[Paper Review] On the asymptotic expansion of Bergman kernel

Xianzhe Dai, Kefeng Liu+1 more|arXiv (Cornell University)|Apr 27, 2004

Geometry and complex manifolds33 references131 citations

TL;DR

This paper establishes the full off-diagonal asymptotic expansion and Agmon-type estimate for the Bergman kernel of the spin^c Dirac operator on high powers of an ample line bundle over compact symplectic manifolds and orbifolds. Using microlocal analysis and heat kernel methods, it proves a uniform asymptotic expansion in powers of $ p^{-1} $, with coefficients that are polynomials in curvature tensors and their derivatives, extending Donaldson's work to orbifolds and linking geometric quantization to stability and index theory.

ABSTRACT

We study the asymptotic of the Bergman kernel of the spin$^c$ Dirac operator on high tensor powers of a line bundle.

Motivation & Objective

• To extend Donaldson's work on Kähler metrics with constant scalar curvature to orbifolds by studying asymptotic Bergman kernel expansions.
• To establish a full off-diagonal asymptotic expansion of the Bergman kernel for the spin^c Dirac operator on high powers of an ample line bundle over symplectic manifolds and orbifolds.
• To derive uniform Agmon-type estimates for the Bergman kernel, valid across varying Riemannian metrics and geometric data.
• To relate the asymptotic expansion of the Bergman kernel to the heat kernel and index theory, particularly in the context of geometric quantization and stability.
• To show that the coefficients in the expansion are polynomials in curvature tensors $ R^{TX}, R^{ ext{det}}, R^{E}, R^{L} $, their derivatives, and reciprocals of eigenvalues of $ f{J} $.

Proposed method

• Uses the spin^c Dirac operator $ D_p $ acting on $ (0,q) $-forms with values in $ L^p \otimes E $, where $ L $ is a positive line bundle with curvature $ \omega $.
• Applies microlocal analysis and the Boutet de Monvel–Guillemin construction to define Szegö kernels and associated pseudodifferential operators $ D_b $.
• Derives the asymptotic expansion of the Bergman kernel $ B_p(x) $ via a parametrix construction and stationary phase methods in the oscillatory integral setting.
• Establishes uniform estimates by controlling derivatives in $ \mathscr{C}^l $-norms across parameter families bounded in $ \mathscr{C}^s $-norms with lower-bounded metric.
• Relies on the heat kernel expansion of $ \exp(-\frac{u}{p}D_p^2) $ to derive the Bergman kernel asymptotics through spectral and trace methods.
• Uses local normal coordinates and the complex structure $ J $ to express the kernel in terms of $ \mathcal{J}_{x_0} $, the complex structure operator, and computes Gaussian integrals over $ \mathbb{R}^{2n} $ to extract leading-order terms.

Experimental results

Research questions

• RQ1 How does the Bergman kernel of the spin^c Dirac operator behave asymptotically as the tensor power $ p \to \infty $, away from the diagonal?
• RQ2 Can a full off-diagonal asymptotic expansion be constructed for the Bergman kernel on symplectic orbifolds, with explicit curvature-dependent coefficients?
• RQ3 What is the precise form of the Agmon-type estimate for the Bergman kernel, and how uniform is it under variations in the metric and connection data?
• RQ4 How are the asymptotics of the Bergman kernel related to the heat kernel of $ D_p^2 $, and what does this imply for index theory and geometric quantization?
• RQ5 To what extent do the coefficients in the expansion reflect geometric invariants such as curvature and holonomy, especially at orbifold points?

Key findings

• The Bergman kernel admits a full off-diagonal asymptotic expansion: $ B_p(x) = \sum_{r=0}^k b_r(x) p^{n-r} + \mathscr{O}(p^{n-k-1}) $, uniformly in $ \mathscr{C}^l $-norm for any $ k,l \in \mathbb{N} $.
• The coefficients $ b_r(x) \in \operatorname{End}(\Lambda(T^{*(0,1)}X) \otimes E)_x $ are polynomials in $ R^{TX}, R^{\det}, R^E, R^L $, their derivatives up to order $ 2r-1 $ or $ 2r $, and reciprocals of eigenvalues of $ \bf{J} $.
• The leading term is $ b_0(x) = (\det \bf{J})^{1/2} I_{\mathbb{C} \otimes E} $, which encodes the symplectic volume form.
• The expansion is uniform: for any $ k,l $, there exists $ C_{k,l} $ independent of the Riemannian metric $ g^{TX} $, provided the data are bounded in $ \mathscr{C}^s $ and $ g^{TX} $ is uniformly bounded below.
• The asymptotic expansion of the Fubini–Study metric satisfies $ \left| \frac{1}{p} \phi_p^* \omega_{FS} - \omega \right|_{\mathscr{C}^l} \leq C_l \left( \frac{1}{p} + p^{l/2} e^{-c\sqrt{p} d(x,X')} \right) $, showing rapid convergence away from the cut locus.
• At orbifold points $ y_j $, the Bergman kernel has a singular contribution $ \frac{e^{i\theta_j p} g|_E \circ I_{\mathbb{C} \otimes E}}{|G_{y_j}| \det_{\mathbb{C}}(1 - g^{-1}_{T^{(1,0)}X})} \delta_{y_j} $, reflecting group action and monodromy.