Differential forms and quadrics of the canonical image
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Differential forms and quadrics of the canonical image Luca Rizzi1 · Francesco Zucconi2 Received: 23 May 2019 / Accepted: 29 February 2020 © Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We extend the theory of Pirola and Zucconi (J Algebraic Geom 12(3):535–572, 2003). We introduce the new notion of adjoint quadric for canonical images of irregular varieties. Using this new notion, we obtain the infinitesimal Torelli theorem for varieties whose canonical image is a complete intersection of hypersurfaces of degree > 2 and for Schoen surfaces. Finally, we show that a family with fiberwise liftable holomorphic forms such that the fibers have Albanese morphism of degree 1 is birationally trivial if there exist no adjoint quadrics. Keywords Extension class of a vector bundle · Torsion freeness · Infinitesimal Torelli problem · Canonical map · Holomorphic forms · Albanese variety · Families of varieties · Generic Torelli problem Mathematics Subject Classification 14C34 · 14D07 · 14E99 · 14J10 · 14J40
Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 General theory . . . . . . . . . . . . . . . . . . . . . . . 2 Adjoint quadrics and infinitesimal Torelli theorem . . . . . . 2.1 The notion of Adjoint quadric . . . . . . . . . . . . . . . 2.2 Torelli-type problems . . . . . . . . . . . . . . . . . . . 2.2.1 First consequences of the general theory . . . . . . 2.2.2 First consequences of the theory of adjoint quadrics 2.3 Infinitesimal Torelli theorem for Schoen surfaces . . . . 3 Families with birational fibers . . . . . . . . . . . . . . . . . 3.1 Families of morphisms . . . . . . . . . . . . . . . . . . 3.2 Families with liftability conditions . . . . . . . . . . . . 3.3 The theorem . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Francesco Zucconi [email protected] Luca Rizzi [email protected]
1
Graduate School of Mathematical Sciences, The University of Tokyo, Tokyo 153-8914, Japan
2
DMIF, The University of Udine, 33100 Udine, Italy
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L. Rizzi, F. Zucconi
1 Introduction In this paper, we go deeper along the direction indicated in [35] to obtain Torelli-type theorems.
1.1 General theory Let ξ ∈ Ext1 (F , O X ) be an extension
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