Non-deformability of entire curves in projective hypersurfaces of high degree

Olivier Debarre; Gianluca Pacienza; Mihai Păun

Displaying similar documents to “Non-deformability of entire curves in projective hypersurfaces of high degree”

Low pole order frames on vertical jets of the universal hypersurface

Joël Merker (2009)

Annales de l’institut Fourier

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For low order jets, it is known how to construct meromorphic frames on the space of the so-called vertical $k$ -jets $J_{vert}^{k} (𝒳)$ of the universal hypersurface $𝒳 \subset ℙ^{n + 1} \times ℙ^{\frac{(n + 1 + d)!}{((n + 1)! d!)} - 1}$ parametrizing all projective hypersurfaces $X \subset ℙ^{n + 1} (ℂ)$ of degree $d$ . In 2004, for $k = n$ , Siu announced that there exist two constants $c_{n} \geq 1$ and $c_{n}^{'} \geq 1$ such that the twisted tangent bundle $T_{J_{vert}^{n} (𝒳)} \otimes 𝒪_{ℙ^{n + 1}} (c_{n}) \otimes 𝒪_{ℙ^{\frac{(n + 1 + d)!}{((n + 1)! d!)} - 1}} (c_{n}^{'})$ is generated at every point by its global sections. In...

On the real analytic Levi flat hypersurfaces in complex tori of dimension two

Kazuko Matsumoto, Takeo Ohsawa (2002)

Annales de l’institut Fourier

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We compute the Levi form of the logarithm of the distance function for real hypersurfaces in two dimensional complex tori, and discuss the characterization of Levi flat hypersurfaces there.

Logarithmic Surfaces and Hyperbolicity

Gerd Dethloff, Steven S.-Y. Lu (2007)

Annales de l’institut Fourier

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In 1981 J. Noguchi proved that in a logarithmic algebraic manifold, having logarithmic irregularity strictly bigger than its dimension, any entire curve is algebraically degenerate. In the present paper we are interested in the case of manifolds having logarithmic irregularity equal to its dimension. We restrict our attention to Brody curves, for which we resolve the problem completely in dimension 2: in a logarithmic surface with logarithmic irregularity...

Deformation of holomorphic maps onto Fano manifolds of second and fourth Betti numbers 1

Jun-Muk Hwang (2007)

Annales de l’institut Fourier

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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 $H_{4} (X, C)$ . 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 $X$ .

Obstructions to deforming curves on a $3$ -fold, II: Deformations of degenerate curves on a del Pezzo $3$ -fold

Hirokazu Nasu (2010)

Annales de l’institut Fourier

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We study the Hilbert scheme ${Hilb}^{s c} V$ of smooth connected curves on a smooth del Pezzo $3$ -fold $V$ . We prove that any degenerate curve $C$ , any curve $C$ contained in a smooth hyperplane section $S$ of $V$ , does not deform to a non-degenerate curve if the following two conditions are satisfied: (i) $χ (V, ℐ_{C} (S)) \geq 1$ and (ii) for every line $ℓ$ on $S$ such that $ℓ \cap C = \emptyset$ , the normal bundle $N_{ℓ / V}$ is trivial ( $N_{ℓ / V} ≃ {𝒪_{ℙ^{1}}}^{\oplus 2}$ ). As a consequence, we prove an analogue (for ${Hilb}^{s c} V$ ) of a conjecture of J. O. Kleppe, which is concerned with non-reduced components...

Displaying similar documents to “Non-deformability of entire curves in projective hypersurfaces of high degree”

Low pole order frames on vertical jets of the universal hypersurface

On the real analytic Levi flat hypersurfaces in complex tori of dimension two

Logarithmic Surfaces and Hyperbolicity

Deformation of holomorphic maps onto Fano manifolds of second and fourth Betti numbers 1

Obstructions to deforming curves on a 3 -fold, II: Deformations of degenerate curves on a del Pezzo 3 -fold

Obstructions to deforming curves on a $3$ -fold, II: Deformations of degenerate curves on a del Pezzo $3$ -fold