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On the Chern numbers of a smooth threefold. (English) Zbl 1430.14039
A question asked by Hirzebruch is: Which linear combinations of Chern numbers on a smooth complex projective variety are topologically invariant? D. Kotschick [J. Topol. 1, No. 2, 518–526 (2008; Zbl 1145.57023); Adv. Math. 229, No. 2, 1300–1312 (2012; Zbl 1244.57048)] answered the question and showed that a rational linear combination of Chern numbers is a homeomorphism invariant of smooth complex projective varieties if and only if it is a multiple of the Euler characteristic. Kotschick also asked the following question for smooth projective threefolds: Does \(c_1^3=-K_X^3\) take only finitely many values on the projective algebraic structures \(X\) with the same underlying 6-manifold?
Motivated by this question, the authors study the Chern numbers of a smooth projective threefold \(X\). By Minimal Model Program, there is a birational map \(\phi: X\dashrightarrow Y\) into a threefold \(Y\) such that either \(X\) is not uniruled (A variety of dimension \(n\) is uniruled if there is a variety \(T\) of dimension \(n-1\) and a dominant map \(T\times\mathbb{P}^1\dashrightarrow X\)) and \(Y\) is minimal (i.e., the canonical divisor \(K_X\) is nef) or \(X\) is uniruled and \(Y\) admits a Mori fiber space structure (i.e., a morphism \(Y\rightarrow Z\) with connected fibers with relative Picard number equal to one and whose general fiber is a nontrivial Fano variety). Their strategy consists in two steps: first find the upper bound of \(K_Y^3\) then estimate the bound of \(K_X^3-K_Y^3\).
By the Bogomolov-Miyaoka-Yau inequality for terminal threefolds, the authors prove their fist result.
Theorem 1.2. Let \(X\) be a smooth complex projective threefold which is not uniruled. Then \[ \mathrm{vol}(X)\leq 6b_2(X)+36b_3(X), \] where \[ \mathrm{vol}(X)=\lim\sup_{m\rightarrow \infty}\frac{n!h^0(X, mK_X)}{m^n}. \] By combining methods in birational geometry, topology, and arithmetic geometry, the authors prove their main result:
Theorem 1.3. Let \(Y\) be a \(\mathbb{Q}\)-factorial terminal threefold with associate cubic form \(F_Y\), and let \(f:Y\rightarrow X\) be a divisorial contraction to a point or a smooth curve contained in the smoothy locus of \(X\) (in the last case assume also that \(\Delta_{F_Y}\neq 0\)).
(1) If \(f\) contracts a divisor to a point, then \[ |K_Y^3-K_X^3|\leq 2S_W+6(b_3(Y)+1), \] where \(S_Y\) is a topology invariant defined by \(F_Y\). Moreover, the same inequality is true after replacing \(b_3(Y)\) by \(Ib_3(Y)=\dim IH^3(Y, \mathbb{Q})\).
(2) The cubic form \(F_X\) is determined up to finite ambiguity by the cubic form \(F_Y\).
An application of these results is: Let \(X\) be a smooth complex projective threefold which is not uniruled and let \(F_X\) be its associate cubic form. Assume that \(\Delta_{F_X}\neq 0\) and that there exists a birational morphism \(f:X\rightarrow Y\) onto a minimal projective threefold \(Y\), which is obtained as a composition of divisorial contractions to points and blow-downs to smooth curves in smooth loci. Then there is a constant \(C\) depending only on the topology of the 6-manifold underlying \(X\) such that \[ |K_X^3|\leq C. \]

MSC:
14E30 Minimal model program (Mori theory, extremal rays)
14J30 \(3\)-folds
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