In this paper, we extend the construction of the canonical polarized variation of Hodge structures over tube domain considered by Gross (Math Res Lett 1:1-9, 1994) to bounded symmetric domain and introduce a series of invariants of infinitesimal variation of Hodge structures, which we call characteristic subvarieties. We prove that the characteristic subvariety of the canonical polarized variations of Hodge structures over irreducible bounded symmetric domains are identified with the characteristic bundles defined by Mok (Ann Math 125(1): 1987). We verified the generating property of Gross for all irreducible bounded symmetric domains, which was predicted in Gross (Math Res Lett 1:1-9, 1994).

Let k be the algebraic closure of a finite field of odd characteristic p and X a smooth projective scheme over the Witt ring W (k) which is geometrically connected in characteristic zero. We introduce the notion of Higgs-de Rham flow 1 and prove that the category of periodic Higgs-de Rham flows over X/W (k) is equivalent to the category of Fontaine modules, hence further equivalent to the category of crystalline representations of theétale fundamental group π 1 (X K ) of the generic fiber of X, after Fontaine-Laffaille and Faltings. Moreover, we prove that every semistable Higgs bundle over the special fiber X k of X of rank ≤ p initiates a semistable Higgs-de Rham flow and thus those of rank ≤ p − 1 with trivial Chern classes induce k-representations of π 1 (X K ). A fundamental construction in this paper is the inverse Cartier transform over a truncated Witt ring. In characteristic p, it was constructed by Ogus-Vologodsky in the nonabelian Hodge theory in positive characteristic; in the affine local case, our construction is related to the local Ogus-Vologodsky correspondence of Shiho. 45 7. Rigidity theorem for Fontaine modules 51 Appendix A. Semistable Higgs bundles of small ranks are strongly Higgs semistable 55 References 59

It is a fundamental problem in geometry to decide which moduli spaces of polarized algebraic varieties are embedded by their period maps as Zariski open subsets of locally Hermitian symmetric domains. In the present work we prove that the moduli space of Calabi-Yau threefolds coming from eight planes in P 3 does not have this property. We show furthermore that the monodromy group of a good family is Zariski dense in the corresponding symplectic group. Moreover, we study a natural sublocus which we call hyperelliptic locus, over which the variation of Hodge structures is naturally isomorphic to wedge product of a variation of Hodge structures of weight one. It turns out the hyperelliptic locus does not extend to a Shimura subvariety of type III (Siegel space) within the moduli space. Besides general Hodge theory, representation theory and computational commutative algebra, one of the proofs depends on a new result on the tensor product decomposition of complex polarized variations of Hodge structures.

Let k be a perfect field of odd characteristic and X a smooth algebraic variety over k which is W 2 -liftable. We show that the exponent twisiting of the classical Cartier descent gives an equivalence of categories between the category of nilpotent Higgs sheaves of exponent ≤ p − 1 over X/k and the category of nilpotent flat sheaves of exponent ≤ p − 1 over X/k, and it is equivalent up to sign to the inverse Cartier and Cartier transforms for these nilpotent objects constructed in the nonabelian Hodge theory in positive characteristic by Ogus-Vologodsky [9]. In view of the crucial role that Deligne-Illusie's lemma has ever played in their algebraic proof of E 1 degeneration and Kodaira vanishing theorem in abelian Hodge theory, it may not be overly surprising that again this lemma plays a significant role via the concept of Higgs-de Rham flow [5] in establishing p-adic Simpson correspondence in nonabelian Hodge theory and Langer's algebraic proof of Bogomolov inequality for semistable Higgs bundles and Miyaoka-Yau inequality [6].

For each natural odd number n ≥ 3, we exhibit a maximal family of n-dimensional Calabi-Yau manifolds whose Yukawa coupling length is 1. As a consequence, Shafarevich's conjecture holds true for these families. Moreover, it follows from Deligne and Mostow (Publ. Math. IHÉS, 63:5-89, 1986) and Mostow (Publ. Math. IHÉS, 63:91-106, 1986; J. Am. Math. Soc., 1(3): 1988) that, for n = 3, it can be partially compactified to a Shimura family of ball type, and for n = 5, 9, there is a sub Q-PVHS of the family uniformizing a Zariski open subset of an arithmetic ball quotient.