Lefschetz Hyperplane Theorem Research Articles

Lefschetz theorems in flat cohomology and applications

We prove a version of the Lefschetz hyperplane theorem for fppf cohomology with coefficients in any finite commutative group scheme over the ground field. As consequences, we establish new Lefschetz results for the Picard scheme.

Asymptotically holomorphic theory for symplectic orbifolds

We extend Donaldson’s asymptotically holomorphic techniques to symplectic orbifolds. More precisely, given a compact symplectic orbifold such that the symplectic form defines an integer cohomology class, we prove that there exist sections of large tensor powers of the prequantizable line bundle such that their zero sets are symplectic suborbifolds. We then derive a Lefschetz hyperplane theorem for these suborbifolds, that computes their real cohomology up to middle dimension. We also get the hard Lefschetz and formality properties for them, when the ambient orbifold satisfies those properties.

Vanishing cohomology and Betti bounds for complex projective hypersurfaces

We employ the formalism of vanishing cycles and perverse sheaves to introduce and study the vanishing cohomology of complex projective hypersurfaces. As a consequence, we give upper bounds for the Betti numbers of projective hypersurfaces, generalizing those obtained by different methods by Dimca in the isolated singularities case, and by Siersma-Tib\u{a}r in the case of hypersurfaces with a $1$-dimensional singular locus. We also prove a supplement to the Lefschetz hyperplane theorem for hypersurfaces, which takes the dimension of the singular locus into account, and we use it to give a new proof of a result of Kato.

A Lefschetz hyperplane theorem for non-Archimedean Jacobians

We establish a Lefschetz hyperplane theorem for the Berkovich analytifications of Jacobians of curves over an algebraically closed non-Archimedean field. Let J be the Jacobian of a curve X, and let $$W_{d} \subset J$$ be the locus of effective divisor classes of degree d. We show that the pair $$(J^{an},W_d^{an})$$ is d-connected, and thus in particular the inclusion of the analytification of the theta divisor $$\Theta ^{an}$$ into $$J^{an}$$ satisfies a Lefschetz hyperplane theorem for $${\mathbb {Z}}$$ -cohomology groups and homotopy groups. A key ingredient in our proof is a generalization, over arbitrary characteristics and allowing arbitrary singularities on the base, of a result of Brown and Foster for the homotopy type of analytic projective bundles.

On a Lefschetz-type phenomenon for elliptic Calabi-Yaus

We consider 18 families of elliptic Calabi-Yaus which arise in constructing F-theory compactifications of string vacua, and show in each case that the upper Hodge diamond of a crepant resolution of the associated Weierstrass model coincides with the upper Hodge diamond of the (blown up) projective bundle in which the crepant resolution is naturally embedded. Such results are unexpected, as each crepant resolution we consider does not satisfy the hypotheses of the Lefschetz hyperplane theorem. In light of such findings, we suspect that all elliptic Calabi-Yaus satisfy such a ‘Lefschetz-type phenomenon’.

A homotopy exact sequence for overconvergent isocrystals

AbstractIn this article we prove exactness of the homotopy sequence of overconvergent fundamental groups for a smooth and projective morphism in characteristicp. We do so by first proving a corresponding result for rigid analytic varieties in characteristic$0$, following dos Santos [dS15] in the algebraic case. In characteristicp, we then proceed by a series of reductions to the case of a liftable family of curves, where we can apply the rigid analytic result. We then use this to deduce a Lefschetz hyperplane theorem for convergent fundamental groups, as well as a comparison theorem with the étale fundamental group.

On the Weak Lefschetz Principle in Birational Geometry

This is an expository article written for the Notices of the AMS in which we discuss the weak Lefschetz Principle in birational geometry. Our departing point is the influential work of Solomon Lefschetz started in 1924. Indeed, we look at the original formulation of the Lefschetz hyperplane theorem in algebraic topology and build up to recent developments of it in birational geometry. In doing so, the main theme of the article is the following: there are many scenarios in geometry in which analogous versions of the Lefschetz hyperplane theorem hold. These scenarios are somewhat unexpected and have had a profound impact in mathematics.

Cohomology of vector bundles and non-pluriharmonic loci

In this paper, we study cohomology groups of vector bundles on neighborhoods of a non-pluriharmonic locus in Stein manifolds and in projective manifolds. By using our results, we show variants of the Lefschetz hyperplane theorem.

The Motive of a Smooth Theta Divisor

We prove a motivic version of the Lefschetz hyperplane theorem for a smooth ample divisor $\Theta$ on an Abelian variety. We use this to construct a motive $P$ that realizes the primitive cohomology of $\Theta$.

Towards a quantum Lefschetz hyperplane theorem in all genera

An effective algorithm of determining Gromov--Witten invariants of smooth hypersurfaces in any genus (subject to a degree bound) from Gromov--Witten invariants of the ambient space is proposed. The Appendix is joint with E. Schulte-Geers.

THE FOLIATED LEFSCHETZ HYPERPLANE THEOREM

A foliation$(M,{\mathcal{F}})$is said to be$2$-calibrated if it admits a closed 2-form$\unicode[STIX]{x1D714}$making each leaf symplectic. By using approximately holomorphic techniques, a sequence$W_{k}$of$2$-calibrated submanifolds of codimension-$2$can be found for$(M,{\mathcal{F}},\unicode[STIX]{x1D714})$. Our main result says that the Lefschetz hyperplane theorem holds for the pairs$(F,F\cap W_{k})$, with$F$any leaf of${\mathcal{F}}$. This is applied to draw important consequences on the transverse geometry of such foliations.

Quantum periods for 3–dimensional Fano manifolds

The quantum period of a variety X is a generating function for certain Gromov-Witten invariants of X which plays an important role in mirror symmetry. In this paper we compute the quantum periods of all 3-dimensional Fano manifolds. In particular we show that 3-dimensional Fano manifolds with very ample anticanonical bundle have mirrors given by a collection of Laurent polynomials called Minkowski polynomials. This was conjectured in joint work with Golyshev. It suggests a new approach to the classification of Fano manifolds: by proving an appropriate mirror theorem and then classifying Fano mirrors. Our methods are likely to be of independent interest. We rework the Mori-Mukai classification of 3-dimensional Fano manifolds, showing that each of them can be expressed as the zero locus of a section of a homogeneous vector bundle over a GIT quotient V/G, where G is a product of groups of the form GL_n(C) and V is a representation of G. When G=GL_1(C)^r, this expresses the Fano 3-fold as a toric complete intersection; in the remaining cases, it expresses the Fano 3-fold as a tautological subvariety of a Grassmannian, partial flag manifold, or projective bundle thereon. We then compute the quantum periods using the Quantum Lefschetz Hyperplane Theorem of Coates-Givental and the Abelian/non-Abelian correspondence of Bertram-Ciocan-Fontanine-Kim-Sabbah.

Combinatorial stratifications and minimality of 2-arrangements

We prove that the complement of any affine 2-arrangement in R d is minimal, that is, it is homotopy equivalent to a cell complex with as many i-cells as its ith rational Betti number. For the proof, we provide a Lefschetz-type hyperplane theorem for complements of 2-arrangements, and introduce Alexander duality for combinatorial Morse functions. Our results greatly generalize previous work by Falk, Dimca–Papadima, Hattori, Randell and Salvetti–Settepanella and others, and they demonstrate that in contrast to previous investigations, a purely combinatorial approach suffices to show minimality and the Lefschetz Hyperplane Theorem for complements of complex hyperplane arrangements.

The perverse filtration and the Lefschetz hyperplane theorem, II

The perverse filtration in cohomology and in cohomology with compact supports is interpreted, in terms of kernels of restriction maps to suitable subvarieties by using the Lefschetz hyperplane theorem and spectral objects. Various mixed-Hodge-theoretic consequences for intersection cohomology and for the decomposition theorem are derived.

The perverse filtration and the Lefschetz hyperplane theorem

The perverse filtration and the Lefschetz hyperplane theorem

Orbifold quantum Riemann–Roch, Lefschetz and Serre

Given a vector bundle $F$ on a smooth Deligne-Mumford stack $\X$ and an invertible multiplicative characteristic class $\bc$, we define the orbifold Gromov-Witten invariants of $\X$ twisted by $F$ and $\bc$. We prove a "quantum Riemann-Roch theorem" which expresses the generating function of the twisted invariants in terms of the generating function of the untwisted invariants. A Quantum Lefschetz Hyperplane Theorem is derived from this by specializing to genus zero. As an application, we determine the relationship between genus-0 orbifold Gromov-Witten invariants of $\X$ and that of a complete intersection. This provides a way to verify mirror symmetry predictions for complete intersection orbifolds.

Stratified Morse Theory: Past and Present

In 1974, Mark Goresky and Robert MacPherson began their development of intersection homology theory (see [24] in these volumes), and their first paper on this topic appeared in 1980; see [12]. At that time, they were missing a fundamental tool which was available for the study of smooth manifolds; they had no Morse Theory for stratified spaces. Goresky and MacPherson wished to have a Stratified Morse Theory to allow them to prove a Lefschetz hyperplane theorem for the intersection homology of complex singular spaces, just as ordinary Morse Theory yields the Lefschetz Hyperplane Theorem for ordinary homology of complex manifolds ([34], §7). The time was ripe for a stratified version of Morse Theory. In 1970, Mather had given a rigorous proof of Thom’s first isotopy lemma [33]; this result says that proper, stratified, submersions are locally-trivial fibrations. In 1973, Morse functions on singular spaces had been defined by Lazzeri in [25], and the density and stability of Morse functions under perturbations had been proved in [37]. We shall recall these definitions and results in Section 2. What was missing was the analog of the fundamental result of Morse Theory, a theorem describing how the topology of a space is related to the critical points of a proper Morse function. In [16], Goresky and MacPherson proved such a theorem for stratified spaces. Suppose that M is a smooth manifold, that X is a Whitney stratified subset ofM , and that f : X → R is a proper function which is the restriction of a smooth function on M . For all v ∈ R, let X≤v := f−1((−∞, v]). Suppose that a, b ∈ R, a < b, and f−1([a, b]) contains a single (stratified) critical point, p, which is non-degenerate (see Definition 2.3) and contained in the open set f−1((a, b)). Let S be the stratum containing p. Then, the Main Theorem of Stratified Morse Theory (see Theorem 2.16) says that the topological space X≤b is obtained from the space X≤a by attaching a space A to X≤a along a subspace B ⊆ A, where the pair (A,B), the Morse data, is the product of the tangential Morse data of f at p and the normal Morse data of f at p. This result is especially powerful in the complex analytic case, where the normal Morse data depends on the stratum S, but not on the point p or on the particular Morse function f . Detailed proofs of these results appeared in the 1988 book Stratified Morse Theory [16]; we present a summary of a number of these results in Section 2. Even before the appearance of [16], Goresky and MacPherson published two papers, Stratified Morse Theory [15] and Morse Theory and Intersection Homology Theory [14], which contained announcements of many of the fundamental definitions and results of Stratified Morse Theory. In addition, these two papers showed that Stratified Morse Theory has a number of important applications to complex analytic spaces, including homotopy results, the desired Lefschetz Hyperplane Theorem for intersection homology, and the first proof that the (shifted) nearby cycles of a perverse sheaf are again perverse. We shall discuss these results and others in Section 3.

SEMIPOSITIVE BUNDLES AND BRILL–NOETHER THEORY

A Lefschetz hyperplane theorem for the determinantal loci of a morphism between two holomorphic vector bundles E and F over a complex manifold is proved, under the condition that E* ⊗ F is Griffiths k-positive. This result is applied to find some homotopy groups of the Brill–Noether loci for a generic curve. 2000 Mathematics Subject Classification 32Q55, 14M12, 14H51.

A new six-dimensional irreducible symplectic variety

We construct a six-dimensional irreducible symplectic variety with $b_2=8$. Since the known examples of irreducible symplectic varieties have $b_2=7$ or $b_2=23$, our variety is in a new deformation class. The example is obtained as follows. Let $J$ be the Jacobian of a genus-two curve with its natural principal polarization: results of another paper of ours give a symplectic desingularization of the moduli space of semistable rank-two sheaves on $J$ with $c_1=0$ and $c_2=2$. Let $\mathcal {M}_{\mathbf {v}}$ be this symplectic desingularization: there is a natural locally trivial fibration $\mathcal {M}_{\mathbf {v}}\rightarrow J\times \widehat {J}$. Our example is the fiber over $(0,\widehat {0})$ of this map, we denote it by $\widetilde {\mathcal {M}}$. The main body of the paper is devoted to the proof that $\widetilde {\mathcal {M}}$ is irreducible symplectic and that $b_2(\widetilde {\mathcal {M}})=8$. Applying the generalized Lefschetz Hyperplane Theorem we get that low-dimensional homotopy (or homology) groups of $\widetilde {\mathcal {M}}$ are represented by homotopy (or homology) groups of a subset of $\widetilde {\mathcal {M}}$ which has an explicit description. The main problem is to provide the explicit description and to extract the necessary information on homotopy or homology groups.

On the Construction of Contact Submanifolds with Prescribed Topology

We prove the existence of contact submanifolds realizing the Poincare dual of the top Chern class of a complex vector bundle over a closed contact manifold. This result is analogue in the contact categoryto Donaldson's construction of symplectic submanifolds. The main tool in the construction is to show the existence of sequences of sections which are asymptotically holomorphic in an appropiate sense and that satisfya transversalitywith es- timates propertydirectlyin the contact category . The description of the ob- tained contact submanifolds allows us to prove an extension of the Lefschetz hyperplane theorem which completes their topological characterization.