Let $X$ be a smooth complex projective variety with nef $\bigwedge^2 T_X$ and $\dim X \geq 3$. We prove that, up to a finite étale cover $\tilde{X} \to X$, the Albanese map $\tilde{X} \to {\rm Alb}(\tilde{X})$ is a locally trivial fibration whose fibers are isomorphic to a smooth Fano variety $F$ with nef $\bigwedge^2 T_F$. As a bi-product, we see that either $T_X$ is nef or $X$ is a Fano variety. Moreover we study a contraction of a $K_X$-negative extremal ray $φ: X \to Y$. In particular, we prove that $X$ is isomorphic to the blow-up of a projective space at a point if $φ$ is of birational type. We also prove that $φ$ is a smooth morphism if $φ$ is of fiber type. As a consequence, we give a structure theorem of varieties with nef $\bigwedge^2 T_X$.
Let $X$ be a complex smooth projective variety such that the exterior power of the tangent bundle $\bigwedge^{r} T_X$ is nef for some $1\leq r<\dim X$. We prove that, up to an étale cover, $X$ is a Fano fiber space over an Abelian variety. This gives generalizations of the structure theorem of varieties with nef tangent bundle by Demailly, Peternell and Schneider and that of varieties with nef $\bigwedge^{2} T_X$ by the author. Our result also gives an answer to a question raised by Li, Ou and Yang for varieties with strictly nef $\bigwedge^{r} T_X$ when $r < \dim X$.
Yau Mathematical Sciences Center, Tsinghua University（清华大学丘成桐数学科学中心）；School of Mathematical Sciences, Peking University（北京大学数学科学学院）；Institute of Mathematics, State Key Laboratory of Mathematical Sciences, AMSS, Chinese Academy of Sciences（中国科学院数学与系统科学研究院数学研究所）
In this paper, we classify Fano manifolds with elementary contractions of birational type such that the second or third exterior power of tangent bundles are numerically effective.
In this paper, we classify Fano fourfolds whose the second exterior power of tangent bundles are numerically effective with Picard number greater than one.
We study the positivity of exterior powers of the tangent sheaf on toric varieties in order to generalize results by Campana and Peternell about 3-folds with nef second exterior power of the tangent bundle. Using the theory of equivariant vector bundles and the toric MMP, we establish in the smooth case a criterion for the positivity of $Λ^m\mathcal T_X$ in terms of wall relations. As an application, we classify smooth toric varieties of arbitrary dimension $n\ge3$ with $Λ^2\mathcal T_X$ nef and those of dimension $n\ge 4$ with $Λ^3\mathcal T_X$ ample.
Complex contact manifolds arise naturally in differential geometry, algebraic geometry and exterior differential systems. Their classification would answer an important question about holonomy groups. The geometry of such manifold $X$ is governed by the contact lines contained in $X$. These are related to the notion of a variety of minimal rational tangents. In this review we discuss the partial classification theorems of projective complex contact manifolds. Among such manifolds one finds contact Fano manifolds (which include adjoint varieties) and projectivised cotangent bundles. In the firs…
Let $X$ be a smooth complex projective variety. A recent conjecture of S. Kovács states that if t\ he $p^{\text{th}}$-exterior power of the tangent bundle $T_X$ contains the $p^{\text{th}}$-exterior power of an ample vector bundle, then $X$ is either a projective space or a smooth quadric hypersurface. This conjecture is appealing since it is a common generalization of Mori's, Wahl's, Andreatt\ a-Wísniewski's, and Araujo-Druel-Kovács's characterizations of these spaces. In this paper I give a proof affirming this conjecture for varieties with Picard number 1.
We apply the theory of functors of Artin rings to prove the existence of a universal formal solution to a exterior differential system. We compute the tangent space and the obstruction theory of the funtor of Artin ring governing the solutions of the exterior differential system.
Let $M$ and $N$ be two compact complex manifolds. We show that if the tautological line bundle $\mathscr{O}_{T_M^*}(1)$ is not pseudo-effective and $\mathscr{O}_{T_N^*}(1)$ is nef, then there is no non-constant holomorphic map from $M$ to $N$. In particular, we prove that any holomorphic map from a compact complex manifold $M$ with RC-positive tangent bundle to a compact complex manifold $N$ with nef cotangent bundle must be a constant map. As an application, we obtain that there is no non-constant holomorphic map from a compact Hermitian manifold with positive holomorphic sectional curvature …
For a given irreducible projective variety $X$, the closure of the set of all hyperplanes containing tangents to $X$ is the projectively dual variety $X^{\vee}$. We study the singular locus of projectively dual varieties of certain Segre-Plücker embeddings in series of papers. In this work we give a classification of the irreducible components of the singular locus of the dual Grassmannian. Basically, it admits two components: cusp type and node type which are degeneracies of a certain Hessian matrix, and the closure of the set of tangent planes having more than one critical point, respectivel…
In this paper, we study smooth complex projective varieties $X$ such that some exterior power $\bigwedge^r T_X$ of the tangent bundle is strictly nef. We prove that such varieties are rationally connected. We also classify the following two cases. If $T_X$ is strictly nef, then $X$ isomorphic to the projective space $\mathrm{P}^n$. If $\bigwedge^2 T_X$ is strictly nef and if $X$ has dimension at least $3$, then $X$ is either isomorphic to $\mathrm{P}^n$ or a quadric $\mathrm{Q}^n$.
Let $X$ be a Fano manifold. While the properties of the anticanonical divisor $-K_X$ and its multiples have been studied by many authors, the positivity of the tangent bundle $T_X$ is much more elusive. We give a complete characterisation of the pseudoeffectivity of $T_X$ for del Pezzo surfaces, hypersurfaces in the projective space and del Pezzo threefolds.
Pasquier and Perrin discovered that the ${\rm G}_2$-horospherical manifold ${\bf X}$ of Picard number 1 can be realized as a smooth specialization of the rational homogeneous space parameterizing the lines on the 5-dimensional hyperquadric, in other words, it can be deformed nontrivially to the rational homogeneous space. We show that ${\bf X}$ is the only smooth projective variety with this property. This is obtained as a consequence of our main result that ${\bf X}$ can be recognized by its VMRT, namely, a Fano manifold of Picard number 1 is biregular to ${\bf X}$ if and only if its VMRT at …
The purpose of this paper is to translate positivity properties of the tangent bundle (and the anti-canonical bundle) of an algebraic manifold into existence and movability properties of rational curves and to investigate the impact on the global geometry of the manifold $X$. Among the results we prove are these: \quad If $X$ is a projective manifold, and ${\cal E} \subset T_X$ is an ample locally free sheaf with $n-2\ge rk {\cal E}\ge n$, then $X \simeq \EP_n$. \quad Let $X$ be a projective manifold. If $X$ is rationally connected, then there exists a free $T_X$-ample family of (rational) cur…
After establishing suitable notions of stability and Chern classes for singular pairs, we use Kähler-Einstein metrics with conical and cuspidal singularities to prove the slope semistability of orbifold tangent sheaves of minimal log-canonical pairs of log general type. We then proceed to prove the Miyaoka-Yau inequality for all minimal pairs with standard coefficients. Our result in particular provides an alternative proof of the Abundance theorem for threefolds that is independent of positivity results for tangent sheaves.
For ample vector bundles $E$ over compact complex varieties $X$ and a Schur functor $S_I$ corresponding to an arbitrary partition $I$ of the integer $|I|$, one would like to know the optimal vanishing theorem for the cohomology groups $H^{p,q}(X, S_I(E))$, depending on the rank of $E$ and the dimension $n$ of $X$. Three years ago (Nov. 1995), in an unpublished paper one of us (W.N.) proved a vanishing theorem for the situation where the partition $I$ is a hook. Here we give a simpler proof of this theorem. We also treat the same problem under weaker positivity assumptions, in particular under …
The classical HKR-theorem gives an isomorphism of the n-th Hochschild cohomology of a smooth algebra and the n-th exterior power of its module of Kähler differentials. Here we generalize it for simplicial, graded and anticommutative objects in ``good pairs of categories''. We apply this generalization to complex spaces and noetherian schemes and deduce two decomposition theorems for their (relative) Hochschild cohomology (special cases of those were recently shown by Buchweitz-Flenner and Yekutieli). The first one shows that Hochschild cohomology contains tangent cohomology: $\HH^n(X/Y,\sM)=\c…
In the paper we present an alternative approach to the boundedness of Seshadri constants (which measure the local positivity) of nef and big line bundles at a general point of a complex--projective variety. Our approach is based on the construction of divisors in certain adjoint linear systems having isolated singularities of high order. A variant of the method gives bounds for Seshadri constants valid at arbitrary points of a smooth variety; these bounds, however, depend on the line bundle and the positivity of the tangent bundle of the variety in question.
We confirm Beauville's conjecture that claims that if the p-th exterior power of the tangent bundle of a smooth projective variety contains the p-th power of an ample line bundle, then the variety is either the projective space or the p-dimensional quadric hypersurface.
