1}(t),$$

$$\phi (t)\in A^{0,1}(X_0,T^{1,0}{X}_0)$$

$$e^{i_{\phi (t)}}=\sum _{k=0}^{\infty }\frac{1}{k!}{i}_{\phi (t)}^k,i_{\phi (t)}^k=\underset{k\text{times}}{\underbrace{i_{\phi (t)}\circ \cdots \circ i_{\phi (t)}}}.$$

$$e^{i_{\phi (t)}}:A^{n,p}(X_0)\to A^{n,p}(X_t)$$

$$e^{-i_{\phi (t)}}\circ \nabla \circ e^{i_{\phi (t)}}=\nabla -{\mathcal{L}}_{\phi (t)}^{1,0}+i_{\bar{\partial }\phi (t)-\frac{1}{2}\left[\phi (t),\phi (t)\right]}$$

$${\mathcal{L}}_{\phi (t)}^{1,0}:=i_{\phi (t)}\circ {\nabla }^{1,0}-{\nabla }^{1,0}\circ i_{\phi (t)}.$$

$$\bar{\partial }\sigma (t)+\partial \circ i_{\phi (t)}\sigma (t)=0$$

$$e^{i_{\phi }|i_{\bar{\phi }}}:A^{p,q}(X_0)\to A^{p,q}(X_t).$$

$$\left(\bar{\partial }-L_{\phi (t)}^{1,0}\right)\sigma (t)=0$$

$$L_{\phi (t)}^{1,0}:=i_{\phi (t)}\circ \partial -\partial \circ i_{\phi (t)}.$$

$${\bar{\partial }}_t\circ P_t(s)=P_t\circ \left(\bar{\partial }-{\mathcal{L}}_{\varphi (t)}^{1,0}+{\psi }_{E_t}\right)s$$

$${\bar{\partial }}_t\circ P_t(s)=P_t\circ \left(\bar{\partial }-L_{\varphi (t)}^{1,0}\right)s.$$

$$0\to \ker p_{\ast }\to TE\to p^{\ast }(TX)\to 0$$

$$0\to \ker p_{\ast }{|}_{T^{1,0}E}\to T^{1,0}E\to p^{\ast }(T^{1,0}X)\to 0.$$

$$q_{\nabla }:T_{1,0}^{\ast }(E)\to p^{\ast }\left(T_{1,0}^{\ast }(X)\right){q_{\nabla }|}_{p^{\ast }\left(T_{1,0}^{\ast }(X)\right)}=Id$$

$$d{z}_{\alpha }^i\mapsto d{z}_{\alpha }^i,d{v}_{\alpha }^i\mapsto -{\theta }_{\alpha }^{ki}{v}_{\alpha }^k,$$

$$\mathcal{E}\stackrel{p}{\to }\mathcal{X}\stackrel{\pi }{\to }\Delta$$

$${\sigma }_t:={\sigma |}_{X_t}\text{and}{\lambda }_t:={\lambda |}_{E_t}.$$

$$\xi (t):T_{1,0}^{\ast }(E_0)\to T_{0,1}^{\ast }(E_0)$$

$$\varphi (t)\in A^{0,1}(X_0,T^{1,0}{X}_0)$$

$$\psi (t)\in A^{0,1}\left(X_0,End(E_0)\right)$$

$$\varphi (t)={\left(\frac{\partial (z_{\alpha }{)}_t}{\partial (z_{\alpha }{)}_0}\right)}_{ik}^{-1}\frac{\partial (z_{\alpha }{)}_t^k}{\partial ({\bar{z}}_{\alpha }{)}_0^j}d({\bar{z}}_{\alpha }{)}_0^j\otimes \frac{\partial }{\partial (z_{\alpha }{)}_0^i}$$

$${\psi }_{\alpha }^{kl}(t)=(w_{\alpha }^{-1}{)}^{lj}\left((\bar{\partial }-L_{\varphi (t)})w_{\alpha }^{jk}\right)+i_{\varphi (t)}{\theta }_{\alpha }^{kl},$$

$$(v_{\alpha }{)}_t^j=w_{\alpha }^{jl}(v_{\alpha }{)}_0^l$$

$$\bar{\partial }\xi (t)=\frac{1}{2}\left[\xi (t),\xi (t)\right]$$

$$\bar{\partial }\varphi (t)=\frac{1}{2}\left[\varphi (t),\varphi (t)\right]$$

$$\left(\bar{\partial }-{\mathcal{L}}_{\varphi (t)}\right)\psi (t)-\psi (t)\wedge \psi (t)-i_{\varphi (t)}\Theta =0,$$

$${\mathcal{L}}_{\varphi (t)}=i_{\varphi (t)}\circ \nabla -\nabla \circ i_{\varphi (t)}.$$

$$\xi (t)=\varphi (t)+\psi (t)$$

Full text

1. Introduction

The Beltrami differential( or Kuranishi data) plays a central role in analytic deformation theory. Given a holomorphic family of compact complex manifolds, i.e., a holomorphic proper submersion

[TABLE]

from a total complex manifold $\mathcal{X}$ to a small neighbourhood $\Delta\subset\mathbb{C}^{m}$ of [math]. There exists a transversely holomorphic trivialization (see [2, Appendix A], [12, Proposition 9.5] and [9, Theorem IV.31])

[TABLE]

such that each fibre $X_{t}$ is considered as $X_{0}$ with a new complex structure $J_{t}$. The Beltrami differential is a holomorphic tangent bundle valued form

[TABLE]

which describes the varying of complex structures on $X_{0}$.

In [7], Liu-Rao-Yang defined an extension map

[TABLE]

which is a linear isomorphism

[TABLE]

between $(n,q)$-forms on $X_{0}$ and $(n,q)$-forms on $X_{t}$, where $n=\dim_{\mathbb{C}}X_{0}$.

They also proved a formula [7, Theorem 1.3] that

[TABLE]

where $\nabla$ is a Chern connection, and the generalized Lie derivative is defined as

[TABLE]

As a direct corollary [7, Proposition 5.1], the Dolbeault operator $\bar{\partial}_{t}$ on the complex $A^{n,\cdot}(X_{t})$ satisfies the following commutative diagram.

[TABLE]

Then, they calculated an extension formula from a holomorphic form $\sigma\in A^{n,0}(X_{0})$ to a holomorphic form $\sigma(t)\in A^{n,0}(X_{t})$ by solving the obstruction equation

[TABLE]

iteratively.

Recently, the extension map was generalized by Rao-Zhao [10] into any $\bar{\partial}$-closed forms in $A^{p,q}(X_{0})$. They defined a linear isomorphism

[TABLE]

In this situation, the $\bar{\partial}_{t}$-closed equation on $A^{p,q}(X_{t})$ corresponds to the equation

[TABLE]

on $X_{0}$, where the holomorphic Lie derivative is defined as

[TABLE]

The main purpose of this article is to use the pair deformation $\{(X_{t},E_{t})\}$ which was described in [8] to give a correspondence between $E_{t}$-valued $(p,q)$-forms on $X_{t}$ and $E_{0}$-valued $(p,q)$-forms on $X_{0}$.

The correspondence is defined by a commutative diagram

[TABLE]

where $I^{p,q}_{t}$ preserves the $\bar{\partial}_{t}$-complex.

The Dolbeault operator $\bar{\partial}_{t}$ on each fiber $\left(X_{t},E_{t}\right)$ satisfies

Theorem 1.1**.**

(Theorem 4.2) For any $E_{0}$-valued $(p,q)$-form $s\in A^{0}(X_{0},E_{0})$, we have

[TABLE]

When each $E_{t}$ is the trivial line bundle, we get a correspondence between $(p,q)$-forms on $X_{t}$ and $(p,q)$-forms on $X_{0}$.

Theorem 1.2**.**

(Theorem 3.4) For any $(p,q)$-form $s\in A^{p,q}(X_{0})$, we have

[TABLE]

In this situation, the isomorphism map $P_{t}$ is different from that in [10], but the correspondence of Dolbeault operator is the same.

The correspondence supplies a way to transform the Dolbeault complex on each fiber $X_{t}$ into a family of elliptic complexes on $X_{0}$. Then we can use standard techniques from the theory of elliptic operators to solve the varying cohomology problems. A direct application is to use iteration method to extend an element in cohomology class such as [7, Theorem 5.5], [10, Theorem 1.3, 1.4], [11, Theorem 1.5, 1.7] and [8, Theorem 1.5]. On the other aspect, the correspondence of Dolbeault operator can be applied to study the obstruction to the cohomology extension such as in [3], [4] and [1].

2. preliminary

In this section, we will review some basic results of pair deformation in [8].

Let $X$ be a compact complex manifold and $p:E\rightarrow X$ be a holomorphic vector bundle over $X$, then the short exact sequence of tangent bundle

[TABLE]

has a smooth splitting via any linear connection $\nabla$ on $E$(see p.335 in [5]).

If the linear connection is integrable(i.e. $\nabla=\bar{\partial}+\nabla^{1,0}$), then it defines a smooth splitting on holomorphic tangent bundle

[TABLE]

In [8, Proposition 3.2], we use its dual case to get a smooth projection

[TABLE]

on holomorphic cotangent bundle of $E$ via an integrable connection $\nabla$ .

Precisely, under the holomorphic frame coordinate $(U_{\alpha},e_{\alpha},z_{\alpha},v_{\alpha})$ of $E$ (i.e., $e_{\alpha}$ is local frame of $E$ on an open set $U_{\alpha}\subset X$, $z_{\alpha}$ are holomorphic coordinate functions of $X$, and $v_{\alpha}$ are coordinate functions of vertical linear space), the projection $q_{\nabla}$ can be represented locally as

[TABLE]

where $\theta_{\alpha}$ is the connection 1-form of $\nabla$ on $E$.

Recall that a holomorphic family of pairs is a holomorphic vector bundle

[TABLE]

over total space $\mathcal{X}$.

As [8, Proposition 4.1], when $\Delta$ is small enough, there is a compatible trivialization

[TABLE]

such that on each fibre $\left(X_{t},E_{t}\right)$ we have the following commutative diagram

[TABLE]

where

[TABLE]

Given a holomorphic family of pairs $\{(X_{t},E_{t})\}$, the Beltrami differential

[TABLE]

can be represented as two parts

[TABLE]

and

[TABLE]

by [8, Theorem 4.3]. The projection $q_{\nabla}$ via an integrable connection $\nabla$ decides the way to decompose $\xi(t)$ into $\phi(t)$ and $\psi(t)$.

To be specific, if we have a transversely holomorphic trivialization $\sigma_{t}:X_{t}\rightarrow X_{0}$ and its compatible trivialization $\lambda_{t}:E_{t}\rightarrow E_{0}$, then we get the local representations

[TABLE]

and

[TABLE]

where

[TABLE]

under the holomorphic frame coordinate $(U_{\alpha},(e_{\alpha})_{t},(z_{\alpha})_{t},(v_{\alpha})_{t})$ of $(X_{t},E_{t})$.

In [8, Theorem 4.9], the decomposition $\xi(t)=\phi(t)+\psi(t)$ via a Chern connection divides the Maurer-Cartan equation

[TABLE]

into two equations

[TABLE]

and

[TABLE]

where

[TABLE]

We introduce the following two propositions which will be used in later sections.

Proposition 2.1**.**

Given a holomorphic pair deformation $\{(X_{t},E_{t})\}$, if the decomposition

[TABLE]

of Beltrami differential is via an integrable connection $\nabla$ on $E_{0}$, then $\nabla+\psi(t)$ is an integrable connection on $E_{t}$.

Proof.

Under the frame coordinate $\left(U_{\alpha},(e_{\alpha})_{t},(x_{\alpha})_{t},(v_{\alpha})_{t}\right)$ of $(X_{t},E_{t})$, the smooth transition between two holomorphic basis is

[TABLE]

as (2.2).

Then, by the representation of $\psi(t)$ in (2.1) we have

[TABLE]

where $\alpha+i_{\phi(t)}\alpha$ is bundle valued $(1,0)$-form on $X_{t}$. Hence, $\nabla+\psi(t)$ is an integrable connection on $(E_{t},X_{t})$. ∎

It was proved in Section 5 of [8], that the operator $e^{i_{\phi(t)}}$ also defines a linear isomorphism

[TABLE]

between $E_{0}$-valued $(n,q)$-forms on $X_{0}$ and $E_{t}$-valued $(n,q)$-forms on $X_{t}$.

Moreover, we have

Proposition 2.2**.**

Given any $\sigma\in A^{n,q}(X_{0},E_{0})$, then $e^{i_{\phi(t)}}\sigma\in A^{n,q}(X_{t},E_{t})$ and

[TABLE]

Proof.

By Proposition 2.1, $\nabla+\psi_{E_{t}}$ is an integrable connection of $(X_{t},E_{t})$, thus

[TABLE]

for any $\sigma\in A^{n,q}(X_{0},E_{0})$.

Because $\phi(t)$ satisfies Maurer-Cartan equation (2.3), (1.1) becomes

[TABLE]

In particular, when $\sigma\in A^{n,q}(X_{0},E_{0})$, we have

[TABLE]

On the other hand, $\psi_{E_{t}}$ is of type $(0,1)$, which implies

[TABLE]

Hence, we have

[TABLE]

∎

3. The Correspondence between $(p,q)$-forms

In this section, we will use the compatible trivialization of holomorphic cotangent bundle to get a correspondence between $\bar{\partial}_{t}$-complex of $(p,q)$-form on $X_{t}$ and $\bar{\partial}_{0}$-complex of $(p,q)$-forms on $X_{0}$.

Recall that the compatible trivialization of pairs $\{(X_{t},\Omega_{t})\}$ in Section 6 of [8] is

[TABLE]

where $\Omega_{t}$ is the holomorphic cotangent bundle of $X_{t}$, and $(z_{\alpha})_{t}$ is the holomorphic local coordinate of $X_{t}$.

Suppose that $\phi(t)$ is the Beltrami differential of $X_{t}$ and $(z^{i})$ is a fixed holomorphic coordinate of $X_{0}$, then we proved in Section 6 of [8] that the decomposition of Beltrami differential

[TABLE]

via a Chern connection $\nabla$ satisfies

[TABLE]

where

[TABLE]

Suppose that $\Omega^{p}_{t}$ is the $p$-times wedge product of holomorphic cotangent bundle $\Omega_{t}$, and $K_{t}$ is the canonical bundle of $X_{t}$.

Proposition 3.1**.**

For the pair deformations $\{(X_{t},\Omega^{p}_{t})\}$ and $\{(X_{t},K_{t})\}$ we have

[TABLE]

and

[TABLE]

where

[TABLE]

Proof.

By [8, Corollary 4.4], we have

[TABLE]

and

[TABLE]

∎

Moreover, for the pair deformation $\{(X_{t},\Omega_{t}^{p}\otimes K_{t}^{-1})\}$ we have

Corollary 3.2**.**

[TABLE]

At first, we need to establish the commutative diagram

[TABLE]

Proposition 3.3**.**

There is an isomorphism $I^{p,q}_{t}$ such that the above diagram is commutative.

Proof.

Given any $t\in\Delta$, fix a holomorphic coordinate $(z^{i})$ on $X_{t}$, then a $(p,q)$-form $s$ on $X_{t}$ is represented locally as

[TABLE]

the operator $I^{p,q}$ is defined as

[TABLE]

Hence,

[TABLE]

∎

Then, denote

[TABLE]

the diagram

[TABLE]

is commutative.

Theorem 3.4**.**

For any $(p,q)$-form $s\in A^{p,q}(X_{0})$, we have

[TABLE]

Proof.

By Proposition 2.2,

[TABLE]

It suffices to show that

[TABLE]

Denote

[TABLE]

and

[TABLE]

Then

[TABLE]

Moreover,

[TABLE]

implies

[TABLE]

On the other hand,

[TABLE]

Hence,

[TABLE]

The last equality follows from

[TABLE]

∎

4. The Correspondence between bundle valued $(p,q)$-forms

Similar to above section, we need to define the isomorphism $I^{p,q}_{t}$ at first.

Proposition 4.1**.**

There is an isomorphism $I^{p,q}_{t}$ such that the diagram

[TABLE]

is commutative.

Proof.

Fix a holomorphic coordinate $(z^{i})$ on $X_{t}$, and holomorphic frame $\{e_{k}\}$ on $E_{t}$, any $E_{t}$-valued $(p,q)$-form $s$ on $X_{t}$ is represented as

[TABLE]

the operator $I^{p,q}$ is defined as

[TABLE]

then

[TABLE]

∎

Theorem 4.2**.**

Denote

[TABLE]

then for any $E_{0}$-valued $(p,q)$-form $s\in A^{0}(X_{0},E_{0})$ we have

[TABLE]

Proof.

By Proposition 2.2,

[TABLE]

Thus, we just need to prove

[TABLE]

Denote

[TABLE]

and

[TABLE]

Then, we have

[TABLE]

On the other hand,

[TABLE]

Hence, by Theorem 3.4

[TABLE]

∎

Theorem 4.2 generalizes the correspondence (10) in [6], whose commutative diagram is valid for bundle valued $(0,q)$-forms.

The Chern connection $\nabla$ on pair $(X_{0},E_{0})$ is not integrable with respect to $(X_{t},E_{t})$ in general, but we have the following observation.

Corollary 4.3**.**

[TABLE]

for any $E_{0}$-valued $(p,q)$-form $s\in A^{0}(X_{0},E_{0})$.

Proof.

Since

[TABLE]

and

[TABLE]

we have

[TABLE]

Theorem 4.2 implies

[TABLE]

∎

Proposition 4.4**.**

[TABLE]

Proof.

By Lemma 3.2 in [7],

[TABLE]

The last equality is by Maurer-Cartan integrable equation $\bar{\partial}\phi(t)-\frac{1}{2}[\phi(t),\phi(t)]=0$. ∎

The Proposition 4.4 implies that

[TABLE]

Hence,

[TABLE]

Corollary 4.5**.**

When we consider the line bundle case, i.e., each $E_{t}$ is holomorphic line bundle over $X_{t}$, then the second integrable equation in [8]

[TABLE]

is solvable if and only if

[TABLE]

is solvable.

Proof.

[TABLE]

∎

The above assertion implies Proposition 1.4 in [6], which tells that the line bundle is unobstructed along the base family if and only if $c_{1}(L_{0})\in H^{1,1}(X_{t})$.