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Examples of Fano manifolds with non‐pseudoeffective tangent bundle
... 1.3] that the tangent bundle of a toric variety is big. Höring, Liu, and Shao [5,Th. 1.4] give a complete answer to del Pezzo surfaces: If X is a del Pezzo surface of degree d, then Also in the paper [5], they solve these problems for del Pezzo threefolds. ...
... Höring, Liu, and Shao [5,Th. 1.4] give a complete answer to del Pezzo surfaces: If X is a del Pezzo surface of degree d, then Also in the paper [5], they solve these problems for del Pezzo threefolds. In [4], Höring and Liu consider Fano manifolds X with Picard number one, and they prove that if X admits a rational curve with trivial normal bundle and with big T X , then X is isomorphic to the del Pezzo threefold of degree five. ...
... In particular, if X has a standard conic bundle structure π : X → P 2 , we have a natural irreducible effective divisorC on P(T X ) induced from the fibers of π : X → P 2 according to [5,Cor. 2.13], and [C] ∼ ζ + Π * (K X − π * K P 2 ). ...
In this paper, we study the positivity property of the tangent bundle of a Fano threefold X with Picard number 2 . We determine the bigness of the tangent bundle of the whole 36 deformation types. Our result shows that is big if and only if . As a corollary, we prove that the tangent bundle is not big when X has a standard conic bundle structure with non-empty discriminant. Our main methods are to produce irreducible effective divisors on constructed from the total dual VMRT associated to a family of rational curves. Additionally, we present some criteria to determine the bigness of .
... Theorem 1.4 offers a potential approach to address Conjecture 1.1 when the total dual VMRT is a hypersurface in the projectivized tangent bundle. For precise definitions, we direct the reader to [ Recall that when X is a smooth hypersurface of degree d, the joint paper of Höring, Liu and the first author shows that the tangent bundle T X is pseudo-effective if and only if d ≤ 2 (see [16,Theorem 1.4]); when X is a smooth complete intersection of two quadrics, [4] verifies that the tangent bundle T X is Q-effective but not big; when X is a smooth Fano complete intersection of dimension at least 3 and of Fano index 1 or 2, [15, Theorem 1.1] implies that T X is not big. We refer the reader to Theorem 4.5 written by Liu, which shows that a general smooth finite cover over a general complete intersection of two quadrics cannot have a pseudo-effective tangent bundle. ...
... However, this leads to a contradiction to the ampleness of the ramification divisor [16]) that ( * ) D X ≡ aξ − bτ * H X holds where a > 0 is the codegree of the VMRT C x at a general point and b > 0 is an integer. Since the total dual VMRT D X (as a prime divisor in P(T X ) by our assumption) D X is covered by minimal sections ⊆ P(T X ) of K such that ξ · = 0, we have D X · < 0. Hence, D X | D X is not pseudo-effective and thus the irreducible divisor D X is not big; in particular, D X is extremal in the pseudo-effective cone PE(P( ...
... Since X is a complete intersection, X is Fano. If dim(X ) = 2, then the result follows from [16,Theorem 1.2]. So in the following, we assume dim(X ) ≥ 3 and hence X has Picard number 1 by the Lefschetz theorem. ...
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.
... As ξ is big, it follows from [FL22, Theorem 3.4] (cf. [HLS22]) that D X ≡ aξ−bτ * H X holds where a > 0 is the codegree of the VMRT C x at a general point and b > 0 is an integer; moreover, the total dual VMRT D X (as a prime divisor in P(T X ) by our assumption) is extremal in the pseudo-effective cone PE(P( ...
... Observe that H 0 (Y, Sym r (T Y (3))) = 0 for r ≥ 1 (see [HLS22,Theorem 1.3]). This implies that H 0 (X, Sym r T X ) = 0 for any r ≥ 1; hence, T X is not big. ...
... d 1 ≥ · · · ≥ d k . If k = 1, then the result follows from[HLS22, Theorem 1.4]. Suppose that k ≥ 2. Then by the first half of the proof, there exists a smooth complete intersection X of multi-degree (d 1 , ..., d k ) and a finite morphism from X to a smooth hypersurface Y of degree d 1 . ...
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.
... -Hypersurfaces in ސ n (n ≥ 3) [13]. ...
... • Total dual variety of minimal rational tangents: del Pezzo surfaces and del Pezzo threefolds [13]. ...
... On moment map and bigness of tangent bundles of G-varieties 1503 We refer the reader to Section 3B for the definitions of spherical varieties, symmetric varieties and horospherical varieties. Our initial motivation for the present work is trying to produce more examples of Fano manifolds with Picard number 1 and with big tangent bundle, while the only previous known nonhomogeneous examples, up to our knowledge, are the quintic del Pezzo threefold V 3 [13] and the horospherical G 2 -variety X 5 [32]. As the second application of Proposition 1.1, we derive infinitely many (nonhomogeneous) examples of Fano manifolds with Picard number 1 and with big tangent bundle, which are summarised in the following: ...
... still holds: see Remark 4.2.2. This provides an alternative proof of the fact that the tangent bundle of X is not big, which was proved independently in [HLS22] and [Mal21]. ...
... In particular, as soon as d ≥ 4, this shows that Ω H is uniformly q-ample (with uniform bound λ = 1 d−3 if d ≥ N + 1), and hence so is a general hypersurface of degree d ≥ 4. For the notion of (uniform) intermediate ampleness, we refer to Section 4.3. A stronger version of this vanishing theorem was actually proved recently in [HLS22], in a different context and via a different method. In loc.cit, the authors prove that for any smooth hypersurface H ⊂ P N , N ≥ 3, of degree d ≥ 3, and for any m ∈ N, n ∈ Z satisfying the broad inequality m(d − 3) ≥ n, one has the vanishing H 0 (H, S m T X(n)) = (0). ...
... is already surjective: it can be checked directly, or one can notice that the cokernel of this map is actually isomorphic to H 1 (Flag (1,2) C N +1 , L m,0 ), which is zero. This allows to recover the result that the tangent bundle T X is not big, which was recently proved in [Mal21] and independently in [HLS22]. Indeed, one has the isomorphism T X ≃ Ω X (1), and the description of ...
In this paper, we provide two different resolutions of structural sheaves of projectivized tangent bundles of smooth complete intersections. These resolutions allow in particular to obtain convenient (and completely explicit) descriptions of cohomology of twisted symmetric powers of cotangent bundles of complete intersections, which are easily implemented on computer. We then provide several applications. First, we recover the known vanishing theorems on the subject, and show that they are optimal via some non-vanishing theorems. Then, we study the symmetric algebra of global sections of symmetric powers of , where X is a smooth complete intersection of codimension , improving the known results in the literature. We also study partial ampleness of cotangent bundles of general hypersurfaces. Finally, we illustrate how the explicit descriptions of cohomology can be implemented on computer. In particular, this allows to exhibit new and simple examples of family of surfaces along which the canonically twisted pluri-genera do not remain constant.
... To exclude the product cases, we will focus on the case where X is a Fano manifold of Picard number 1 with dimension at least 3 in this paper. Note that in this situation the pseudoeffectivity of the normalized tangent bundle of X implies that the tangent bundle of X is big and it is expected that the bigness of the tangent bundle is already a rather restrictive property (see [HLS20]). We expect the following classification: ...
... Indeed, if we assume that the VMRT of X at a general point is not dual defective, then the pseudoeffectivity of the normalized tangent bundle can be interpreted as information on the cohomological class of the total dual VMRT (cf. [HR04,HLS20]). This allows us to relate Conjecture 1.2 to Conjecture 1.6. ...
... In general, it is quite difficult to compute these cohomological groups due to the lack of tools. However, recently it is observed in [HLS20] that the problem can be translated into the calculation of the cohomological class of the total dual VMRT if the VMRT is not dual defective. By combining this with the geometry of stratified Mukai flops, we will completely settle Problem 1.13 for rational homogeneous spaces of Picard number 1 with E being the tangent bundle, which reads as follows: ...
We propose a conjectural list of Fano manifolds of Picard number 1 with pseudoeffective normalized tangent bundles, which we prove in various situations by relating it to the complete divisibility conjecture of Russo and Zak on varieties with small codegree. Furthermore, the pseudoeffective thresholds and hence the pseudoeffective cones of the projectivized tangent bundles of rational homogeneous spaces of Picard number 1 are explicitly determined by studying the total dual VMRT 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.
... • (projectivised) moment map rational homogeneous spaces [Ric74] • twisted symmetric vector fields toric varieties [Hsi15] intersection of two quadrics in P 4 and cubic surfaces in P 3 [Mal21] hypersurfaces in P n (n ≥ 3) [HLS20] • total dual VMRT del Pezzo surfaces and del Pezzo threefolds [HLS20] -Fano manifolds of Picard number 1 and with 0-dimensional VMRT [HL21a] 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 [FL21] -Fano threefolds with Picard number 2 [KKL22] The main body of this paper will be devoted to pursue furthermore the criterion for the bigness of Λ via moment map. Let G be a connected algebraic group with Lie algebra g and let X be a smooth projective G-variety. ...
... • (projectivised) moment map rational homogeneous spaces [Ric74] • twisted symmetric vector fields toric varieties [Hsi15] intersection of two quadrics in P 4 and cubic surfaces in P 3 [Mal21] hypersurfaces in P n (n ≥ 3) [HLS20] • total dual VMRT del Pezzo surfaces and del Pezzo threefolds [HLS20] -Fano manifolds of Picard number 1 and with 0-dimensional VMRT [HL21a] 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 [FL21] -Fano threefolds with Picard number 2 [KKL22] The main body of this paper will be devoted to pursue furthermore the criterion for the bigness of Λ via moment map. Let G be a connected algebraic group with Lie algebra g and let X be a smooth projective G-variety. ...
... We refer the reader to Section 3.2 for the definitions of spherical varieties, symmetric varieties and horospherical varieties. Our initial motivation for the present work is trying to produce more examples of Fano manifolds of Picard number 1 and with big tangent bundle, while the only previous known non-homogeneous examples, up to our knowledge, are the quintic del Pezzo threefold V 3 [HLS20] and the horospherical G 2 -variety X 5 [PP10]. As a direct application of Proposition 1.1 and Theorem 1.2, we derive infinitely many (non-homogeneous) examples of Fano manifolds of Picard number 1 and with big tangent bundle, which are summarised in the following: ...
Let G be a connected algebraic group and let X be a smooth projective G-variety. In this paper, we prove a sufficient criterion to determine the bigness of the tangent bundle TX using the moment map . As an application, the bigness of the tangent bundles of certain quasi-homogeneous varieties are verified, including symmetric varieties, horospherical varieties and equivariant compactifications of commutative linear algebraic groups. Finally, we study in details the Fano manifolds X with Picard number 1 which is an equivariant compactification of a vector group . In particular, we will determine the pseudoeffective cone of and show that the image of the projectivised moment map along the boundary divisor D of X is projectively equivalent to the dual variety of the VMRT of X.
... In general it is difficult to give a numerical characterization for pseudoeffective or bigness of the tangent bundle, even in low dimension with low rank of Picard group. It has been shown by Hsiao Also in the paper [HLS20], they solve these problems for del Pezzo threefolds. In [HL21], Höring and Liu consider Fano manifolds X with Picard number one, and they prove that if X admits a rational curve with trivial normal bundle and with big T X then X is isomorphic to the del Pezzo threefold of degree five. ...
... LetD be the total dual VMRT on P(T V5 ) associated to the family of lines on V 5 . Then, according to[HLS20, Theorem 5.4],[D] ∼ 3η − Φ * H .for H = O V5 (1) and the tautological class η = O P(T V 5 ) (1) of Φ : P(T V5 ) → V 5 . ...
... Total Dual VMRTs and some Criteria to Disprove BignessIn this section, we briefly introduce the theory related to the total dual VMRT, which is firstly introduced by[HR04] in a study of Hecke curves on the moduli of vector bundles on curves. Later,[OSW16] generalizes the theory to the case of minimal rational curves, and[HLS20] develops explicit formulas in the case where a variety has zero dimensional VMRTs in a study of the tangent bundle of del Pezzo manifolds. ...
In this paper, we study on the property of bigness of the tangent bundle of a Fano threefold X with Picard number 2. We determine the bigness of the tangent bundle of whole 36 deformation types. Our result shows that is big if and only if . And as a corollary we prove that the tangent bundle is not big when X has a standard conic bundle structure with non-empty discriminant. Our main methods are to produce irreducible effective divisors on constructed from the total dual VMRT associated to a family of rational curves. Additionally, we present some criteria to determine the bigness of .
... In this paper we go in a direction that is closer to the idea of higher Fano manifolds [dJS07,AC12]: we assume that the tangent bundle is big, i.e., the canonical class ζ := O P(X) (1) on the projectivised tangent bundle P(T X ) is a big divisor class. While this implies that ζ is "semipositive" over some Zariski open subset ∅ = U 0 ⊂ P(T X ), its negative locus typically maps surjectively onto X. Nevertheless the examples considered in our earlier work with F. Shao [HLS20] indicate that assuming bigness should lead to strong restrictions on the manifold. We prove the first classification result for Fano manifolds with big tangent bundles: ...
... In view of Theorem 1.1 and the examples considered in [HLS20], we expect that prime Fano manifolds with big tangent bundle should have a rather high index. Our computations yield a first step in this direction: ...
... Definition 5.4): over a general point x ∈ X the fibreČ x is a finite union of hyperplanes, one for each curve in K passing through the point x. We observed in [HLS20] that the bigness assumption leads to restrictions on the cohomology class ofČ ⊂ P(T X ). In fact Proposition 5.11 shows that X has index two, i.e., we have −K X = 2A with A an ample Cartier divisor. ...
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.
... (3) Such a situation is rather exceptional: Most varieties do not admit nonzero symmetric tensors (for instance, hypersurfaces of degree ≥ 3 [HLS22]); when they do, even for varieties as simple as quadrics, the algebra of symmetric tensors is fairly complicated (see, for instance, [BLi24]). We do not have a conceptual explanation for the particularly simple behavior in our case. ...
... Remark 7.2 The variety C is an example of a total dual VMRT [HLS22]. For the proof of the theorem, we will combine this tool with the birational transformation of PT * X defined by a double cover. ...
Let X be an n -dimensional (smooth) intersection of two quadrics, and let be its cotangent bundle. We show that the algebra of symmetric tensors on X is a polynomial algebra in n variables. The corresponding map is a Lagrangian fibration, which admits an explicit geometric description; its general fiber is a Zariski open subset of an abelian variety, which is a quotient of a hyperelliptic Jacobian by a 2 -torsion subgroup. In dimension 3 , is the Hitchin fibration of the moduli space of rank 2 bundles with fixed determinant on a curve of genus 2 .
... The bigness of T X is a rather restrictive property for projective manifolds (see for instance [HLS22,KKL22,HL23,SZ23] and the references therein). The known interesting examples of projective manifolds with big tangent bundles include the quintic del Pezzo threefold [HLS22] and smooth projective horospherical varieties ([Liu23, § 3B2]), which contain toric varieties [Hsi15] and rational homogeneous spaces [Ric74]. ...
... The bigness of T X is a rather restrictive property for projective manifolds (see for instance [HLS22,KKL22,HL23,SZ23] and the references therein). The known interesting examples of projective manifolds with big tangent bundles include the quintic del Pezzo threefold [HLS22] and smooth projective horospherical varieties ([Liu23, § 3B2]), which contain toric varieties [Hsi15] and rational homogeneous spaces [Ric74]. Applying Theorem 1.6 to these examples yields the following list of examples of symplectic orbifold cones. ...
The algebra of symmetric tensors of a projective manifold X leads to a natural dominant affinization morphism It is shown that is birational if and only if is big. We prove that if is birational, then is a symplectic variety endowed with the Schouten--Nijenhuis bracket if and only if is of Fano type, which is the case for smooth projective toric varieties, smooth horospherical varieties with small boundary and the quintic del Pezzo threefold. These give examples of a distinguished class of conical symplectic varieties, which we call symplectic orbifold cones.
... However, for surfaces S with negative Kodaira dimension, the known answer is still far from satisfactory. Höring, Liu, and Shao [5,Theorem 1.4] extended the result of Mallory in [13] and provided a complete answer for del Pezzo surfaces: if S is a del Pezzo surface of degree d, then For relatively minimal ruled surfaces, Kim showed that a projective bundle P C (E) over a smooth projective curve C has a big tangent bundle if and only if E is unstable or C = P 1 [11]. ...
... If f : S → B is a sequence of blowing ups of a conic bundle, then the total dual VMRTC on P(T S ) associated to the family of conics satisfies [5] ...
In this paper, we study the property of bigness of the tangent bundle of a smooth projective rational surface with nef anticanonical divisor. We first show that the tangent bundle of S is not big if S is a rational elliptic surface. We then study the property of bigness of the tangent bundle of a weak del Pezzo surface S. When the degree of S is 4, we completely determine the bigness of the tangent bundle through the configuration of -curves. When the degree d of S is less than or equal to 3, we get a partial answer. In particular, we show that is not big when the number of -curves is less than or equal to , and is big when d=3 and S has the maximum number of -curves. The main ingredient of the proof is to produce irreducible effective divisors on , using Serrano's work on the relative tangent bundle when S has a fibration, or the total dual VMRT associated to a conic fibration on S.
... The corresponding version for tangent bundles on surfaces has been widely studied. For example, Höring, Liu and Shao [7] proved that for a smooth del Pezzo surface S, the tangent bundle T S is pseudoeffective (resp. big) if and only if the degree d " K 2 S is at least 4 (resp. ...
... Also, Jia, Lee and Zhong [9] showed that if S is a smooth non-uniruled projective surface, then T S is pseudoeffective if and only if S is minimal and c 2 pSq " 0. In particular, the non-vanishing conjecture for tangent bundles holds for del Pezzo surfaces (cf. [7,Theorem 1.2]) and for non-uniruled surfaces (cf. [9,Corollary 1.4]). ...
In this paper, we prove the non-vanishing conjecture for cotangent bundles on isotrivial elliptic surfaces. Combined with the result by H\"{o}ring and Peternell, it completely solves the question for surfaces with Kodaira dimension at most 1.
... Fano threefolds are of course rather special manifolds, but they are a natural testing ground for Conjecture 1.1: since Z M is affine, the tangent bundle T M is big [10, Prop.4.2], a very restrictive property for manifolds that are not rational homogeneous [12]. Thus, it seems likely that potential counterexamples to Conjecture 1.1 share at least some properties of rational homogeneous spaces, e.g. ...
... where Q is a line bundle. Since by (12) one has e * C ω * (10,2), we obtain that Q O C×l (7, −1). Letẽ C : U C → S ⊂ P(T M ) be the map determined by the quotient e * C T M → Q. Thenẽ C factors e C and by the universal property of the tautological bundle one hasẽ * C ζ M O C×l (7, −1). ...
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.
... For example, a conjecture proposed by Campana and Peternell asks whether the homogeneous varieties are the only smooth Fano varieties X with nef T X , and the conjecture is settled for dimension three [3], four [4,8,16] (see also [19,Corollary 4.4]), and five [10,24]. Recently, a series of works done by Höring, Liu, Shao [6], and Höring, Liu [5] investigates smooth Fano varieties X with big T X as follows. ...
... These results make use of a special divisor on the projective bundle P X (T X ), called the total dual VMRTC (see [9,21]). In [6], they find a formula forC , which can be written as follows in the case where X attains a conic bundle structure X → Y . ...
... Let us note that in the case where X = {q 1 = 0} ∩ {q 2 = 0} ⊂ P 4 (which is not covered by the above theorem), the conclusion of Theorem 0.0.4 still holds: see Remark 3.2.2. This provides a proof of the fact that the tangent bundle of X is not big, which was proved independently in [HLS22] and [Mal21]. ...
... Indeed, for any m ∈ N, the map α * (q 1 ) : C[Y, X] m,0 −→ C[Y, X] m−1,1 is already surjective: it can be checked directly, or one can notice that the cokernel of this map is actually isomorphic to H 1 (Flag (1,2) C N +1 , L m,0 ), which is zero. This allows to recover the result that the tangent bundle T X is not big, which was recently proved in [Mal21] and independently in [HLS22]. Indeed, one has the isomorphism T X ≃ Ω X (1), and the description of m∈N H 0 (X, S m Ω X (m)) ≃ C[q 1 , q 2 ] readily implies that T X is not big (using the description of bigness via the asymptotic growth of the dimension of the space of global section of larger and larger symmetric powers of the tangent bundle). ...
In this paper, we first generalize the Euler exact sequence, allowing to obtain an explicit and tractable description of cohomology groups of twisted symmetric powers of cotangent bundles of projective spaces. Then, via the construction of a particular rank 2 vector bundle on the projectivized tangent bundle P( T P N ), we obtain a resolution of the structural sheaf O P(TX) of the projectivized tangent bundle of a smooth complete intersection X ⊂ P N . Combining these two facts, we obtain explicit descriptions of cohomology groups of twisted symmetric powers of cotangent bundles of smooth complete intersections, which, in some particular cases, are simple enough to be fully understood. As a first application, we prove a non-vanishing theorem, which shows that the classic vanishing theorem of Bruckmann--Rackvitz is optimal. As a second application, we study explicitly the algebrawhere X is a smooth complete intersection whose dimension is greater than its codimension.
... In [Par11], examples of pseudo-effective tangent bundles are constructed in a similar way. On the other hand, Höring-Liu-Shao in [HLS20] also found examples of pseudo-effective tangent bundles by a different method based on VMRT (varieties of minimal rational tangents). Note that the definition of pseudoeffective vector bundles in [HLS20] is weaker than our definition of Proposition 4.1. ...
... On the other hand, Höring-Liu-Shao in [HLS20] also found examples of pseudo-effective tangent bundles by a different method based on VMRT (varieties of minimal rational tangents). Note that the definition of pseudoeffective vector bundles in [HLS20] is weaker than our definition of Proposition 4.1. The remaining problem in the classification of surfaces is as follows: ...
We provide supplements and open problems related to structure theorems for maximal rationally connected fibrations of certain positively curved projective varieties, including smooth projective varieties with semi-positive holomorphic sectional curvature, pseudo-effective tangent bundle, and nef anti-canonical divisor.
... In characteristic 0, a polynomial ring is obviously D-simple, so the above immediately establishes D-simplicity of the homogeneous coordinate ring of the degree-5 del Pezzo. This in turn implies that the tangent bundle of the degree-5 del Pezzo is big, and we thus recover one of the results of [HLS22]. In characteristic p, the coordinate ring is strongly F -regular (for infinitely many p at least), and thus D-simple (see [Smi95, Theorem 2.2.(4)]). ...
... In this section, we discuss the situation for smooth hypersurfaces of degree d ≥ 3 and prime Fano threefolds (i.e., Fano threefolds with Picard number 1). The hypersurface case is completely settled, without finiteness or homogeneity assumptions, by combining results of [HLS22] and [Hsi15]. Although nothing in this section is original except for the study of the quadric threefold, the conclusions do not appear elsewhere in the literature and so we include them here. ...
We study the question of when a ring can be realized as a direct summand of a regular ring by examining the case of homogeneous coordinate rings. We present very strong obstacles to expressing a graded ring with isolated singularity as a finite graded direct summand. For several classes of examples (del Pezzo surfaces, hypersurfaces), we give a complete classification of which coordinate rings can be expressed as direct summands (not necessarily finite), and in doing so answer a question of Hara about the FFRT property of the quintic del Pezzo. We also examine what happens in the case where the ring does not have isolated singularities, through topological arguments: as an example, we give a classification of which coordinate rings of singular cubic surfaces can be written as finite direct summands of regular rings.
... Hence Hsiao's question has a positive answer if the Campana-Peternell conjecture holds. Recently, Höring, Liu and Shao ( [HLS22]) proved that the tangent bundle of a del Pezzo manifold X of dimension 2 or 3 is big if and only if the degree of X is at least 5 and the tangent bundle of a smooth hypersurface X in a projective space is big if and only if the degree of X is at most 2. Mallory also considered the bigness of tangent bundles of del Pezzo surfaces in [Mal21]. Their results imply that the bigness of tangent bundles is a rather restrictive property. ...
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.
... Fano threefolds are of course rather special manifolds, but they are a natural testing ground for Conjecture 1.1: since Z M is affine, the tangent bundle T M is big [GW20,Prop.4.2], a very restrictive property for manifolds that are not rational homogeneous [HLS22]. Thus it seems likely that potential counterexamples to Conjecture 1.1 share at least some properties of rational homogeneous spaces, e.g. ...
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.
In this paper, we discuss the ℂ-algebra H0(X, S•TX) for a smooth complex projective variety X. We compute it in some simple examples, and give a sharp bound on its Krull dimension. Then we propose a conjectural characterization of non-uniruled projective manifolds with pseudo-effective tangent bundle.
In this paper, we develop a theory of pseudo-effective sheaves on normal projective varieties. As an application, by running the minimal model program, we show that projective klt varieties with pseudo-effective tangent sheaf can be decomposed into Fano varieties and Q-abelian varieties.
Comment: completely revised version, 13 pages
Let A be an elliptic curve, and let be the Serre vector bundle on A. A famous example of Demailly-Peternell-Schneider shows that the tautological class of contains a unique closed positive current. In this survey we start by generalising this statement to arbitrary compact K\"ahler manifolds. We then give an application to abelian fibrations where the total space X has pseudoeffective cotangent bundle and raise some questions about nonvanishing properties of these bundles.
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.
In this paper, we develop a theory of pseudo-effective sheaves on normal projective varieties. As an application, by running the minimal model program, we show that projective klt varieties with pseudo-effective tangent sheaf can be decomposed into Fano varieties and Q-abelian varieties.
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.
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 .
In this paper we study the algebraic ranks of foliations on -factorial normal projective varieties. We start by establishing a Kobayashi-Ochiai's theorem for Fano foliations in terms of algebraic rank. We then investigate the local positivity of the anti-canonical divisors of foliations, obtaining a lower bound for the algebraic rank of a foliation in terms of Seshadri constant. We describe those foliations whose algebraic rank slightly exceeds this bound and classify Fano foliations on smooth projective varieties attaining this bound. Finally we construct several examples to illustrate the general situation, which in particular allow us to answer a question asked by Araujo and Druel on the generalised indices of foliations.
In this paper, we develop the theory of singular Hermitian metrics on vector bundles. As an application, we give a structure theorem of a projective manifold X with pseudo-effective tangent bundle; X admits a smooth fibration to a flat projective manifold Y such that its general fibre is rationally connected. Moreover, by applying this structure theorem, we classify all the minimal surfaces with pseudo-effective tangent bundle and study general nonminimal surfaces, which provide examples of (possibly singular) positively curved tangent bundles.
In this paper we study the positivity of the cotangent bundle of projective manifolds. We conjecture that the cotangent bundle is pseudoeffective if and only the manifold has non-zero symmetric differentials. We confirm this conjecture for most projective surfaces that are not of general type.
Using recent results of Bayer-Macr\`i, we compute in many cases the pseudoeffective and nef cones of the projectivised cotangent bundle of a smooth projective K3 surface. We then use these results to construct explicit families of smooth curves on which the restriction of the cotangent bundle is not semistable (and hence not nef). In particular, this leads to a counterexample to a question of Campana-Peternell.
Comment: Published version
In this paper we partly extend the Beauville-Bogomolov decomposition theorem to the singular setting. We show that any complex projective variety of dimension at most five with canonical singularities and numerically trivial canonical class admits a finite cover, \'etale in codimension one, that decomposes as a product of an Abelian variety, and singular analogues of irreducible Calabi-Yau and irreducible holomorphic symplectic varieties.
We discuss various results on Hilbert schemes of lines and conics and automorphism groups of smooth Fano threefolds with Picard rank 1. Besides a general review of facts well known to experts, the paper contains some new results, for instance, we give a description of the Hilbert scheme of conics on any smooth Fano threefold of index 1 and genus 10. We also show that the action of the automorphism group of a Fano threefold X of index 2 (respectively, 1) on an irreducible component of its Hilbert scheme of lines (respectively, conics) is faithful if the anticanonical class of X is very ample with a possible exception of several explicit cases. We use these faithfulness results to prove finiteness of the automorphism groups of most Fano threefolds and classify explicitly all Fano threefolds with infinite automorphism group. We also discuss a derived category point of view on the Hilbert schemes of lines and conics, and use this approach to identify some of them.
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.
We point out a connection between bigness of the tangent bundle of a smooth projective variety X over ℂ and simplicity of the section rings of X as modules over their rings of differential operators. As a consequence, we see that the tangent bundle of a smooth projective toric variety or a (partial) flag variety is big. Some other applications and related questions are discussed.
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.
Let
be a double cover branched along a smooth hypersurface of degree
. We study the varieties of minimal rational tangents
at a general point
x
of
X
. We describe the homogeneous ideal of
and show that the projective isomorphism type of
varies in a maximal way as
x
varies over general points of
X
. Our description of the ideal of
implies a certain rigidity property of the covering morphism
. As an application of this rigidity, we show that any finite morphism between such double covers with
must be an isomorphism. We also prove that Liouville-type extension property holds with respect to minimal rational curves on
X
.
This paper using a geometric approach produces vanishing and nonvanishing results concerning the spaces of twisted symmetric differentials
on subvarieties
, with k ≤ m. Emphasis is given to the case of k = m which is special and whose nonvanishing results on the dimensional range dim X > 2/3(N − 1) are related to the space of quadrics containing X and the variety of all tangent trisecant lines of X. The paper ends with an application showing that the twisted symmetric plurigenera,
along smooth families of projective varieties Xt are not invariant even for α arbitrarily large.
A line bundle over a complex projective variety is called big and 1-ample if a large multiple of it is generated by global sections and a morphism induced by the evaluation of the spanning sections is generically finite and has at most 1-dimensional fibers. A vector bundle is called big and 1-ample if the relative hyperplane line bundle over its projectivisation is big and 1-ample.
The main theorem of the present paper asserts that any complex projective manifold of dimension 4 or more, whose tangent bundle is big and 1-ample, is equal either to a projective space or to a smooth quadric. Since big and 1-ample bundles are ‘almost’ ample, the present result is yet another extension of the celebrated Mori paper ‘Projective manifolds with ample tangent bundles’ (Ann. of Math. 110 (1979) 593–606).
The proof of the theorem applies results about contractions of complex symplectic manifolds and of manifolds whose tangent bundles are numerically effective. In the appendix we re-prove these results. 2000 Mathematics Subject Classification 14E30, 14J40, 14J45, 14J50.
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.
In this paper, we develop the theory of singular Hermitian metrics on vector bundles. As an application, we give a structure theorem of a projective manifold X with pseudo-effective tangent bundle; X admits a smooth fibration to a flat projective manifold Y such that its general fibre is rationally connected. Moreover, by applying this structure theorem, we classify all the minimal surfaces with pseudo-effective tangent bundle and study general nonminimal surfaces, which provide examples of (possibly singular) positively curved tangent bundles.
We provide enumerative formulas for the degrees of varieties parameterizing hypersurfaces and complete intersections which contain projective subspaces and conics. Besides, we find all cases where the Fano scheme of the general complete intersection is irregular of dimension at least 2, and for the Fano surfaces we deduce formulas for their holomorphic Euler characteristic.
This book can form the basis of a second course in algebraic geometry. As motivation, it takes concrete questions from enumerative geometry and intersection theory, and provides intuition and technique, so that the student develops the ability to solve geometric problems. The authors explain key ideas, including rational equivalence, Chow rings, Schubert calculus and Chern classes, and readers will appreciate the abundant examples, many provided as exercises with solutions available online. Intersection is concerned with the enumeration of solutions of systems of polynomial equations in several variables. It has been an active area of mathematics since the work of Leibniz. Chasles' nineteenth-century calculation that there are 3264 smooth conic plane curves tangent to five given general conics was an important landmark, and was the inspiration behind the title of this book. Such computations were motivation for Poincaré's development of topology, and for many subsequent theories, so that intersection theory is now a central topic of modern mathematics.
In this paper, we pose several conjectures on structures and images of maximal rationally connected fibrations of smooth projective varieties admitting semi-positive holomorphic sectional curvature. Toward these conjectures, we prove that the numerical dimension of images of such fibrations is zero under the assumption of the abundance conjecture. As an application, we show that any compact Kaehler surface with semi-positive holomorphic sectional curvature is rationally connected, or a complex torus, or a ruled surface over an elliptic curve.
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.
We study how the geometry of a projective variety X is reflected in the positivity properties of the diagonal , considered as a cycle on . We analyze when the diagonal is big, when it is nef, and when it is rigid. In each case, we give several implications for the geometric properties of X. For example, when the diagonal is big, we prove that the Hodge groups vanish for . We also classify varieties of low dimension where the diagonal is nef and big.
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.
Graph theory has experienced a tremendous growth during the 20th century. One of the main reasons for this phenomenon is the applicability of graph theory in other disciplines such as physics, chemistry, psychology, sociology, and theoretical computer science. This book aims to provide a solid background in the basic topics of graph theory. It covers Dirac's theorem on k-connected graphs, Harary-Nashwilliam's theorem on the hamiltonicity of line graphs, Toida-McKee's characterization of Eulerian graphs, the Tutte matrix of a graph, Fournier's proof of Kuratowski's theorem on planar graphs, the proof of the nonhamiltonicity of the Tutte graph on 46 vertices and a concrete application of triangulated graphs. The book does not presuppose deep knowledge of any branch of mathematics, but requires only the basics of mathematics. It can be used in an advanced undergraduate course or a beginning graduate course in graph theory.
Algebraic geometry has benefited enormously from the powerful general machinery developed in the latter half of the twentieth century. The cost has been that much of the research of previous generations is in a language unintelligible to modern workers, in particular, the rich legacy of classical algebraic geometry, such as plane algebraic curves of low degree, special algebraic surfaces, theta functions, Cremona transformations, the theory of apolarity and the geometry of lines in projective spaces. The author's contemporary approach makes this legacy accessible to modern algebraic geometers and to others who are interested in applying classical results. The vast bibliography of over 600 references is complemented by an array of exercises that extend or exemplify results given in the book.
We study the geometry of the secant and tangential variety of a cominuscule
and minuscule variety, e.g. a Grassmannian or a spinor variety. Using methods
inspired by statistics we provide an explicit local isomorphism with a product
of an affine space with a variety which is the Zariski closure of the image of
a map defined by generalized determinants. In particular, equations of the
secant or tangential variety correspond to relations among generalized
determinants. We also provide a representation theoretic decomposition of
cubics in the ideal of the secant variety of any Grassmannian.
The first edition of this well-known and very popular standard text on compact complex surfaces was published in 1984, back then written by W. Barth, C. Peters and A. Van de Ven (1984; Zbl 0718.14023). Apart from the modern treatises on complex surfaces by I. R. Shafarevich, B. G. Averbukh, J. R. Vajnberg, A. B. Zizhchenko, Yu. I. Manin, B. G. Mojshezon, G. N. Tyurina, A. N. Tyurin and A. Beauville [Astérisque 54, 1–172 (1978; Zbl 0394.14014)], the first edition of the book under review offered the only up-to-date account of the subject in textbook form. Moreover, for almost twenty years it has been by far the most comprehensive textbook on complex surfaces from the modern point of view. The first edition contained eight main chapters on about 300 pages, concluding with the classification of K3 surfaces and Enriques surfaces. The book under review is the second, substantially enlarged edition of this standard text, this time with K. Hulek as fourth co-author. In fact, in the two decades after the appearance of the first edition of the book, several crucial developments in the theory of complex surfaces have taken place, and the authors have taken the opportunity to update the original text by including some of those recent achievements. The most important progress in the theory of complex surfaces has been made in regard of a better understanding of their differentiable structure (as real 4-manifolds), not at least in view of their appearance in mathematical physics. The new invariants discovered by Donaldson, on the one hand, and by Seiberg and Witten, on the other hand, stand for these spectacular recent developments. Other far-reaching achievements have been obtained by means of the study of nef-divisors on surfaces, parallel to progress made in the birational classification of higher-dimensional algebraic varieties, and also the Kähler structures on complex surfaces are now much better understood. Finally, I. Reider’s new approach to adjoint mappings, Bogomolov’s inequality for the Chern classes of rank-2 vector bundles on surfaces, and the mirror symmetry properties of K3 surfaces represent just as important new insights in the theory of complex surfaces. Well, all these recent developments have been worked into the present new edition, in one way or another, and the text has grown to 436 pages, that is by more than forty percent. Apart from the correction of some minor irregularities in the first edition, the authors have left the well-proven original text basically intact. The enlargement of the material has been contrived by the addition of the new Chapter 9 entitled “Topological and Differentiable Structure of Surfaces”, including an introduction to Donaldson and Seiberg-Witten invariants (mainly by instructive examples), and by substantially enhancing Chapter 4 (“Some General Properties of Surfaces”). This chapter comes now with twelve (instead of eight) sections and includes the above-mentioned topics such as the nef cone, Bogomolov’s inequality, Reider’s method, and the existence of Kähler metrics on surfaces. There are also some other refining polishings and rearrangements in Chapter 5 (“Examples”) and Chapter 8 (“K3-Surfaces and Enriques Surfaces”). As to the contents of Chapter 8, three sections on special topics have been added, too, discussing the mirror symmetry phenomenon for projective K3-surfaces, special curves on K3-surfaces and an application to hyperbolic geometry, respectively. Needless to say, the bibliography has been updated and tremendously enlarged, thereby reflecting the vast activity in the field during the past twenty years. Now as before, the text is enriched by numerous instructive examples, but there are still no exercises for self-control, stimulus for further reading, or different outlooks. All in all, the second, enlarged edition of this (meanwhile classic) textbook on complex surfaces has gained a good deal of topicality and disciplinary depth, while having maintained its high degree of systematic methodology, lucidity, rigor, and cultured style. No doubt, this book remains a must for everyone dealing with complex algebraic surfaces, be it a student, an active researcher in complex geometry, or a mathematically ambitioned (quantum) physicist.
One of the major discoveries of the last two decades of the twentieth century in algebraic geometry is the realization that the theory of minimal models of surfaces can be generalized to higher dimensional varieties. This generalization, called the minimal model program or Mori's program, has developed into a powerful tool with applications to diverse questions in algebraic geometry and beyond. This book provides the a comprehensive introduction to the circle of ideas developed around the program, the prerequisites being only a basic knowledge of algebraic geometry. It will be of great interest to graduate students and researchers working in algebraic geometry and related fields.
This paper gives a computation of the irregularity of the Fano surface of lines on the double cover branched in a quartic. A tangent bundle theorem is proved for?, from which it follows that determines uniquely. It is shown that the Abel-Jacobi map is an isogeny.
Bibliography: 8 titles.
This paper determines under which conditions an algebraic variety can be uniquely (up to equivalence) represented in the form of a conic bundle. The results are used to show that many conic bundles over rational varieties are nonrational, and to construct examples of nonrational algebraic threefolds whose three-dimensional integral cohomology group is trivial.
Bibliography: 16 titles.
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→∞.
A smooth projective algebraic 3-fold V over the field C is called a Fano 3-fold if the anticanonical divisor — K v is ample. The integer g = g(V) = ½(- K v ) 3 is called the genus of the Fano 3-fold V . The maximal integer r ≧ 1 such that ϑ(— K v )≃ ℋ r for some (ample) invertible sheaf ℋ ε Pic V is called the index of the Fano 3-fold V . Let V be a Fano 3-fold of the index r = 2 and the genus g = 21 which has the second Betti number b 2 (V) = 1. Then V can be embedded in P 6 with degree 5, by the linear system |ℋ| , where ϑ(— K v ) ≃ ℋ 2 (see Iskovskih [5]). We denote this Fano 3-fold V by V 5 .
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.
We study the geometry of projective manifolds whose tangent bundles are nef on sufficiently general curves (i.e. the tangent bundle is generically nef) and show that manifolds whose anticanonical bundles are semi-ample have this property. Furthermore we introduce a notion of sufficient nefness and investigate the relation with manifolds whose anticanonical bundles are nef.
- Beltrametti M. C.
del Pezzo surfaces of degree four
- Kunyavskiĭ B. È
Surfaces de del pezzo: V - modèles anticanoniques.Séminaire sur les singularités des surfaces1976-1977
- M Demazure