Weighted Fano varieties and infinitesimal Torelli problem

Fatighenti, Enrico; Rizzi, Luca; Zucconi, Francesco

doi:10.1016/j.geomphys.2018.09.020

Cited by 7 publications

(6 citation statements)

References 51 publications

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“…For example, as complete intersections in (weighted) projective spaces one finds only the cubic fourfold, see [PS18]. More examples are found if one allows terminal and Q-factorial singularities, see [FRZ19] but no new examples of IHS are produced anyway. In [FM18] we conjectured that even taking complete intersection in Grassmannian one does not get any new example other than a complete intersection with four linear hypersurfaces in the Grassmannian Gr(2, 8) and the above mentioned examples.…”

Section: Introductionmentioning

confidence: 99%

Fano varieties of K3 type and IHS manifolds

Fatighenti¹,

Mongardi²

2019

Preprint

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Section: Introductionmentioning

confidence: 99%

Fano varieties of K3 type and IHS manifolds

Fatighenti¹,

Mongardi²

2019

Preprint

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“…As in the hypersurface case, the only vanishings that are not automatic are for H p, p (Y ). Indeed, using (9) and ( 10) one gets the division in even and odd case, similarly to what we did in Theorem 3.8. Moreover by Künneth formula, I p, p−1 (Y ) = I p, p−1 (G).…”

Section: Lemma 48 Hmentioning

confidence: 54%

“…Some more examples are found if we allow mild singularities-e.g. cyclic quotientsee [9]. Most of these examples are deeply linked with hyperkähler geometry and derived category problems.…”

Section: Lemma 48 Hmentioning

confidence: 99%

A note on a Griffiths-type ring for complete intersections in Grassmannians

Fatighenti

Mongardi

2021

Math. Z.

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We calculate a Griffiths-type ring for smooth complete intersections in Grassmannians. This is the analogue of the classical Jacobian ring for complete intersections in projective space and allows us to explicitly compute their Hodge groups. IntroductionGriffiths' theory of residues is a powerful tool in algebraic geometry. It identifies the Hodge groups of a smooth projective hypersurface X with some special homogeneous slices of a graded ring, the Jacobian ring associated with the defining equation of X (see [12] for the original result). Its very explicit nature has led to proofs of several well-known theorems, for example, the Torelli theorem or the Noether-Lefschetz theorem in some special cases, including e.g. threefolds.This result has been generalised to the case of complete intersection in toric varieties, thanks to the work of Batyrev and Cox, Dimca, Konno, Mavlyutov, and many others [1,6,17,20]. In this case, the generalised Jacobian ring is as explicit as in the hypersurface case (given in terms of generators and relations). Another generalisation was subsequently given by Green in [11], who investigated the case of hypersurfaces of sufficiently high degrees in an arbitrary variety. However, the latter was less explicit than the former case. In a very recent work which was developed in parallel with ours, Huang-Lian-Yau-Yu [13] generalised Green's description to zero loci of homogeneous vector bundles. Their approach and ours share many techniques, which were also used in the first author's PhD thesis [7]. However, they differ both in the scope of the results, which hold in greater generality in [13], and in the explicit computability of Hodge structures, which is explained in greater detail in the present Enrico Fatighenti

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“…• hypersurfaces in projective space [7,11] • hypersurfaces in weighted projective space [39] • complete intersections in projective space [37,41,43] • zerosets of sections of vector bundles [16] • certain cyclic covers of a Hirzebruch surface [29] • complete intersections in certain homogeneous Kähler manifolds [28] • some weighted complete intersections [44] • certain Fano quasi-smooth weighted hypersurfaces [14] • elliptic surfaces [27] The methods used in many of these studies have in common that they describe the cohomology groups relevant for the infinitesimal Torelli map as components of a so-called Jacobi ring and argue that the map can be interpreted as multiplication by some element in this ring. We generalize this method to the case of quasismooth complete intersections in weighted projective space.…”

Section: Introductionmentioning

confidence: 99%

Infinitesimal Torelli for weighted complete intersections and certain Fano threefolds

Licht¹

2022

Preprint

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We generalize the classical approach of describing the infinitesimal Torelli map in terms of multiplication in a Jacobi ring to the case of quasi-smooth complete intersections in weighted projective space. As an application, we prove the infinitesimal Torelli theorem for hyperelliptic Fano threefolds of Picard rank 1, index 1, degree 4 and study the action of the automorphism group on cohomology. The results of this paper are used to prove Lang-Vojta's conjecture for the moduli of such Fano threefolds in a follow-up paper.

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Weighted Fano varieties and infinitesimal Torelli problem

Cited by 7 publications

References 51 publications

Fano varieties of K3 type and IHS manifolds

Fano varieties of K3 type and IHS manifolds

A note on a Griffiths-type ring for complete intersections in Grassmannians

Infinitesimal Torelli for weighted complete intersections and certain Fano threefolds

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