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Let X and X be compact Hausdorff spaces, and X , X be Banach lattices. Let X denote the Banach lattice of all continuous X -valued functions on X equipped with the pointwise ordering and the sup norm. We prove that if there exists a Riesz isomorphism X such that X is non-vanishing on X if and only if X is non-vanishing on X , then X is homeomorphic to X , and X is Riesz isomorphic to X . In this case, X can be written as a weighted composition operator: X , where X is a homeomorphism from X onto X , and X is a Riesz isomorphism from X onto X for every X in X . This generalizes some known results obtained recently.

Let E and E be two Hilbert E -modules over E -algebras E and E , respectively. Let E be a surjective linear isometry from E onto E and E a map from E into E . We will prove in this paper that if the E -algebras E and E are commutative, then E preserves the inner products and E is a module map, ie, there exists a E -isomorphism E between the E -algebras such that <div class="gsh_dspfr"> E

We prove that the first Chern form of the moduli space of polarized Calabi-Yau manifolds, with the Hodge metric or the Weil-Petersson metric, represent the first Chern class of the canonical extensions of the tangent bundle to the compactification of the moduli space with normal crossing divisors.