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Trilinear forms and Chern classes of Calabi–Yau threefolds. (English) Zbl 1299.14035
Let \(X\) be a projective threefold with \(K_X\) trivial and \(H^1(X,\mathcal{O}_X)=0\), i.e. a Calabi-Yau threefold (in the strict sense), and let \(H\) be a very ample divisor on \(X\). It is a well-known open problem whether the number of distinct topological types that \(X\) can have is finite or not. On the other hand, it is known that if the value of the selfintersection \(H^3\) is fixed, then there are only finitely many diffeomorphism types. The goal of this paper is to give an explicit bound for the Euler characteristic of \(X\) in terms of \(H^3\).
By the Chern-Gauss-Bonnet theorem, the Euler characteristic of \(X\) equals the Chern number \(c_3(X)\). The authors prove that \[ -72 H^3 -160\leq c_3(X)\leq 12 H^3+80. \]

MSC:
14J32 Calabi-Yau manifolds (algebro-geometric aspects)
14F45 Topological properties in algebraic geometry
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References:
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