Computing Green currents via the heat kernel

Abstract

Let X be a compact Kahler manifold. For an analytic cycle Y ⊂ X of complex codimension k the Poincare duality assigns a class in the de Rham cohomology which can be represented by a harmonic (k, k)-form μY,∞. A Green current gY connects both objects by the equality ddgY = [μY,∞]− δY . Thus we can reconstruct Y and its harmonic projection μY,∞ from this equation, because Y is the locus where ddgY is not smooth, while μY,∞ is determined by its values on the dense open set X Y . Furthermore, the above equality tells us that δY and μY,∞ differ only by some exact current. Green currents for varieties of arbitrary dimension were systematically developed by H. Gillet and C. Soule in [3]. These currents play a central role in Arakelov theory (see [10] and [7]). On the one hand the existence of Green currents follows more or less formally from the Hodge decomposition for currents. To do intersection theory, on the other hand, we need currents which behave well. Green forms and Green currents of logarithmic type are examples of these currents (see [2], and [3]). Here we present a method to find a canonical Green current gY which depends only on the Kahler manifold (X,ω) and the cycle Y . This current gY is given by (k, k)-forms with L coefficients which are smooth on X Y . We start with repeating some basic results on Kahler manifolds in 1 which can be found in the textbooks [5] or [11]. Furthermore, we recall the main properties of the heat kernel before we consider the heat flow of Dirac currents in section 2 obtaining a Green operator for these currents. In 3 we define the canonical harmonic Green current and show that its harmonic projection is zero. In section 4 we apply the heat kernel to intersections of properly intersecting cycles. This is needed to derive a formula for the ∗-product of Green currents in 5. However, we have to restrict to cycles that intersect properly. Hence our results are weaker than those of [3]. As we used a different approach, we decided to present them here. In section 6 we give examples which we found by a good guess (6.1) or because we know all eigen functions of the Laplace operator (6.2) or from the explicit formula of the heat kernel (6.3). The last example shows how our techniques can be extended to non-compact Kahler manifolds for which a heat kernel exists. We conclude in section 7 with some questions and remarks. Acknowledgment. The author wishes to thank W. Gubler, J. Kramer, and U. Kuhn for many discussions which contributed to this paper. The comments of the referee helped to clarify the exposition.