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Defect of a nodal hypersurface. (English) Zbl 0983.14017
In this paper, the author gives interesting formulas for the Hodge numbers of a nodal hypersurface in a smooth complex projective fourfold. Let $$X$$ be a smooth complex projective fourfold and let $$Y$$ be a nodal hypersurface in $$X$$ such that:
A1: the line bundle $${\mathcal M}:={\mathcal O}_X(Y)$$ is ample,
A2: $$H^2\Omega^1_X=0,$$
A3: $$H^3(\Omega^1_X \otimes{\mathcal M}^{-1}) =0$$.
Denote by $$\widetilde Y$$ a big resolution of $$Y$$ and $$\widehat Y$$ a small one.
Theorem 1: \begin{aligned} & h^{11}(\widetilde Y)=h^{11}(X) +\mu+\delta,\\ & h^{12}(\widetilde Y)=h^0({\mathcal M}^{\otimes 2}\otimes K_X)+ h^3{\mathcal O}_x-h^0( {\mathcal M}\otimes K_X)-h^3\Omega^1_X-h^4(\Omega^1_X\otimes{\mathcal M}^{-1})-\mu +\delta\end{aligned} Theorem 2: $h^{11}(\widehat Y)=h^{11}(\widetilde Y)-\mu,\quad h^{12} (\widehat Y)=h^{12}(\widetilde Y),$ where $$\mu$$ is the number of nodes and $$\delta$$ the defect of $$Y$$.

##### MSC:
 14J17 Singularities of surfaces or higher-dimensional varieties 14C30 Transcendental methods, Hodge theory (algebro-geometric aspects) 14E15 Global theory and resolution of singularities (algebro-geometric aspects) 14B05 Singularities in algebraic geometry 14J30 $$3$$-folds
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