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In this paper we study harmonic maps from super Riemann surfaces in complex projective spaces and projective spaces associated with the super skew-field D . In both cases, we develop the theory of Gauß transforms and study the notion of isotropy, in particular its relation to holomorphic differentials on the super Riemann surface. Moreover, we give a definition of finite type harmonic maps for a special class of maps into C P n | n + 1 and thus obtain a classification for certain harmonic super tori. Furthermore, we investigate the equations satisfied by the underlying objects and give an example of a harmonic super torus in D P 2 whose underlying map is not harmonic. [ABSTRACT FROM AUTHOR]

Abstract: A holomorphy potential is a complex valued function whose complex gradient, with respect to some Kähler metric, is a holomorphic vector field. Given holomorphic vector fields on a compact complex manifold, form, for a given Kähler metric, a product of the following type: a function of the scalar curvature multiplied by functions of the holomorphy potentials of each of the vector fields. It is shown that the stipulation that such a product be itself a holomorphy potential for yet another vector field singles out critical metrics for a particular functional. This may be regarded as a generalization of the extremal metric variation of Calabi, where and the functional is the square of the -norm of the scalar curvature. The existence question for such metrics is examined in a number of special cases. Examples are constructed in the case of certain multifactored product manifolds. For the SKR metrics investigated by Derdzinski and Maschler and residing in the complex projective space, it is shown that only one type of nontrivial criticality holds in dimension three and above. [Copyright &y& Elsevier]

Abstract: We define holomorphic structures on canonical line bundles of the quantum projective space and identify their space of holomorphic sections. This determines the quantum homogeneous coordinate ring of the quantum projective space. We show that the fundamental class of is naturally presented by a twisted positive Hochschild cocycle. Finally, we verify the main statements of the Riemann–Roch formula and the Serre duality for and . [Copyright &y& Elsevier]

We consider complex manifolds with a class of holomorphic coordinate functions satisfying the condition that each transition function is given by the standard action on CP2n−1 of some element in Sp(2n,C)/Z2. We show that such a manifold has a natural contact structure. Given any contact manifold, one can associate with it a symplectic manifold. It is shown that the symplectic manifolds arising from complex manifolds with special coordinate functions of the above type admit a canonical quantization. [Copyright &y& Elsevier]