Infinitesimal Variation of Harmonic Forms and Lefschetz Decomposition

Infinitesimal part of Weak Lefschetz using Milnor $K-$ theory

Let $X$ be a smooth projective variety over a field of characteristic $0$ and $L$ an ample divisor. In this paper we study the Weak Lefschetz conjecture for Chow groups using the technique employed by Grothendieck in his study of the problem for Picard groups, and using Bloch's formula to interpret Chow groups in terms of Milnor $K$- theory we prove the infinitesimal part of the conjecture.

Jacobian 代数与超平面截面的变分

Jacobian algebras and variation of hyperplane sections

通过 Milnor 代数研究至多孤立奇点超曲面的超平面截面模空间的变分，给出超平面截面映射为有限态的无穷小商判据，并推广了光滑情形的 Beauville-Patel-Riedl-Tseng 理论。

Infinitesimal variation functions for families of smooth varieties

In this paper we introduce some {\it variation functions} associated to the rank of the Infinitesimal Variations of Hodge Structure for a family of smooth projective complex curves. We give some bounds and inequalities and, in particular, we prove that if $X$ is a smooth plane curve $X$, then there exists a first order deformation $ξ\in H^1(T_X)$ which deforms $X$ as plane curve, such that $ξ\cdot:H^0(ω_X)\to H^1(\mathcal{O}_{X})$ is an isomorphism. We also generalize the notions of variation functions to the higher dimensional case and we analyze the link between IVHS and the Weak and Strong …

Hyperkaehler manifolds with torsion, supersymmetry and Hodge theory

Let M be a hypercomplex Hermitian manifold, (M,I) the same manifold considered as a complex Hermitian with a complex structure I induced by the quaternions. The standard linear-algebraic construction produces a canonical nowhere degenerate (2,0)-form on (M,I). It is well known that M is hyperkaehler if and only if the form Ωis closed with respect to the de Rham differential. The M is called HKT (hyperkaehler with torsion) if this form is closed with respect to the Dolbeault differential (this condition is weaker. Conjecturally, all compact hypercomplex manifolds admit an HKT-metrics. We exploi…