Refined analytic torsion for twisted de Rham complexes

Let $E$ be a flat complex vector bundle over a closed oriented odd dimensional manifold $M$ endowed with a flat connection $\nabla$. The refined analytic torsion for $(M,E)$ was defined and studied by Braverman and Kappeler. Recently Mathai and Wu defined and studied the analytic torsion for the twisted de Rham complex with an odd degree closed differential form $H$, other than one form, as a flux and with coefficients in $E$. In this paper we generalize the construction of the refined analytic torsion to the twisted de Rham complex. We show that the refined analytic torsion of the twisted de Rham complex is independent of the choice of the Riemannian metric on $M$ and the Hermitian metric on $E$. We also show that the twisted refined analytic torsion is invariant (under a natural identification) if $H$ is deformed within its cohomology class. We prove a duality theorem, establishing a relationship between the twisted refined analytic torsion corresponding to a flat connection and its dual. We also define the twisted analogue of the Ray-Singer metric and calculate the twisted Ray-Singer metric of the twisted refined analytic torsion. In particular we show that in case that the Hermtitian connection is flat, the twisted refined analytic torsion is an element with the twisted Ray-Singer norm one.Comment: 30 Page

AN ANALYSIS OF CURRENT PROBLEMS IN CHINA'S AGRICULTURE DEVELOPMENT: AGRICULTURE, RURAL AREAS AND FARMERS

China is the most populous country in the world. Of its 1.3 billion people, 22% of the world population, about 67% are living in rural areas. Although China is the third largest country in terms of area, the arable land is only 7% of the global amount. With relatively meager endowment, it is undoubtedly a daunting task for the agricultural sector to provide adequate supply to fulfil huge needs for food and other agricultural products. In addition, agriculture development in China confronts with challenges to raise the average income and standard of living of the rural population in the long run. Since China's economic reform was launched in 1978, the "People's Commune" system was dismantled and replaced by the "Household Responsibility" system. Agricultural production has achieved rapid growth and income per capita in the rural area has risen 10 times in 20 years. During this transformation process, a number of serious problems have been emerging in the agricultural sector. They include the diminishing size of the arable land, enlarging of income disparity and stagnating of productivity growth, which have been exacerbated by the population growth and increasing demands for agricultural products. The agricultural sector is also plagued by environmental degradation and confronted by township enterprise development. Furthermore, China's recent accession into the World Trade Organization (WTO) brings more tremendous challenges to its agriculture. This paper is intended to provide a concise analysis of the problems and possible policy options associated with current agriculture development. It reveals that the main problems are market partition, inefficiency in government administration in supply and distribution, and price distortions of agricultural products, originating from China's development strategy of preferred industrialization in the industry sector and urban development. This paper also explores and assesses a few government policy options for the alleviation of these problems. Policy options focus on deepening market-oriented reforms, including price deregulation, market integration and property (land) reforms, which also reflect the requirements of the Agriculture Agreement of WTO. Policy options also focus on improvement of government supported programs in investment and subsidies aimed at boosting productivity, narrowing the inequality of income distribution and easing the barriers for mobility of surplus labor into the industry and service sectors in urban areas.International Development,

Twisted Cappell-Miller holomorphic and analytic torsions

Recently, Cappell and Miller extended the classical construction of the analytic torsion for de Rham complexes to coupling with an arbitrary flat bundle and the holomorphic torsion for $\bar{\partial}$-complexes to coupling with an arbitrary holomorphic bundle with compatible connection of type $(1,1)$. Cappell and Miller studied the properties of these torsions, including the behavior under metric deformations. On the other hand, Mathai and Wu generalized the classical construction of the analytic torsion to the twisted de Rham complexes with an odd degree closed form as a flux and later, more generally, to the $\mathbb{Z}_2$-graded elliptic complexes. Mathai and Wu also studied the properties of analytic torsions for the $\mathbb{Z}_2$-graded elliptic complexes, including the behavior under metric and flux deformations. In this paper we define the Cappell-Miller holomorphic torsion for the twisted Dolbeault-type complexes and the Cappell-Miller analytic torsion for the twisted de Rham complexes. We obtain variation formulas for the twisted Cappell-Miller holomorphic and analytic torsions under metric and flux deformations.Comment: 21 page