# bochner-kodaira formula on kahler manifold

Let $$(M,\omega)$$ be a compact Kahler manifold. Given a vector field $$V = V^i \partial_{z_i} + V^{\overline{i}} \partial_{\overline{z_i}}$$ on $$M$$, how do I show the Bochner-Kodaira formula:

$$||\nabla V||^2 = ||\overline{\nabla} V||^2 + \int_M R_{\overline{j}i}V^i \overline{V^{\overline{j}}} \omega^n$$

$$\nabla V$$ is $$(1,0)$$-form $$TM^{\mathbb{C}}$$-valued given in local coordinate by:

$$\nabla V = (\partial_jV^i + V^a\Gamma^i_{ja}) dz_j \otimes \partial_{z_i} + \partial_{j}V^{\overline{i}} dz_j \otimes \partial_{\overline{z_i}}$$

Similarly, $$\overline{\nabla}$$ is defined in local coordinate by:

$$\overline{\nabla} V = (\partial_{\overline{j}}V^{\overline{i}} + V^{\overline{a}}\overline{\Gamma^i_{ja}}) d\overline{z_j} \otimes \partial_{\overline{z_i}} + \partial_{\overline{j}}V^{\overline{i}} d\overline{z_j} \otimes \partial_{z_i}$$

• Instead of proving directly the integral formula, you could try to compare $\nabla^*\nabla$ and $\bar{\nabla}^*\bar{\nabla}$ using a system of normal Kähler coordinates Commented Jul 3, 2023 at 2:33
• I tried to write out both terms $||\nabla V||$ and $||\overline{\nabla}V||$ in normal coordinate, but it is not clear to me how to make the Ricci tensor terms appear. Commented Jul 3, 2023 at 4:37
• I'm curious, what is your reference for this identity? Commented Jul 3, 2023 at 11:10
• It is equation (3.3) in this paper arxiv.org/pdf/math/0412185.pdf Commented Jul 3, 2023 at 16:20

I think that, in the paper you linked, the authors were using slightly different conventions: $$V$$ is a $$(1,0)$$-vector field on a Kähler manifold, so that $$\nabla_aV^b=\partial_aV^b+\Gamma^b_{ac}V^c$$ and $$\nabla_{\bar{a}}V^b=\partial_{\bar{a}}V^b$$, and $$\nabla$$ is the $$(1,0)$$-part of a connection while $$\bar{\nabla}$$ is the $$(0,1)$$-part. Now, the pairings can be rewritten as $$\lVert \nabla V\rVert^2=\left\langle V,\nabla^*\nabla V\right\rangle_{L^2(g)}$$ $$\lVert \bar{\nabla}V\rVert^2=\lVert \bar{\partial}V\rVert^2=\left\langle V,\bar{\partial}^*\bar{\partial} V\right\rangle_{L^2(g)}$$ so the formula you seek follows from the usual Nakano-Bochner identity applied to the holomorphic vector bundle $$T^{1,0}M$$ with Hermitian metric $$g$$. You can find it, together with a proof, as Corollary $$1.3$$ in Demailly's notes.