Canonical lifting of vector fields

Question

Consider a compact Kahler manifold $M$ of complex dimension $n$ and a holomorphic vector field $X$ defined on it. Let $L$ be the line bundle $\Lambda^nT^{1,0}M$. Then is there a canonical way of lifting $X$ to another vector field $X^* $on $L$?

In general, is there such a procedure by which we can canonically lift vector fields on a manifold to other vector bundles defined on the same manifold?

Answer 1

Yes, there is a way to lift $X$ to the total space of $L := \bigwedge^n T^{1,0} M$ because this particular line bundle $π: L \to M$ is invariant under biholomorphisms.

Since $M$ is compact, $X$ (or rather its real part) can be integrated to a one-parameter family of biholomorphisms $(Φ_t)_{t \in \mathbb{R}}: M \to M$ and these lift to biholomorphisms $\widetilde{Φ}_t : \bigwedge^n T^{1,0} M \to \bigwedge^n T^{1,0} M$ "covering" $Φ_t$, that is \begin{equation} π ∘ \widetilde{Φ}_t = Φ_t \tag{1} \end{equation} (The way you lift them is just putting $\widetilde{Φ}_t(Ω) = \left( Φ_t \right)_* Ω$ for $Ω \in \bigwedge^n T_x^{1,0} M$).

So taking the time derivative of $\widetilde{Φ}_t$ we get a vector field $X^*(p) = \left.\frac{d}{d t}\right|_0 \widetilde{Φ}_t(p)$ and by equation (1) we see that $X^*$ is a lift of $X$.

Answer 2

For a general method, may be we can use the two facts :

1. Let $R\longrightarrow M$ be the frame bundle of $M$, with its solder form $\theta$ (a point $e\in R$ is a basis $(v_1,\ldots,v_n)$ of $T_{p(e)}M$, and $\theta(e)(\tilde{v})$ is the vector of coordinates of $Tp(\tilde{v})$ in $e$). For any local automorphism $\varphi$ on an open subset $U\subset M$, there exist a canonical lifting $\tilde{\varphi}$ on $p^{-1}(U)$ defined by the condition : $$\tilde{\varphi}^\star \theta = \theta$$ which express that the action of $\tilde{\varphi}$ is to transport a basis of $T_x M$ to the basis of $T_{\varphi(x)} M$ corresponding to the image through $T\varphi$.

Passing to the identity component, there is the notion of canonical lift of a local vector field (using its flow as in the previous answer).

2. For any $P$-principal bundle $p:R\longrightarrow M$ and a $P$-module $\mathbb{V}$ (eg : $P=GL_n(\mathbb{C})$ and $\mathbb{V}= \overset{n}{\bigwedge}\mathbb{C}^n$), there is the associated vector bundle $R(\mathbb{V})$ on $M$ representing $P$-equivariant functions on $R$ with values in $\mathbb{V}$.

In the case $\mathbb{V}=\overset{n}{\bigwedge}\mathbb{C}^n$, $R(\mathbb{V})=L$.

Local vector fields on $R$ induce local vector fields on $R(\mathbb{V})$.

When $\mathbb{V}$ is some tensor of $\mathbb{C}^n$ and its dual representation, the geometric meaning is to apply $T\varphi$ to a basis of $T_xM$. This gives the corresponding action on basis of tensors of this vector space. It preserves permutations and is linear so it passes to quotients such as exterior product.