Environmental fragility based on the spatial distribution of soil resistance for the en-vironmental protection area (EPA) of Botucatu, Sao Paulo, Brazil

Effective soil management requires an understanding of the physical, chemical and spatial distribution features of soil. Based on the spatial distribution of soil resistance to mechanical penetration, this study sought to construct an environmental fragility index of this resistance and apply it to an environmental fragility map of the Environmental Protection Area (EPA) of Botucatu, Sao Paulo, Brazil. Methodologies consisting of an empirical analysis of the environment, geostatistics, a multi-criteria decision analysis and algebraic maps were used. Measurements of soil resistance to mechanical penetration, sloping, soil type and land use were integrated into an environmental fragility map. The results showed that 32.5% of the sample area fell into the low fragility categories and 67.57%, into the middling and very high fragility categories. Our conclusion was that soil resistance to mechanical penetration, which is a natural feature found in various types of soils, can therefore be included as one of the criteria in a fragility analysis. We found evidence suggesting that soil resistance to mechanical penetration has a direct relationship with sloping and land use, namely, in cases where different types of use and their management exert change on the soil’s natural resistance to the extent of rendering it fragile.

The Hodge Theory of Maps

This chapter summarizes the classical results of Hodge theory concerning algebraic maps. Hodge theory gives nontrivial restrictions on the topology of a nonsingular projective variety, or, more generally, of a compact Kähler manifold: the odd Betti numbers are even, the hard Lefschetz theorem, the formality theorem, stating that the real homotopy type of such a variety is, if simply connected, determined by the cohomology ring. Similarly, Hodge theory gives nontrivial topological constraints on algebraic maps. This chapter focuses on the latter, as it considers how the existence of an algebraic map f : X → Y of complex algebraic varieties is reflected in the topological invariants of X.