Bergman kernels and eigenvalue estimate of ∂-laplacian

الأصل · EN

Let (X,ω) be a compact Kähler manifold. Let (L,h) be a hermitian holomorphic line bundle over X, such that Θₗ,ₕ≥ -εω for a small ε>0, E be a holomorphic line bundle over X. For k∈ N+, denote by Xₖ:=(X,ωᵏ) the Kähler manifold X with new scaled metric ωᵏ=kω. Estimates of the number of eigenvalues smaller than λ of the -Laplacian on forms on Xₖ with values in Lᵏ⊗ E are presented for 0≤ λ<k. In particular, when λ=0, we get a numeric bound for the cohomology groups.