## complex manifold geography in dimension 2 and 3

- B. Hunt
- J. Differential Geometry,
- 1989

- Published 1996

Let X be a 3-fold of general type with Chern class ci(X) = ci(TX) ∈ H(X,Z). Then cup product gives the Chern numbers c31(X), c1c2(X), c3(X) ∈ Z. One natural question to ask is: Which point in P2(Q) does correspond to the Chern numbers [c31(X), c1c2(X), c3(X)] for some 3-fold X? By using Fermat cover (desingularization of the branched cover of CP over an arrangement of planes), Hunt [H] was able to show that there are two triangles inside which every point corresponds to a 3-fold. Extending Hunt’s idea, Liu [C3, L] obtained a larger region in the affine chart c1c2 6= 0 (LMCN in the chart attached). In [C2], the author gave an explicit description of the “SCI zone”, the limit points of the Chern ratios (c31/c1c2, c3/c1c2)(X) of all the complete intersection 3-folds X . However the only known bound of the region of all possible Chern numbers, under the assumption that either X is minimal or the canonical divisor KX is ample, is 0 ≤ c31/c1c2 ≤ 8/3. (The right inequality is Yau’s) In this paper, we study the bounds of the Chern ratios of 3-folds in P. Let X ⊂ P be a 3-fold with hyperplane class H, canonical class K. What we can show is the following

@inproceedings{Chang1996arXI,
title={ar X iv : a lg - g eo m / 9 61 10 26 v 1 2 2 N ov 1 99 6 On the density of ratios of Chern numbers of embedded threefolds},
author={Mei - Chu Chang},
year={1996}
}