### Kähler metrics of constant scalar curvature on bundles over CPn-1.

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Let $(M,J,\Omega )$ be a closed polarized complex manifold of Kähler type. Let $G$ be the maximal compact subgroup of the automorphism group of $(M,J)$. On the space of Kähler metrics that are invariant under $G$ and represent the cohomology class $\Omega $, we define a flow equation whose critical points are the extremal metrics,those that minimize the square of the ${L}^{2}$-norm of the scalar curvature. We prove that the dynamical system in this space of metrics defined by the said flow does not have periodic orbits, and that its only...

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We provide a simple characterization of codimension two submanifolds of ${\mathbb{P}}^{n}\left(\mathbb{R}\right)$ that are of algebraic type, and use this criterion to provide examples of transcendental submanifolds when $n\ge 6$. If the codimension two submanifold is a nonsingular algebraic subset of ${\mathbb{P}}^{n}\left(\mathbb{R}\right)$ whose Zariski closure in ${\mathbb{P}}^{n}\left(\u2102\right)$ is a nonsingular complex algebraic set, then it must be an algebraic complete intersection in ${\mathbb{P}}^{n}\left(\mathbb{R}\right)$.

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