On the decomposition theorem for intersection de Rham complexes

We establish a positive characteristic analogue of intersection cohomology for polarized variations of Hodge structure. This includes: a) the decomposition theorem for the intersection de Rham complex; b) the $E_1$-degeneration theorem for the intersection de Rham complex of a periodic de Rham bundle: c) the Kodaira vanishing theorem for the intersection cohomology groups of a periodic Higgs bundle.

Polarized Variation of Hodge Structures of Calabi-Yau Type and Characteristic Subvarieties Over Bounded Symmetric Domains

In this paper we extend the construction of the canonical polarized variation of Hodge structures over tube domain considered by B. Gross in cite{G} 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 N. Mok in cite{M}. We verified the generating property of B. Gross for all irreducible bounded symmetric domains, which was predicted in cite{G}.

An Arakelov Inequality in Characteristic p and Upper Bound of p-Rank Zero Locus

In this paper we show an Arakelov inequality for semi-stable families of algebraic curves of genus $ggeq 1$ over characteristic $p$ with nontrivial Kodaira-Spencer maps. We apply this inequality to obtain an upper bound of the number of algebraic curves of $p-$rank zero in a semi-stable family over characteristic $p$ with nontrivial Kodaira-Spencer map in terms of the genus of a general closed fiber, the genus of the base curve and the number of singular fibres. An extension of the above results to smooth families of Abelian varieties over $k$ with $W_2$-lifting assumption is also included.

On the monodromy of the moduli space of Calabi-Yau threefolds coming from eight planes in $mathbb{P}^3$

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 $mathbb{P}^3$ does {em 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.

Computational details on the disproof of modularity

The purpose of these notes is to provide the details of the Jacobian ring computations carried out in [1], based on the computer algebra system Magma [2].

Non-damped Acoustic Plasmon and Superconductivity in Single Wall Carbon Nanotubes

We show that non-damped acoustic plasmons exist in single wall carbon nanotubes (SWCNT) and propose that the non-damped acoustic plasmons may mediate electron-electron attraction and result in superconductivity in the SWCNT. The superconducting transition temperature Tc for the SWCNT (3,3) obtained by this mechanism agrees with the recent experimental result (Z. K. Tang et al, Science 292, 2462(2001)). We also show that it is possible to get higher Tc up to 99 K by doping the SWCNT (5,5).