## Algebraic Geometry: A Volume in Memory of Paolo Francia ( De by Mauro C. Beltrametti, Fabrizio Catanese, Ciro Ciliberto

By Mauro C. Beltrametti, Fabrizio Catanese, Ciro Ciliberto

The papers during this quantity conceal a large spectrum of algebraic geometry, from causes conception to numerical algebraic geometry and are in general fascinated by greater dimensional kinds and minimum version application and surfaces of normal kind. part of the articles grew out of a convention in reminiscence of Paolo Francia held in Genova in September 2001 with nearly 70 contributors.

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On algebraic 1-motives related to Hodge cycles 41 The category MHS 00 is abelian and MHS is a fully faithful abelian subcategory of MHS 00 • Note that similar categories of infinite dimensional mixed Hodge structures already appeared in the literature, see Hain [20] and Morgan [31). , whose objects and morphisms are obtained by formally add to MHS (small) filtered colimits of objects in MHS with colimit morphisms. Consider the case A = Q. In this case, in the category MHS 00 we have infinite products of those families of objects {H; }; EI such that the induced families of filtrations {W;};EI and {F;};EI are finite.

Recall that Jannsen [23] formulated an homological version of the Hodge conjecture for singular varieties. Moreover, Bloch in a letter to Jannsen (see the Appendix A in [23] cf. Section 5), gave a counterexample to a naive cohomological Hodge conjecture for curves on a singular 3-fold. , F 1 n H 2 (X, Z) is generated by c 1 of line bundles on X, holds true in the singular setting "because one has the exponential". Anyways, jointly with V. , let LP H*(X, Z) be the filtration induced by the Leray spectral sequence along the canonical continuous map Xan --+ XZar, is F 1 n L 1 H 2 (X, Z) given by c 1 of line bundles on X?

3 ( 1932). [So] A. J. Sommese, Submanifolds of abelian varieties, Math. Ann. 233 ( 1978), 229-256. [Sp] R. Speiser, Cohomological dimension and abelian varieties, Amer. J. Math. 95 (1973), 1-34. [Z] 0. Zariski, Theory and applications of holomorphic functions on algebraic varieties over arbitrary ground fields. Mem. Amer. Math. Soc. 5, Amer. Math. , Providence, Rl, 1951. L. O. ro On algebraic 1-motives related to Hodge cycles Luca Barbieri-Viale Abstract. The goal of this paper is to introduce Hodge 1-motives of algebraic varieties and to state a corresponding cohomological Orothendieck-Hodge conjecture, generalizing the classical Hodge conjecture to arbitrarily singular proper schemes.