Abstract
It is well known that for a smooth, projective variety \(X\) over \({\mathbb {C}}\) we have \(N^{p}H^{i}(X,{\mathbb {Q}})\subset F^{p} H^{i}(X,{\mathbb {C}})\cap H^{i}(X,{\mathbb {Q}})\), where \(N^{\bullet }\) and \(F^{\bullet }\) are respectively the coniveau and Hodge filtrations. In general this inclusion is strict. We introduce a natural subspace \(S^{p,i}\subset F^{p}H^{i}(X,{\mathbb {C}})\) such that \(N^{p}H^{i}(X,{\mathbb {Q}})= S^{p,i}\cap H^{i}(X,{\mathbb {Q}})\) holds true for any \(i,p\). The main technical tool is the use of semi-algebraic sets, which are available by the triangulation of complex projective varieties.
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Acknowledgments
I wish to thank Claire Voisin for her advice and warm encouragement. Moreover, I owe to Elisabetta Fortuna a suggestion for the proof of Lemma 5 and several explanations concerning the semi-algebraic geometry; I thank her heartily for all this. I would also thank Fabrizio Catanese for an useful conversation, and Valentina Beorchia and Francesco Zucconi for their constant help and encouragement. Finally I acknowledge with thanks the careful critical reading of an earlier versions of this paper by the referee. Several of his suggestions have been incorporated.
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The author was supported by funds of the Università degli Studi di Trieste-Finanziamento di Ateneo per i progetti di ricerca scientifica-FRA 2011.
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Portelli, D. A remark on the generalized Hodge conjecture. Math. Z. 278, 1–17 (2014). https://doi.org/10.1007/s00209-014-1301-y
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DOI: https://doi.org/10.1007/s00209-014-1301-y