No full-text available
To read the full-text of this research,
you can request a copy directly from the authors.
Abstract
We propose a conjectural list of Fano manifolds of Picard number 1 with pseudoeffective normalised tangent bundles, which we prove in various situations by relating it to the complete divisibility conjecture of Francesco Russo and Fyodor L. Zak on varieties with small codegree. Furthermore, the pseudoeffective thresholds and, hence, the pseudoeffective cones of the projectivised tangent bundles of rational homogeneous spaces of Picard number 1 are explicitly determined by studying the total dual variety of minimal rational tangents (VMRTs) and the geometry of stratified Mukai flops. As a by-product, we obtain sharp vanishing theorems on the global twisted symmetric holomorphic vector fields on rational homogeneous spaces of Picard number 1 .
... where we used the fact that b > 0 due to the bigness of T X (cf. [6,Theorem 3.4]). Therefore, c ⊆ D X and hence Supp f * D Y ⊆ Supp D X . ...
... Denote by D X ⊆ P(T X ) and D Y ⊆ P(T Y ) the total dual VMRTs of K and G respectively. By our assumption, both D X and D Y are irreducible hypersurfaces.(6) Let H X be the ample generator of the Picard group Pic(X ).First, we prove Theorem 1.4. ...
... ) a general codimension k (with k ≤ 3) linear section of a 10-dimensional spinor variety;(2) a general hyperplane section of Gr(2, 6); (3) a general hyperplane section of the symplectic Grassmannian SG(3,6).On the other hand, it follows from [5, Theorem 1.2, Propositions 4.7 and 4.8] that all of the above varieties except for Case (1) with k = 3 are almost homogeneous, and hence they admit no non-isomorphic surjective endomorphisms (see[23, Theorem 1.4]). The only case left when X is a smooth linear section of a ...
Let be a surjective morphism of Fano manifolds of Picard number 1 whose VMRTs at a general point are not dual defective. Suppose that the tangent bundle is big. We show that f is an isomorphism unless Y is a projective space. As applications, we explore the bigness of the tangent bundles of complete intersections, del Pezzo manifolds, and Mukai manifolds, as well as their dynamical rigidity.
... -Fano manifolds with Picard number 1 and with zero-dimensional variety of minimal rational tangents [12]. -Moduli spaces SU C (r, d) of stable vector bundles of rank r and degree d over a projective curve C of genus g such that r ≥ 3, g ≥ 4 and (r, d) = 1 [8]. ...
... Moreover, the projectivised moment map G X : (ސT * X ) → (ސM G X ) is everywhere well-defined. Let (ސT * X ) ε −→ (ސ M G X ) → (ސM G X ) be the Stein factorisation; see [8,Seciton 5.A]. Then ε is actually the birational morphism defined by |m | with m ≫ 1, where is the tautological divisor of (ސT * X ). ...
... Now we proceed to calculate the codegree of the VMRT of the equivariant compactifications X of vector groups given in Example 4.5. If X is an irreducible Hermitian symmetric space, the pseudoeffective cone of (ސT * X ) and hence the value µ(T X, −K X ) are determined in [35] and [8]. In particular, if the VMRT is not dual defective, it turns out that its codegree is equal to the rank of X in the sense of [ Table 1. ...
... Lemma 3. 6 We have ...
Let M be a smooth Fano threefold such that a canonical extension of the tangent bundle is an affine manifold. We show that M is rational homogeneous.
... Let X be a Fano manifold of Picard number one and with index one. Since T X is big and the VMRT C x ⊂ P(Ω X,x ) is not dual defective, by [FL21,Theorem 3.4], the general members of K have anti-canonical degree at most two. Hence, we must have −K X · K = 2 and then Theorem 5.19 implies that X is of index two, which is a contradiction. ...
Let X be a Fano manifold with Picard number one such that the tangent bundle T_X is big. If X admits a rational curve with trivial normal bundle, we show that X is isomorphic to the del Pezzo threefold of degree five.
Let f : X → Y be a surjective morphism of Fano manifolds of Picard number 1 whose VMRTs at a general point are not dual defective. Suppose that the tangent bundle T X is big. We show that f is an isomorphism unless Y is a projective space. As applications, we study the bigness of the tangent bundles of complete intersections, del Pezzo manifolds, and Mukai manifolds, as well as their dynamical rigidity.
Let X be an irreducible Hermitian symmetric space of compact type (IHSS for short). In this paper, we give the irreducible decomposition of Sym r T X. As a by-product, we give a cohomological characterization of the rank of X. Moreover, we introduce pseudoeffective thresholds to measure the bigness of tangent bundles of smooth complex projective varieties precisely and calculate them for irreducible Hermitian symmetric spaces of compact type.
Let S be a non-uniruled (i.e., non-birationally ruled) smooth projective surface. We show that the tangent bundle is pseudo-effective if and only if the canonical divisor is nef and the second Chern class vanishes, i.e., . Moreover, we study the blow-up of a non-rational ruled surface with pseudo-effective tangent bundle.
Let M be a smooth Fano threefold such that a canonical extension of the tangent bundle is an affine manifold. We show that M is rational homogeneous.
Let X be a Fano manifold with Picard number one such that the tangent bundle TX is big. If X admits a rational curve with trivial normal bundle, we show that X is isomorphic to the del Pezzo threefold of degree five.
Let X be an irreducible Hermitian symmetric space of compact type (IHSS for short). In this paper, we give the irreducible decomposition of Sym r T X. As a by-product, we give a cohomological characterization of the rank of X. Moreover, we introduce pseudoeffective thresholds to measure the bigness of tangent bundles of smooth complex projective varieties precisely and calculate them for irreducible Hermitian symmetric spaces of compact type.
We present a general construction of hypersurfaces with vanishing hessian, starting from any irreducible non-degenerate variety whose dual variety is a hypersurface and based on the so called Dual Cayley Trick. The geometrical properties of these hypersurfaces are different from the known series constructed until now. In particular, their dual varieties can have arbitrary codimension in the image of the associated polar map.
Given a reflexive sheaf on a mildly singular projective variety, we prove a flatness criterion under certain stability conditions. This implies the algebraicity of leaves for sufficiently stable foliations with numerically trivial canonical bundle such that the second Chern class does not vanish. Combined with the recent work of Druel and Greb-Guenancia-Kebekus this establishes the Beauville-Bogomolov decomposition for minimal models with trivial canonical class.
Let X be a projective manifold of dimension n. Suppose that contains an ample subsheaf. We show that X is isomorphic to . As an application, we derive the classification of projective manifolds containing a -bundle as an ample divisor by the recent work of D.~Litt.
It is a well-known fact that families of minimal rational curves on rational
homogeneous manifolds of Picard number one are uniform, in the sense that the
tangent bundle to the manifold has the same splitting type on each curve of the
family. In this note we prove that certain --stronger-- uniformity conditions
on a family of minimal rational curves on a Fano manifold of Picard number one
allow to prove that the manifold is homogeneous.
According to a well-known theorem of Brieskorn and Slodowy, the intersection
of the nilpotent cone of a simple Lie algebra with a transverse slice to the
subregular nilpotent orbit is a simple surface singularity. At the opposite
extremity of the nilpotent cone, the closure of the minimal nilpotent orbit is
also an isolated symplectic singularity, called a minimal singularity. For
classical Lie algebras, Kraft and Procesi showed that these two types of
singularities suffice to describe all generic singularities of nilpotent orbit
closures: specifically, any such singularity is either a simple surface
singularity, a minimal singularity, or a union of two simple surface
singularities of type . In the present paper, we complete the picture
by determining the generic singularities of all nilpotent orbit closures in
exceptional Lie algebras (up to normalization in a few cases). We summarize the
results in some graphs at the end of the paper.
In most cases, we also obtain simple surface singularities or minimal
singularities, though often with more complicated branching than occurs in the
classical types. There are, however, six singularities which do not occur in
the classical types. Three of these are unibranch non-normal singularities: an
-variety whose normalization is , an
-variety whose normalization is , and a
two-dimensional variety whose normalization is the simple surface singularity
. In addition, there are three 4-dimensional isolated singularities each
appearing once. We also study an intrinsic symmetry action on the
singularities, in analogy with Slodowy's work for the regular nilpotent orbit.
Consider the vector space of square matrices of order r and the corresponding projective space ℙ = ℙ
r
2−1. The points of ℙ are in a one-to-one correspondence with the square matrices modulo multiplication by a nonzero constant. Consider the Segre subvariety corresponding to the matrices of rank one and a filtration
In 1991 Campana and Peternell proposed, as a natural algebro-geometric
extension of Mori's characterization of the projective space, the problem of
classifying the complex projective Fano manifolds whose tangent bundle is nef,
conjecturing that the only varieties satisfying these properties are rational
homogeneous. In this paper we review some background material related to this
problem, with special attention to the partial results recently obtained by the
authors.
We show that the augmented base locus coincides with the exceptional locus (i.e., null locus) for any nef R-Cartier divisor on any scheme projective over a field (of any characteristic). Next we prove a semi-ampleness criterion in terms of the exceptional locus generalizing a result of Keel. We also discuss some problems related to augmented base loci of log divisors.
We study a number of conditions equivalent to homology to0 of vanishing cycles on a non-singular complex projective surface, and we classify such surfaces.
We classify all smooth projective horospherical varieties with Picard number 1. We prove that the automorphism group of any such variety X acts with at most two orbits and that this group still acts with only two orbits on X blown up at the closed orbit. We characterize all smooth projective two-orbits varieties with Picard number 1 that satisfy this latter property. Mathematics Subject Classification. 14J45 14L30 14M17 Keywords. Two-orbits varieties, horospherical varieties, spherical varieties.
In this paper we classify rank two Fano bundles \cE on Fano manifolds
satisfying . The classification is obtained
via the computation of the nef and pseudoeffective cones of the
projectivization {\mathbb{P}}(\cE), that allows us to obtain the
cohomological invariants of X and \cE. As a by-product we discuss Fano
bundles associated to congruences of lines, showing that their varieties of
minimal rational tangents may have several linear components.
Let M be a complex projective manifold with the property that for any compact Riemann surface C and holomorphic map f : C → M the pullback of the tangent bundle of M is semistable. We prove that in this case M is a curve or a finite étale quotient of an abelian variety answering a conjecture of Biswas.
This is a continuation of math.AG/0408274, where we have described the relative movable cone for a Springer resolution of the closure of a nilpotent orbit in a complex simple Lie algebra. But, in general, the movable cone does not coincide with the whole space of numerical classes of divisors on the Springer resolution. The purpose of this paper is, to describe the remainder. We shall first construct a deformation of the nilpotent orbit closure in a canonical manner according to Brieskorn and Slodowy, and next describe all its crepant simultaneous resolutions. This construction enables us to divide the whole space into a finite number of chambers. Moreover, by using this construction, one can generalize the main result of math.AG/0408274 to arbitrary Richardson orbits whose Springer maps have degree > 1. New Mukai flops, different from those of type A,D,E_6, will appear in the birational geometry for such orbits.
I give three descriptions of the Mukai flop of type , one in terms of Jordan algebras, one in terms of projective geometry over the octonions, and one in terms of O-blow-ups. Each description shows that it is very similar to certain flops of type A. The Mukai flop of type is also described.
The purpose of this paper is to define and study systematically some asymptotic invariants associated to base loci of line bundles on smooth projective varieties. We distinguish an open dense subset of the real big cone, called the stable locus, consisting of the set of classes on which the asymptotic base locus is locally constant. The asymptotic invariants define continuous functions on the big cone, whose vanishing characterizes, roughly speaking, the unstable locus. We show that for toric varieties at least, there exists a polyhedral decomposition of the big cone on which these functions are polynomial.
For a uniruled projective manifold, we prove that a general rational curve of minimal degree through a general point is uniquely determined by its tangent vector. As applications, among other things we give a new proof, using no Lie theory, of our earlier result that a holomorphic map from a rational homogeneous space of Picard number 1 onto a projective manifold different from the projective space must be a biholomorphic map.
Suppose that X X is a projective manifold whose tangent bundle T X T_X contains a locally free strictly nef subsheaf. We prove that X X is isomorphic to either a projective space or a projective bundle over a hyperbolic manifold of general type. Moreover, if the fundamental group π 1 ( X ) \pi _1(X) is virtually solvable, then X X is isomorphic to a projective space.
Let X be a Fano manifold with Picard number one such that the tangent bundle TX is big. If X admits a rational curve with trivial normal bundle, we show that X is isomorphic to the del Pezzo threefold of degree five.
This two-volume book on "Positivity in Algebraic Geometry" contains a contemporary account of a body of work in complex algebraic geometry loosely centered around the theme of positivity. Topics in Volume I include ample line bundles and linear series on a projective variety, the classical theorems of Lefschetz and Bertini and their modern outgrowths, vanishing theorems, and local positivity. Volume II begins with a survey of positivity for vector bundles, and moves on to a systematic development of the theory of multiplier ideals and their applications. At least a third of the book is devoted to concrete examples, applications, and pointers to further developments. Most of the material in the present Volume II has not previously appeared in book form, and substantial parts are worked out here in detail for the first time. Both volumes are also available as hardcover editions as Vols. 48 and 49 in the series "Ergebnisse der Mathematik und ihrer Grenzgebiete."
This two volume work on Positivity in Algebraic Geometry contains a contemporary account of a body of work in complex algebraic geometry loosely centered around the theme of positivity. Topics in Volume I include ample line bundles and linear series on a projective variety, the classical theorems of Lefschetz and Bertini and their modern outgrowths, vanishing theorems, and local positivity. Volume II begins with a survey of positivity for vector bundles, and moves on to a systematic development of the theory of multiplier ideals and their applications. A good deal of this material has not previously appeared in book form, and substantial parts are worked out here in detail for the first time. At least a third of the book is devoted to concrete examples, applications, and pointers to further developments.
Volume I is more elementary than Volume II, and, for the most part, it can be read without access to Volume II.
A rational curve C in a nonsingular variety X is standard if under the normalization f: P¹ → C ⊂ X, the vector bundle f*T(X) decomposes as O(2)⊕O(1)p⊕Oq for some nonnegative integers satisfying p + q = dim X - 1. For a Fano manifold X of Picard number one and a general point x ∈ X, a general rational curve of minimal degree through x is standard. It has been asked whether all rational curves of minimal degree through a general point x are standard. Our main result is a negative answer to this question.
In continuation of [HM1], [HM4] and [Hw], we work on the following conjecture.
Projective threefolds whose discriminant locus has degree < 8 are classi- fied. This is done by combining the class formula with many recent results on surfaces and threefolds with low sectional genus. Moreover classical results on surfaces concerning the class and the degree are partially extended to a special class of threefolds.
We classify smooth complex projective surfaces of degree d and class mu, satisfying either (i) mu - d less-than-or-equal-to 16, or (ii) mu less-than-or-equal-to 25. All these surfaces are rational or ruled. Indeed, we prove that the smallest value of the class mu of a non-ruled surface is 30 and in fact there are at least two surfaces S, both of degree d = 10 and sectional genus g = 6, with Kodaira dimension kappa(S) = 0 and class mu = 30. Finally, we classify the smooth k-folds (k greater-than-or-equal-to 3) whose sectional surface has class mu less-than-or-equal-to 23.
Beauville asked if a compact Kaehler manifold with splitting tangent bundle has a universal covering that is a product of manifolds. We use Mori theory and elementary results about holomorphic foliations to study this problem for projective uniruled varieties. In particular we obtain an affirmative answer for rationally connected varieties in any dimension and uniruled varieties in dimension 4.
We determine the varieties of linear spaces on rational homogeneous varieties, provide explicit geometric models for these
spaces, and establish basic facts about the local differential geometry of rational homogeneous varieties.
Using currents with minimal singularities, we introduce pointwise minimal multiplicities for a real pseudo-effective (1,1)-cohomology class α on a compact complex manifold X,which are the local obstructions to the numerical effectivity of α. The negative part of α is then defined as the real effective divisor N(α) whose multiplicity along a prime divisor D is just the generic multiplicity of α along D, and we get in that way a divisorial Zariski decomposition of α into the sum of a class Z(α) which is nef in codimension 1 and the class of its negative part N(α), which is an exceptional divisor in the sense that it is very rigidly embedded in X. The positive parts Z(α) generate a modified nef cone, and the pseudo-effective cone is shown to be locally polyhedral away from the modified nef cone, with extremal rays generated by exceptional divisors. We then treat the case of a surface and a hyper-Kähler manifold in some detail. Using the intersection form (respectively the Beauville–Bogomolov form), we characterize the modified nef cone and the exceptional divisors. The divisorial Zariski decomposition is orthogonal, and is thus a rational decomposition, which fact accounts for the usual existence statement of a Zariski decomposition on a projective surface, which is thus extended to the hyper-Kähler case. Finally, we explain how the divisorial Zariski decomposition of (the first Chern class of) a big line bundle on a projective manifold can be characterized in terms of the asymptotics of the linear series |kL| as k→∞.
We prove that two Springer maps of the same degree over a nilpotent orbit closure are connected by stratified Mukai flops, and the latter is obtained by contractions of extremal rays of a natural resolution of the nilpotent orbit closure.
We consider birational projective contractions f:X -> Y from a smooth symplectic variety X over the complex numbers. We first show that exceptional rational curves on X deform in a family of dimension at least 2n-2. Then we show that these contractions are generically coisotropic, provided X is projective. Then we specialize to contractions with 1-dimensional exceptional fibres. We classify them in a natural way in terms of (\Gamma, G), where \Gamma is a Dynkin diagram of type A_l, D_l or E_l and G is a permutation group of automorphisms of \Gamma. The 1-dimensional fibres do not degenerate, except if the contraction is of type (A_{2l},S_2). In that case they do not degenerate in codimension 1. Furthermore we show that the normalization of any irreducible component of Sing(Y) is a symplectic variety. We also provide examples for contractions of any type (\Gamma, G).
Let C be a smooth projective curve of genus over the complex numbers and be the moduli space of stable vector bundles of rank r with a fixed determinant of degree d. In the projectivized cotangent space at a general point E of , there exists a distinguished hypersurface consisting of cotangent vectors with singular spectral curves. In the projectivized tangent space at E, there exists a distinguished subvariety consisting of vectors tangent to Hecke curves in through E. Our main result establishes that the hypersurface and the variety are dual to each other. As an application of this duality relation, we prove that any surjective morphism , where is another curve of genus g, is biregular. This confirms, for , the general expectation that a Fano variety of Picard number 1, excepting the projective space, has no non-trivial self-morphism and that morphisms between Fano varieties of Picard number 1 are rare. The duality relation also gives simple proofs of the non-abelian Torelli theorem and the result of Kouvidakis-Pantev on the automorphisms of .
In this paper we give a short and elementary proof of a characterization of the projective space in terms of rational curves which was conjectured by Mori and Mukai. A proof was first given in a preprint by K. Cho, Y. Miyaoka and N. Shepherd-Barron. While our proof here is shorter and involves substantial technical simplifications, the essential ideas are taken from that preprint.
Let X be a projective variety which is covered by a family of rational curves of minimal degree. The classic bend-and-break argument of Mori asserts that if x and y are two general points, then there are at most finitely many curves in that family which contain both x and y. In this work we shed some light on the question as to whether two sufficiently general points actually define a unique curve. As an immediate corollary to the results of this paper, we give a characterization of projective spaces which improves on the known generalizations of Kobayashi-Ochiai's theorem.