Infinitesimal Deformation Of Complex Analytic Structures And Cohomology Theory Of Sheaves

The theory of sheaves is an appropriate language whenever one studies the interaction of local and global properties of a topological space especially manifold.For the most relevant aspects like infinitesmal deformation and sheaf cohomology,It woud be very natural to use the powerful sheaf machinery.This paper mainly discusses the infinitesimal deformation of complex analytic structures from three aspects.How to integrate these seemly different deformation theories is considered.In particular,we introduce two new terminologies called the Schouten-Nijenhuis Lie bracket and the Beltrami-Maurer-Cartan equation which makes the Beltrami differential equation be compatible with the classical Maurer-Cartan equation,so that we further conveniently study the solvability of some differential equations induced by deformation,including the differential equation related to Siu's conjecture on pluricanonical invariant under additional conditions.The details are as follows.1.Following the idea of Kodaira Spencer,we discuss the infinitesimal deformation of fibers of complex analytic families on complex manifolds.The existence theorem of infinitesimal deformation of complex analytic families is discussed as H2(M,?)=0.Borrowed from the original proof and the technique for proving the convergence of global canonical family by Liu-Rao-Yang,we rework a more detailed version for proving the existence theorem of deformation.2.Focusing on the power series expansion of the first-order deformation of complex structures,we discuss its recursive sestem of coefficient equations.If we choose a class of manifolds with trivial canonical bundles,such as Calabi-Yau manifolds,we show that there exists a formal solution by the generalized Tian-Todorov lemma and(?)-lemma based on the introduced Schouten nijenhuis Lie bracket by us.3.Discuss the solvable problem of differential equation corresponding to Siu's conjecture on the pluricanonical invariant under additional conditions.Based on the induced Beltrami Maurer Cartan equation and the method from Liu-Zhu,we study the holomorphic solvability of the differential equation (?)??-V'(i??)+m(div?)?? with the deformation operator(?)under the restricted conditions from the extended(p,q)-form instead of the special(n,0)-form.Some results and their proofs are given.