Hodge theory and deformations of affine cones of subcanonical projective varieties

Natale, Carmelo Di; Fatighenti, Enrico; Fiorenza, Domenico

doi:10.1112/jlms.12073

Cited by 11 publications

(23 citation statements)

References 26 publications

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“…Bertin [2] showed how to compute the Hodge numbers of some Calabi-Yau threefolds, and this method can be applied to some of the families in Table 1. Another way to find Hodge numbers appears in [12], and this works by computing the graded pieces of the deformation space T 1 . Unfortunately, the complexity of these computations becomes very high when the degree is greater than 17.…”

Section: Preliminariesmentioning

confidence: 99%

Arithmetically Gorenstein Calabi--Yau threefolds in $\mathbb{P}^7$

Coughlan¹,

Lukasz²,

Kapustka³

et al. 2016

ERA-MS

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Section: Preliminariesmentioning

confidence: 99%

Arithmetically Gorenstein Calabi--Yau threefolds in $\mathbb{P}^7$

Coughlan¹,

Lukasz²,

Kapustka³

et al. 2016

ERA-MS

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“…, where m is the integer such that ω X ∼ = O X (m). These cohomology spaces both coincide with their primitive part, since H 2 (X , O X (k)) = 0 for any k. In the case of projective hypersurfaces Griffiths' theory implies [5,Corollary 3.13]. These spaces can be shown to be isomorphic a priori, without deducing it from the previous theorem.…”

Section: Lemma 33 Let X Be a Smooth Hypersurface Of Degree D In The Grassmannianmentioning

confidence: 97%

“…The latter is a classical object in algebraic geometry and deformation theory. We refer to [5] for an overview of its properties that are needed in the present paper. When X is an arbitrary smooth projective variety of codimension c there is a link between the deformations of A X and the Hodge theory of X .…”

Section: Theorem 21mentioning

confidence: 99%

“…When X is an arbitrary smooth projective variety of codimension c there is a link between the deformations of A X and the Hodge theory of X . An assumption needed for the result is the subcanonicality property for X , that is we can write ω X as O X (c), some c ∈ Z. Theorem 2.2 (Theorem 1.1 in [5]) Let X be a smooth projectively normal variety of dimension n > 1, and let m ∈ Z be the integer such that ω…”

Section: Theorem 21mentioning

confidence: 99%

“…Recall from [26] that T 1 A X can be defined as Ext 1 ( 1 A X , O A X ) under the assumption of projective normality of X . We refer to [5] for a collection of properties relevant in this context. In particular, recall that for a smooth projective hypersurface the module T 1…”

Section: Hypersurfaces In Grassmanniansmentioning

confidence: 99%

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

Fatighenti

Mongardi

2021

Math. Z.

Self Cite

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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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Fano 3-folds in format, Tom and Jerry

Brown

Kasprzyk

Qureshi

2017

European Journal of Mathematics

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This is a first introduction to unprojection methods, and more specifically to Tom and Jerry unprojections. These two harmless tricks deserve to be better known, since they answer many practical questions about constructing codimension 4 Gorenstein subschemes. In particular, we discuss here the two smoothing components of the anticanonical cone over P(1, 2, 3).Section 2 treats the "6 × 6 extrasymmetric format", that describes the Segre embedding of P 2 ×P 2 and some of its degenerations. One can view this as just algebraic manipulations, or as a typical case of Tom unprojection. In a similar vein, Section 3 treats the "Double Jerry construction", that describes the Segre embedding of P 1 ×P 1 ×P 1 and some of its degenerations. In Section 4 we put these two unprojection constructions together as a versal deformation of the anticanonical cone over P(1, 2, 3) over a reducible base, with the obstructions also controlled by the matrix format. We conclude with some general remarks, mnemonics, slogans, and FAQ. We do not pretend any generality, or any theoretical treatment of Gorenstein codimension 4 (compare [G4]).

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Hodge theory and deformations of affine cones of subcanonical projective varieties

Cited by 11 publications

References 26 publications

Arithmetically Gorenstein Calabi--Yau threefolds in $\mathbb{P}^7$

Arithmetically Gorenstein Calabi--Yau threefolds in $\mathbb{P}^7$

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

Fano 3-folds in format, Tom and Jerry

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