### Additive vector fields, algebraicity and rationality.

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We formulate the equivalence problem, in the sense of É. Cartan, for families of minimal rational curves on uniruled projective manifolds. An important invariant of this equivalence problem is the variety of minimal rational tangents. We study the case when varieties of minimal rational tangents at general points form an isotrivial family. The main question in this case is for which projective variety $Z$, a family of minimal rational curves with $Z$-isotrivial varieties of minimal rational tangents...

Let $X$ be a uniruled projective manifold and let $x$ be a general point. The main result of [2] says that if the $(-{K}_{X})$-degrees (i.e., the degrees with respect to the anti-canonical bundle of $X$) of all rational curves through $x$ are at least $dimX+1$, then $X$ is a projective space. In this paper, we study the structure of $X$ when the $(-{K}_{X})$-degrees of all rational curves through $x$ are at least $dimX$. Our study uses the projective variety ${\mathcal{C}}_{x}\subset \mathbb{P}{T}_{x}\left(X\right)$, called the VMRT at $x$, defined as the union of tangent directions to the rational curves...

Let $X$ be a Fano manifold with ${b}_{2}=1$ different from the projective space such that any two surfaces in $X$ have proportional fundamental classes in ${H}_{4}(X,\mathbf{C})$. Let $f:Y\to X$ be a surjective holomorphic map from a projective variety $Y$. We show that all deformations of $f$ with $Y$ and $X$ fixed, come from automorphisms of $X$. The proof is obtained by studying the geometry of the integral varieties of the multi-valued foliation defined by the variety of minimal rational tangents of $X$.

We show that the Beauville’s integrable system on a ten dimensional moduli space of sheaves on a K3 surface constructed via a moduli space of stable sheaves on cubic threefolds is algebraically completely integrable, using O’Grady’s construction of a symplectic resolution of the moduli space of sheaves on a K3.

For each 2-dimensional complex torus $T$, we construct a compact complex manifold $X\left(T\right)$ with a ${\u2102}^{2}$-action, which compactifies ${\left({\u2102}^{*}\right)}^{4}$ such that the quotient of ${\left({\u2102}^{*}\right)}^{4}$ by the ${\u2102}^{2}$-action is biholomorphic to $T$. For a general $T$, we show that $X\left(T\right)$ has no non-constant meromorphic functions.

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