Hey guys :) I have a question on a theorem which is known as Tian-Yau-Zelditch theorem now. If there is some reference for following question, please let me know

Let $(M,\omega)$ be a compact Kaehler manifold with pre-quantum Line bundle $L$ and $s_0,...,s_{N_k}$ be an orthonormal basis for $H^0(M,L^{\otimes k})$ then

$$\text{dim}_\mathbb C H^0(M, L^{\otimes k})=\int_M \sum_{\alpha=0}^{N_k}\left | s_\alpha\right |_h^2(z)dV_z=a_0k^n+a_1k^{n-1}+...+a_n$$ which $\left | . \right |_h$ is the fiberwise norm with respect to the hermitian metric $h$.

There are geometric interpretations for the first three coefficients $a_0, a_1$ and $a_2$ as follows,

$$a_0=\text{Vol}(M,\omega)$$ , $$a_1=\frac{1}{2}\int_M R dV_\omega$$

$$a_2=\int_M(\frac{1}{3}\Delta R+\frac{1}{24}\left | Riem \right |^2-\frac{1}{6}\left | Ric \right |^2+\frac{1}{8}R^2 )dV$$

I am wondering if there is any geometric interpretation for $a_3$?


I am not sure what a pre-quantum line bundle is, here is an answer in the "classical" (holomorphic) case. Please let me know if this does not answer your question.

It is a polynomial in the derivatives of the metric. An explicit form can be found on page 237 of "On the lower order terms of the asymptotic expansion of Tian-Yau-Zelditch", by Z. Lu. His key insight was to use so-called K-coordinates to attain a more detailed expansion than Tian. Essentially this is a normal form for the potential of the metric. I am not aware of any use of the explicit form of the lower order terms than the first term, which is the scalar curvature, so I am not sure why an explicit form for the third term would be useful. Higher order terms can be calculated using Lu's method, but the calculations are difficult.

Edit: This seems to be open in the "pre-quantum" setting. The term $a_3$ is computable, using the method of Ma-Marinescu "Generalized Bergman kernels on symplectic manifolds", as stated by them at the start section $2$. However, they don't carry out the calculation of $a_i$ for $i>2$. As mentioned above, to the best of my knowledge, the explicit form of the terms $a_i$ hasn't actually been used for $i>1$.

  • $\begingroup$ A prequantum line bundle is a line bundle with connection $A$ over a (pre)symplectic manifold $(X,\omega)$ such that $\omega$ is the curvature $2$-form of $A$. $\endgroup$ – user62675 May 31 '14 at 15:27

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