A Finiteness Theorem for Polarized Manifolds

There are many previous finiteness theorems about diffeomorphism types in Riemannian geometry. Cheeger’s finiteness theorem asserts that given constants D, υ, and Λ, there are only finitely many n-dimensional compact differential manifoldX admitting Riemannian metric g such that diamg(X) 6 D, Volg(X) > υ and the sectional curvature |Sec(g)| 6 Λ. This theorem can be proved as a corollary of the Cheeger-Gromov convergence theorem (cf. [5, 11]), which shows that if (Xk, gk) is a family compact Riemannian manifolds with the above bounds, then a subsequence of (Xk, gk) converges to a C1,α-Riemannian manifold Y in the C1,α-sense, and furthermore, Xk is diffoemorphic to Y for k ≫ 1. In [1], Cheeger’s finiteness theorem is generalized to the case where the hypothesis on the sectional curvature bound is replaced by the weaker bounds of Ricci curvature |Ric(g)| 6 λ and the L n 2 -norm of curvature ∥Sec(g)∥ L n 2 6 Λ. Furthermore, if n = 4 and g is an Einstein metric, then the integral bound of curvature can be replaced by a bound for the Euler characteristic. We call (X,L) a polarized n-manifold, ifX is a compact complex manifold with an ample line bundle L. In [6], a finiteness theorem for polarized manifolds is obtained. More precisely, Theorem 3 of [6] asserts that for any two constants V > 0 and Λ > 0, there are finite many polynomials P1, · · · , Pl such that if (X,L) is a polarized n-manifold with c1(L) 6 V and −c1(X) · c1(L) 6 Λ, then one Pi is the Hilbert polynomial of (X,L), i.e. Pi(ν) = χ(X,L ν). Consequently, polarized n-manifolds with the above bounds have only finitely many possible deformation types and finitely many possible diffeomorphism types. For any constants λ > 0 and D > 0, denote