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Register a new userOn the motive of OGradys ten-dimensional hyper-Kahler varieties
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We investigate how the motive of hyper-Kahler varieties is controlled by weight-2 (or surface-like) motives via tensor operations. In the first part, we study the Voevodsky motive of singular moduli spaces of semistable sheaves on K3 and abelian surfaces as well as the Chow motive of their crepant resolutions, when they exist. We show that these motives are in the tensor subcategory generated by the motive of the surface, provided that a crepant resolution exists. This extends a recent result of Bulles to the OGrady-10 situation. In the non-commutative setting, similar results are proved for the Chow motive of moduli spaces of stable objects of the K3 category of a cubic fourfold. As a consequence, we provide abundant examples of hyper-Kahler varieties of OGrady-10 deformation type satisfying the standard conjectures. In the second part, we study the Andr{e} motive of projective hyper-Kahler varieties. We attach to any such variety its defect group, an algebraic group which acts on the cohomology and measures the difference between the full motive and its weight-2 part. When the second Betti number is not 3, we show that the defect group is a natural complement of the Mumford--Tate group inside the motivic Galois group, and that it is deformation invariant. We prove the triviality of this group for all known examples of projective hyper-Kahler varieties, so that in each case the full motive is controlled by its weight-2 part. As applications, we show that for any variety motivated by a product of known hyper-Kahler varieties, all Hodge and Tate classes are motivated, the motivated Mumford--Tate conjecture holds, and the Andre motive is abelian. This last point completes a recent work of Soldatenkov and provides a different proof for some of his results.
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We prove the generalized Franchetta conjecture for the locally complete family of hyper-Kahler eightfolds constructed by Lehn-Lehn-Sorger-van Straten (LLSS). As a corollary, we establish the Beauville-Voisin conjecture for very general LLSS eightfolds. The strategy consists in reducing to the Franchetta property for relative fourth powers of cubic fourfolds, by using the recent description of LLSS eightfolds as moduli spaces of semistable objects in the Kuznetsov component of the derived category of cubic fourfolds, together with its generalization to the relative setting due to Bayer-Lahoz-Macr`i-Nuer-Perry-Stellari. As a by-product, we compute the Chow motive of the Fano variety of lines on a smooth cubic hypersurface in terms of the Chow motive of the cubic hypersurface.
This note is an erratum to the paper Tautological classes on moduli spaces of hyper-Kahler manifolds. Thorsten Beckman and Mirko Mauri have pointed to us a gap in the proof of cite[Theorem 8.2.1]{Duke}. We do not know how to correct the proof. We can only recover a partial statement. This gap affects the proof of one of the two main results of cite{Duke}, we explain how to correct it.
We compute the motive of the variety of representations of the torus knot of type (m,n) into the affine groups $AGL_1(C)$ and $AGL_2(C)$. For this, we stratify the varieties and show that the motives lie in the subring generated by the Lefschetz motive q=[C].
Building on an idea of Borcherds, Katzarkov, Pantev, and Shepherd-Barron (who treated the case $e=14$), we prove that the moduli space of polarized K3 surfaces of degree $2e$ contains complete curves for all $egeq 62$ and for some sporadic lower values of $e$ (starting at $14$). We also construct complete curves in the moduli spaces of polarized hyper-Kahler manifolds of $mathrm{K3}^{[n]}$-type or $mathrm{Kum}_n$-type for all $nge 1$ and polarizations of various degrees and divisibilities.
Let $Xsubset mathbb{P}^r$ be an integral and non-degenerate variety. Let $sigma _{a,b}(X)subseteq mathbb{P}^r$, $(a,b)in mathbb{N}^2$, be the join of $a$ copies of $X$ and $b$ copies of the tangential variety of $X$. Using the classical Alexander-Hirschowitz theorem (case $b=0$) and a recent paper by H. Abo and N. Vannieuwenhoven (case $a=0$) we compute $dim sigma _{a,b}(X)$ in many cases when $X$ is the $d$-Veronese embedding of $mathbb{P}^n$. This is related to certain additive decompositions of homogeneous polynomials. We give a general theorem proving that $dim sigma _{0,b}(X)$ is the expected one when $X=Ytimes mathbb{P}^1$ has a suitable Segre-Veronese style embedding in $mathbb{P}^r$. As a corollary we prove that if $d_ige 3$, $1le i le n$, and $(d_1+1)(d_2+1)ge 38$ the tangential variety of $(mathbb{P}^1)^n$ embedded by $|mathcal{O} _{(mathbb{P} ^1)^n}(d_1,dots ,d_n)|$ is not defective and a similar statement for $mathbb{P}^ntimes mathbb{P}^1$. For an arbitrary $X$ and an ample line bundle $L$ on $X$ we prove the existence of an integer $k_0$ such that for all $tge k_0$ the tangential variety of $X$ with respect to $|L^{otimes t}|$ is not defective.
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