Kodaira–Saito vanishing via Higgs bundles in positive characteristic

The goal of this paper is to give a new proof of a special case of the Kodaira–Saito vanishing theorem for a variation of Hodge structure on the complement of a divisor with normal crossings. The proof does not use the theory of mixed Hodge modules, but instead reduces it to a more general vanishing theorem for semistable nilpotent Higgs bundles, which is then proved by using some facts about Higgs bundles in positive characteristic.

Trigonal Deformations of Rank One and Jacobians

We study the infinitesimal deformations of a trigonal curve that preserve the trigonal series and such that the associate infinitesimal variation of Hodge structure is of rank $1.$ We show that if $g\geq 8$ or $g=6,7$ and the curve is Maroni general, this locus is zero dimensional. Moreover, we complete the result [10, Theorem 1.6]. We show in fact that if $g\geq 6$, the hyperelliptic locus ${{\mathcal{M}}}^1_{g,2}$ is the only $2g-1$-dimensional sub-locus ${{\mathcal{Y}}}$ of the moduli space ${{\mathcal{M}}}_g$ of curves of genus $g$, such that for the general element $[C]\in{{\mathcal{Y}}}$, its Jacobian $J(C)$ is dominated by a hyperelliptic Jacobian of genus $g^{\prime}\geq g$.

Functoriality of motivic lifts of the canonical construction

Let $$(G,{\mathfrak {X}})$$ ( G , X ) be a Shimura datum and K a neat open compact subgroup of $$G(\mathbb {A}_f)$$ G ( A f ) . Under mild hypothesis on $$(G,{\mathfrak {X}})$$ ( G , X ) , the canonical construction associates a variation of Hodge structure on $$\text {Sh}_K(G,{\mathfrak {X}})(\mathbb {C})$$ Sh K ( G , X ) ( C ) to a representation of G. It is conjectured that this should be of motivic origin. Specifically, there should be a lift of the canonical construction which takes values in relative Chow motives over $$\text {Sh}_K(G,{\mathfrak {X}})$$ Sh K ( G , X ) and is functorial in $$(G,{\mathfrak {X}})$$ ( G , X ) . Using the formalism of mixed Shimura varieties, we show that such a motivic lift exists on the full subcategory of representations of Hodge type $$\{(-1,0),(0,-1)\}$$ { ( - 1 , 0 ) , ( 0 , - 1 ) } . If $$(G,{\mathfrak {X}})$$ ( G , X ) is equipped with a choice of PEL-datum, Ancona has defined a motivic lift for all representations of G. We show that this is independent of the choice of PEL-datum and give criteria for it to be compatible with base change. Additionally, we provide a classification of Shimura data of PEL-type and demonstrate that the canonical construction is applicable in this context.

Hyperbolicity of Varieties Supporting a Variation of Hodge Structure

We generalize former results of Zuo and the 1st author showing some hyperbolicity properties of varieties supporting a variation of Hodge structure. Our proof only uses the special curvature properties of period domains. In particular, in contrast to the former approaches, it does not use any result on the asymptotic behavior of the Hodge metric.

The Spread Philosophy in the Study of Algebraic Cycles

This chapter discusses the spread philosophy in the study of algebraic cycles, in order to make use of a geometry by considering a variation of Hodge structure where D is the Hodge domain (or the appropriate Mumford–Tate domain) and Γ‎ is the group of automorphisms of the integral lattice preserving the intersection pairing. If we have an algebraic cycle Z on X, taking spreads yields a cycle Ƶ on X. Applying Hodge theory to Ƶ on X gives invariants of the cycle. Another related situation is algebraic K-theory. For example, to study Kₚsuperscript Milnor(k), the geometry of S can be used to construct invariants.