A remark on the generalized Hodge conjecture |
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Authors: | Dario Portelli |
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Institution: | 1. Dip. Matematica e Geoscienze, Università di Trieste, via Valerio 12/a, 34127?, Trieste, Italy
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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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