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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
##### Keywords:
Calabi-Yau threefolds; Euler characteristic
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##### References:
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