Additive vector fields, algebraicity and rationality.
Characterization of the complex projective space by holomorphic vector fields.
Rigidity of homogeneous contact manifolds under Fano deformation.
Nondeformability of the complex hyperquadric.
Equivalence problem for minimal rational curves with isotrivial varieties of minimal rational tangents
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 , a family of minimal rational curves with -isotrivial varieties of minimal rational tangents...
Varieties of minimal rational tangents of codimension 1
Let be a uniruled projective manifold and let be a general point. The main result of [2] says that if the -degrees (i.e., the degrees with respect to the anti-canonical bundle of ) of all rational curves through are at least , then is a projective space. In this paper, we study the structure of when the -degrees of all rational curves through are at least . Our study uses the projective variety , called the VMRT at , defined as the union of tangent directions to the rational curves...
Deformation of holomorphic maps onto the blow-up of the projective plane
Deformation of holomorphic maps onto Fano manifolds of second and fourth Betti numbers 1
Let be a Fano manifold with different from the projective space such that any two surfaces in have proportional fundamental classes in . Let be a surjective holomorphic map from a projective variety . We show that all deformations of with and fixed, come from automorphisms of . 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 .
Uniruled projective manifolds with irreducible reductive G-structures.
Deformation rigidity of the rational homogeneous space associated to a long simple root
Hecke curves and Hitchin discriminant
Algebraic complete integrability of an integrable system of Beauville
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.
A compactification of with no non-constant meromorphic functions
For each 2-dimensional complex torus , we construct a compact complex manifold with a -action, which compactifies such that the quotient of by the -action is biholomorphic to . For a general , we show that has no non-constant meromorphic functions.
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