Motivic hyper-Kähler resolution conjecture, I : Generalized Kummer varieties

Fu, Lie; Tian, Zhiyu; Vial, Charles

doi:10.2140/gt.2019.23.427

Cited by 53 publications

(70 citation statements)

References 59 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…(Indeed, the varieties

Y_{j}

K_{j}

and

X_{j}

have an MCK decomposition, thanks to Corollary , resp. , resp. ).…”

Section: Some Corollariesmentioning

confidence: 99%

“…There is an inclusion

p^{*} E^{*} false(X false) \subset A_{(0)}^{*} false(Y false) .

(Indeed, we have seen in Corollary that

{(p_{j})}^{*} A^{2} false(D_{j} false) \subset A_{(0)}^{2} false(Y_{j} false)

. Furthermore, it is known that

c_{r} false(K_{j} false) \in A_{(0)}^{r} false(K_{j} false), c_{r} false(X_{j} false) \in A_{(0)}^{r} false(X_{j} false)

[, Proposition 7.13], resp. [, Theorem 2].…”

Section: Some Corollariesmentioning

confidence: 99%

“…The property of having an MCK decomposition is severely restrictive, and is closely related to Beauville's "(weak) splitting property" [3]. For more ample discussion, and examples of varieties with an MCK decomposition, we refer to [34,Section 8], as well as [10,35,37]. Proof.…”

Section: Mck Decompositionmentioning

confidence: 99%

See 2 more Smart Citations

Algebraic cycles and EPW cubes

Laterveer

2018

Mathematische Nachrichten

View full text Add to dashboard Cite

Let $X$ be a hyperk\"ahler variety with an anti-symplectic involution $\iota$. According to Beauville's conjectural "splitting property", the Chow groups of $X$ should split in a finite number of pieces such that the Chow ring has a bigrading. The Bloch-Beilinson conjectures predict how $\iota$ should act on certain of these pieces of the Chow groups. We verify part of this conjecture for a $19$-dimensional family of hyperk\"ahler sixfolds that are "double EPW cubes" (in the sense of Iliev-Kapustka-Kapustka-Ranestad). This has interesting consequences for the Chow ring of the quotient $X/\iota$, which is an "EPW cube" (in the sense of Iliev-Kapustka-Kapustka-Ranestad).Comment: 32 pages, to appear in Math. Nachrichten, feedback welcom

show abstract

“…(Indeed, the varieties

Y_{j}

K_{j}

and

X_{j}

have an MCK decomposition, thanks to Corollary , resp. , resp. ).…”

Section: Some Corollariesmentioning

confidence: 99%

“…There is an inclusion

p^{*} E^{*} false(X false) \subset A_{(0)}^{*} false(Y false) .

(Indeed, we have seen in Corollary that

{(p_{j})}^{*} A^{2} false(D_{j} false) \subset A_{(0)}^{2} false(Y_{j} false)

. Furthermore, it is known that

c_{r} false(K_{j} false) \in A_{(0)}^{r} false(K_{j} false), c_{r} false(X_{j} false) \in A_{(0)}^{r} false(X_{j} false)

[, Proposition 7.13], resp. [, Theorem 2].…”

Section: Some Corollariesmentioning

confidence: 99%

See 1 more Smart Citation

Algebraic cycles and EPW cubes

Laterveer

2018

Mathematische Nachrichten

View full text Add to dashboard Cite

show abstract

“…We note that, in general, there will not exist a crepant resolution of X/G (a local obstruction to its existence is discussed in [4,5], among other places). For select verifications of Ruan's conjecture(s) one can see [37,55,22,23], for example.…”

Section: 2mentioning

confidence: 99%

The Hochschild cohomology ring of a global quotient orbifold

Negron

Schedler

2020

Advances in Mathematics

View full text Add to dashboard Cite

We study the cup product on the Hochschild cohomology of the stack quotient [X/G] of a smooth quasi-projective variety X by a finite group G. More specifically, we construct a G-equivariant sheaf of graded algebras on X whose G-invariant global sections recover the associated graded algebra of the Hochschild cohomology of [X/G], under a natural filtration. This sheaf is an algebra over the polyvector fields T poly X on X, and is generated as a T poly X -algebra by the sum of the determinants det(N X g ) of the normal bundles of the fixed loci in X. We employ our understanding of Hochschild cohomology to conclude that the analog of Kontsevich's formality theorem, for the cup product, does not hold for Deligne-Mumford stacks in general. We discuss relationships with orbifold cohomology, extending Ruan's cohomological conjectures. This employs a trivialization of the determinants in the case of a symplectic group action on a symplectic variety X, which requires (for the cup product) a nontrivial normalization missing in previous literature. G .(1.3)In the unpublished manuscript [2], a version of the theorem above appears in the affine setting (in terms of the obvious explicit formula for cup product, rather than (i)-(iv); see Section 7.5). We note that the vector space identification (1.3) is due to Ginzburg-Kaledin in the affine case, and Arinkin-Cȃldȃraru-Hablicsek in general [3].The subalgebra of (iii) is denoted SA(X ) := ⊕ g∈G det(N X g ) throughout. We see from (ii), or more precisely Theorem 7.2 below, that this subalgebra provides all of the novel features of the Hochschild cohomology of the orbifold X , as compared with that of a smooth variety. Furthermore, as the points of the fixed spaces X g account for the points of X which admit automorphisms, this algebra also represents a direct contribution of the stacky points of X to the Hochschild cohomology.An explicit description of the multiplicative structure on the subalgebra SA(X ) is given in Theorem 7.5, and the multiplicative structure on the entire cohomology HH • (X ) is subsequently obtained from Theorem 7.2. The algebra identification (1.3) appears in Corollary 7.8.

show abstract

“…The property of having an MCK decomposition is severely restrictive, and is closely related to Beauville's "(weak) splitting property" [6]. For more ample discussion, and examples of varieties admitting an MCK decomposition, we refer to [45,Chapter 8], as well as [52], [46], [19], [20, Sections 5 and 6], [37]. Proof.…”

Section: In That Casementioning

confidence: 99%

Algebraic cycles and very special cubic fourfolds

Laterveer

2019

Indagationes Mathematicae

View full text Add to dashboard Cite

This note is about the Chow ring of Verra fourfolds. For a general Verra fourfold, we show that the Chow group of homologically trivial 1-cycles is generated by conics. We also show that Verra fourfolds admit a multiplicative Chow-Künneth decomposition, and draw some consequences for the intersection product in the Chow ring of Verra fourfolds. 1 2 ROBERT LATERVEERCorollary (=Corollary 5.3). Let X be any Verra fourfold. The subalgebrainjects into cohomology via the cycle class map. (Here, c j (T X ) are the Chow-theoretic Chern classes.) (This can be generalized to self-products X m , cf. Corollary 5.6.) The construction of a multiplicative Chow-Künneth decomposition relies once more on the spread argument, by first showing (Theorem 4.8) that any Verra fourfold can be motivically related to a genus 2 K3 surface. This close link to K3 surfaces also has the following consequence:Corollary (=Corollary 5.9). Let X, X ′ be two Verra fourfolds that are isogenous (i.e., there exists an isomorphism of Q-vector spaces H 4 (X, Q) ∼ = H 4 (X ′ , Q) compatible with the Hodge structures and with cup product). Then there is an isomorphism of Chow motivesFurther consequences concern a multiplicative decomposition in the derived category (Corollary 5.12), the generalized Hodge conjecture for self-products, and finite-dimensionality of the motive of certain special Verra fourfolds (Corollary 5.14).Conventions. In this note, the word variety will refer to a reduced irreducible scheme of finite type over C. For any n-dimensional variety X, we will write A i (X) for the Chow group of dimension i cycles on X with Q-coefficients, and A j (X) for the operational Chow cohomology of [17] with Q-coefficients (this can be identified with A n−j (X) for smooth X).The notations A j hom (X) and A j AJ (X) will indicate the subgroups of homologically trivial (resp. Abel-Jacobi trivial) cycles.For a morphism between smooth varieties f : X → Y , we will write Γ f ∈ A * (X × Y ) for the graph of f , and t Γ f ∈ A * (Y × X) for the transpose correspondence.The contravariant category of Chow motives (i.e., pure motives with respect to rational equivalence as in [43], [39]) will be denoted M rat .We will write H * (X) = H * (X, Q) for singular cohomology with Q-coefficients. VERRA FOURFOLDSDefinition 2.1 ([50]). A Verra fourfold is a double cover of P 2 × P 2 branched along a smooth divisor of bidegree (2, 2). Remark 2.2. The moduli space of Verra fourfolds is 19-dimensional [25, Section 0.3]. Lemma 2.3. A Verra fourfold X is a Fano variety (i.e., the canonical bundle is anti-ample). In particular, A 4 (X) = Q and the Hodge conjecture is true for X. Proof. The canonical bundle formula shows that X is Fano. Fano varieties are rationally connected [13], [30], and so A 4 (X) = Q. The Hodge conjecture is known for fourfolds with trivial Chow group of 0-cycles [10].

show abstract

Motivic hyper-Kähler resolution conjecture, I : Generalized Kummer varieties

Cited by 53 publications

References 59 publications

Algebraic cycles and EPW cubes

Algebraic cycles and EPW cubes

The Hochschild cohomology ring of a global quotient orbifold

Algebraic cycles and very special cubic fourfolds

Contact Info

Product

Resources

About