A linear bound on the Euler number of threefolds of Calabi–Yau and of general type

Abstract

Let X ⊂ P N be either a threefold of Calabi-Yau or of general type (embedded with rK X ). In this article we give lower and upper bounds, linear on the degree of X and N, for the Euler number of X. As a corollary we obtain the boundedness of the region described by the Chern ratios c 3 c 1 c 2 , c 3 1 c 1 c 2 of threefolds with ample canonical bundle and a new upper bound for the number of nodes of a complete intersection threefold.

On the construction problem for Hodge numbers

“…Remark 30. Instead of using [8], but still relying on [15], Chang-Lopez prove in [6] that there is a computable constant C > 0 such that C ⋅c 1 c 2 (X) ≤ c 3 (X) holds for all three-folds X with ample canonical bundle. Computing C explicitly shows that it is about four times smaller then the analogous constant which appears in (A.5).…”

“…Thanks to Burt Totaro for drawing my attention to the construction problem and for informing me about the results in [4,6]. I am grateful to my advisor Daniel Huybrechts for stimulating discussions and to Ciaran Meachan, Dieter Kotschick and Burt Totaro for useful comments.…”

Examples of Calabi–Yau Threefolds as Covers of Almost-Fano Threefolds

“…We refer the interested reader to [7,8,16] (and the references quoted there) about bounds on e(X) for some classes of Calabi-Yau threefolds.…”

On the Chern number inequalities satisfied by all smooth complete intersection threefolds with ample canonical class

“…As it contains no c 3 term, one may naturally wonder whether there exists a Chern number inequality involving c 3 . This is possible because of the following result: Theorem 1.1 (Chang-Lopez, Corollary 1.3 [2]). The region described by the Chern ratios ( However, the result, as well as its proof, does not produce a new Chern number inequality, even for the subclass of smooth complete intersections.…”

“…Q(1; 5) = ( 1 16 , 43 8 ), Q(2; 2, 3) = ( 1 10 , 19 5 ), Q(3; 2, 3, 3) = ( 18 , 13 4 ), Q(3; 2, 2, 2) = ( 1 3 , 23 12 ), and Q(n; 1, • • • , 1) = ( 2(−4+n) 2 12−5n+n 2 , −24+14n−3n 2 +n 3 3(−4+n)(12−5n+n 2 ) ) with n ≥ 5. Remark 1.3.…”