Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2012.05969

K3 surfaces and cubic fourfolds with Abelian motive

Hanine Awada·Michele Bolognesi·Robert Laterveer·Claudio Pedrini
View original source ↗Read source document ↗Export metadata ↓

Abstract

We show that cubic fourfolds with lattice of algebraic 2-cycles of rank greater than 19 have abelian and finite dimensional (in the sense of Kimura) Chow motive. This also implies Abelianity and finite dimensionality of the motive of related hyperKahler varieties, such as the Fano variety of lines and the LLSvS 8fold. A similar remark allows us to show the Abelianity of the motive of an infinity of LSV 10folds, and of other hyperKahler 10folds associated to the twisted intermediate Jacobian fibration of cubic fourfolds with an associated K3 surface. After that, starting from certain 4-dimensional families of K3 surfaces, we construct two families of Fano varieties whose Chow motive is finite dimensional. Varieties from the first family are some quadric surface fibrations, and contain the finite dimensional transcendental motive of a K3 surface. Varieties from the second family are singular cubic fourfolds, and their motives are Schur-finite and Abelian in Voevodsky's triangulated category of motives.

Explore related subjects

math.AG

Keep this discovery

BibTeXRIS

Hanine Awada, Michele Bolognesi, Robert Laterveer, Claudio Pedrini. 2020-12-10. K3 surfaces and cubic fourfolds with Abelian motive. https://arxiv.org/abs/2012.05969

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Perverse Euler Characteristics of Hermitian Locally Symmetric Spaces

We prove that finite-volume locally Hermitian symmetric spaces of noncompact type have nonnegative perverse Euler characteristics. To show this, we obtain a nefness result for the logarithmic cotangent bundle of a smooth toroidal compactification. Combining this with a positivity criterion for Euler characteristics of perverse sheaves, we deduce the nonnegativity result. We further prove that the inequality is strict for perverse sheaves with full support. As applications, we get nonnegativity results for perverse Euler characteristics on various moduli spaces.

math.AG↗
arXiv2026

Coupled Pklt Tuples and Varieties of Pklt Type

We introduce asymptotic multiplier ideal sheaves and log canonical thresholds associated with tuples of pseudoeffective divisors on a projective klt pair. We prove that the threshold of a coupled potentially klt tuple is computed by a quasi-monomial valuation. For varieties of potentially klt type, we prove that every big divisor admits a birational Zariski decomposition with semiample positive part. We also prove finite generation of multisection rings of big divisors and give a criterion for a variety of potentially klt type to be a Mori dream space.

math.AG↗
arXiv2026

Graded Betti numbers of general curves of large degree

Let $C$ be a smooth projective complex curve of genus $g$ and gonality $k$, and $L$ be a very ample line bundle on $C$. When $L$ has sufficiently large degree, the vanishing and nonvanishing of the Koszul cohomology groups $K_{p,q}(C,L)$ have been determined previously, but the exact values of the graded Betti numbers $\kappa_{p,q}(C, L)$ remain largely unknown. In this paper, we give explicit closed formulas for all graded Betti numbers $\kappa_{p,q}(C, L)$ when the Brill--Noether locus $W_k^1(C)$ has the expected dimension and $H^1(C, L \otimes \omega_C^{-1})=0$. Consequently, we determine the complete Betti table for a general curve when $\deg L \geq 4g-3$ or when $\deg L \geq 3g-3$ and $L$ is general. We also explicitly compute the Boij--S\"{o}derberg coefficient of the section ring $R(C, L)$ governing asymptotic purity, and show eventual monotonicity of the remaining coefficients: they decrease for hyperelliptic curves and increase under a natural generic reducedness assumption on the relevant Brill--Noether loci.

math.AG↗