Abelian varieties of Weil type and Kuga-Satake varieties

霍奇相似性、代数类与 Kuga-Satake 簇

Hodge similarities, algebraic classes, and Kuga-Satake varieties

本文引入超凯勒流形超越格的霍奇相似性概念，证明特定 K3 曲面族平方的霍奇猜想，并利用 Kuga-Satake 构造的函子性推出广义 Kummer 形变型超凯勒流形的霍奇相似性均为代数的。

Generalized Kuga-Satake theory and rigid local systems, I: the middle convolution

This note explains an approach to producing examples of 'generalized Kuga-Satake theory' based on establishing special cases of Simpson's conjecture that rigid local systems are motivic. This strategy is then carried out, using work of Bogner and Reiter, in a low-dimensional example, producing the first non-trivial examples of generalized Kuga-Satake theory outside of the Tannakian category of motives generated by abelian varieties. For a more robust collection of examples, see the sequel "Generalized Kuga-Satake theory and rigid local systems, II: rigid Hecke eigensheaves."

Picard秩14的K3曲面族上的Kuga-Satake构造

Kuga-Satake construction on families of K3 surfaces of Picard rank 14

利用IV$_6$型与II$_4$型Hermite对称域间的等距，通过Kuga-Satake构造建立了Picard秩14的K3曲面模空间与具有全确定四元数乘法的极化阿贝尔8-fold模空间之间的几何映射，并展示了该映射在具体族上的实现。

Kuga-Satake Abelian Varieties in Positive Characteristic

Kuga and Satake associate with every polarized complex K3 surface (X,L) a complex abelian variety called the Kuga-Satake abelian variety of (X,L). We use this construction to define morphisms between moduli spaces of polarized K3 surface with certain level structures and moduli spaces of polarized abelian varieties with level structure over C. In this note we study these morphisms. We prove first that they are defined over finite extensions of Q. Then we show that they extend in positive characteristic. In this way we give an indirect construction of Kuga-Satake abelian varieties over an arbit…

Generalized Kuga-Satake theory and rigid local systems, II: rigid Hecke eigensheaves

This paper uses rigid Hecke eigensheaves, building on Yun's work on the construction of motives with exceptional Galois groups, to produce the first robust examples of `generalized Kuga-Satake theory' outside the Tannakian category of motives generated by abelian varieties. To strengthen our description of the `motivic' nature of Kuga-Satake lifts, we digress to establish a result that should be of independent interest: for any quasi-projective variety over a (finitely-generated) characteristic zero field, the associated weight-graded of its intersection cohomology arises from a motivated moti…

Kuga-Satake construction and cohomology of hyperkahler manifolds

Let M be a simple hyperkahler manifold. Kuga-Satake construction gives an embedding of H^2(M,C) into the second cohomology of a torus, compatible with the Hodge structure. We construct a torus T and an embedding of the graded cohomology space H^*(M,C) \to H^{*+l}(T,C) for some l, which is compatible with the Hodge structures and the Poincare pairing. Moreover, this embedding is compatible with an action of the Lie algebra generated by all Lefschetz sl(2)-triples on M.

Effective computation of Picard groups and Brauer-Manin obstruction of degree two K3 surfaces over number fields

Using the Kuga-Satake correspondence we provide an effective algorithm for the computation of the Picard and Brauer groups of K3 surfaces of degree 2 over number fields.

The Hodge conjecture for self-products of certain K3 surfaces

We use a result of van Geemen to determine the endomorphism algebra of the Kuga--Satake variety of a K3 surface with real multiplication. This is applied to prove the Hodge conjecture for self-products of double covers of $\PP^2$ which are ramified along six lines.

广义Kummer型的六重折叠与K3曲面

Sixfolds of generalized Kummer type and K3 surfaces

本文证明广义Kummer型超Kähler六重折叠自然关联K3^[3]型流形，并在射影情形下双有理等价于特定K3曲面上的稳定层模空间，进而证明该曲面的Kuga-Satake对应是代数的，构造出无限多满足Kuga-Satake Hodge猜想的K3曲面族。

Abelian Fourfolds of Weil type and certain K3 Double Planes

Double planes branched in 6 lines give a famous example of K3 surfaces. Their moduli are well understood and related to abelian fourfolds of Weil type. We compare these two moduli interpretations and in particular divisors on the moduli spaces. On the K3 side, this is achieved with the help of elliptic fibrations. We also study the Kuga-Satake correspondence on these special divisors.

Fourfolds of Weil type and the spinor map

Recent papers by Markman and O'Grady give, besides their main results on the Hodge conjecture and on hyperkaehler varieties, surprising and explicit descriptions of families of abelian fourfolds of Weil type with trivial discriminant. They also provide a new perspective on the well-known fact that these abelian varieties are Kuga Satake varieties for certain weight two Hodge structures of rank six. In this paper we give a pedestrian introduction to these results. The spinor map, which is defined using a half-spin representation of SO(8), is used intensively. For simplicity, we use basic repres…

具有有限维 motive 的9维K3曲面族

A 9-dimensional family of K3 surfaces with finite dimensional motive

该研究利用与四次三维立方体的关联，证明了一类二次曲面三重覆盖K3曲面的Kimura有限维motive，并确立其Kuga-Satake Hodge猜想与Voisin零闭链猜想，进而得到带3阶非辛自同构K3曲面模空间2个9维不可约分支的对应结果。

Picard数为16的K3曲面幂的Hodge猜想

The Hodge conjecture for powers of K3 surfaces of Picard number 16

本文针对一般Picard数为16的K3曲面族，证明在Kuga–Satake对应为代数的条件下，族内任意K3曲面的所有幂都满足Hodge猜想。

Half twists of Hodge structures of CM-type

To a Hodge structure V of weight k with CM by a field K we associate Hodge structures V_{-n/2} of weight k+n for n positive and, under certain circumstances, also for n negative. We show that these `half twists' come up naturally in the Kuga-Satake varieties of weight two Hodge structures with CM by an imaginary quadratic field.

Hodge structure of K3 type with real multiplication and Simple Abelian Fourfolds with Definite Quaternionic Multiplication

In this paper we give a general construction of transcendental lattices for K3 surfaces with real multiplication by arbitrary field up to degree 6 along with formula for their discriminants. We also show that all simple Abelian fourfolds with definite quaternionic multiplication can be realized as Kuga-Satake varieties of K3 surfaces with Picard rank 16 and real multiplication by a quadratic field by keeping track of the arithmetic input on both sides.

Compact tori associated to hyperkaehler manifolds of Kummer type

Let X be a hyperkaehler manifold of Kummer type. We study the 4 dimensional intermediate Jacobian J^3(X) constructed out of the 3rd cohomology of X, motivated by the desire to understand the Kuga-Satake torus associated to X. We prove that there is an embedding of Hodge structures of H^2(X) into H^2(J^3(X)). It follows that if X is projective with polarization L, the projective Kuga-Satake torus of (X,L) is isogenous to the fourth power of J^3(X). By studying the cohomology ring of X, we find that there is a natural smooth quadric Q(X) in the projectivization of H^3(X), with a natural choice o…

Laplace eigenfunctions on Riemannian symmetric spaces and Borel-Weil Theorem

We indicate a geometric relation between Laplace-Beltrami spectra and eigenfunctions on compact Riemannian symmetric spaces and the Borel-Weil theory using ideas from symplectic geometry and geometric quantization. This is done by associating to each compact Riemannian symmetric space, via Marsden-Weinstein reduction, a generalized flag manifold which covers the space parametrizing all of its maximal totally geodesic tori. In the process we notice a direct relation between the Satake diagram of the symmetric space and the painted Dynkin diagram of its associated flag manifold. We consider in d…