arXiv research

We use Morse theory to prove that the Lefschetz Hyperplane Theorem holds for compact smooth Deligne-Mumford stacks over the site of complex manifolds. For $Z \subset X$ a hyperplane section, $X$ can be obtained from $Z$ by a sequence of deformation retracts and attachments of high-dimensional finite disc quotients. We …

The Lefschetz hyperplane section theorem asserts that an affine variety is homotopy equivalent to a space obtained from its generic hyperplane section by attaching some cells. The purpose of this paper is to describe attaching maps of these cells for the complement of a complex hyperplane arrangement defined over real …

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 i-th rational Betti number. For the proof, we provide a Lefschetz-type hyperplane theorem for complements of 2-arrangements, and introduce Alexander duality for co…

Extends Donaldson's techniques to symplectic orbifolds, proving existence of sections and computing cohomology.

A foliation $(M,\mathcal{F})$ is said to be $2$--calibrated if it admits a closed 2-form $ω$ making each leaf symplectic. By using approximately holomorphic techniques, a sequence $W_k$ of $2$--calibrated submanifolds of codimension--$2$ can be found for $(M, \mathcal{F}, ω)$. Our main result says that the Lefschetz hy…

The study describes handle decompositions and Kirby diagrams for line arrangements.

The main goal of this work is to present a detailed study of the foundations of Complex Geometry, highlighting its geometrical, topological and analytical aspects. Beginning with a preliminary material, such as the basic results on holomorphic functions in one or more variables and the definition and first examples of …

We prove a Lefschetz hyperplane theorem for the determinantal loci of a morphism between two holomorphic vector bundles $E$ and $F$ over a complex manifold under the condition that $E^*\ox F$ is Griffiths $k$-positive. We apply this result to find some homotopy groups of the Brill-Noether loci for a generic curve.

We construct symplectic submanifolds of symplectic manifolds with contact border. The boundary of such submanifolds is shown to be a contact submanifold of the contact border. We also give a topological characterization of the constructed submanifolds by means of a ``relative Lefschetz hyperplane Theorem''. We sketch s…

We obtain infinitely many (non-conjugate) representations of 3-manifold fundamental groups into a lattice in the holomorphic isometry group of complex hyperbolic space. The lattice is an orbifold fundamental group of a branched covering of the projective plane along an arrangement of hyperplanes constructed by Hirzebru…

The Hard Lefschetz Theorem extends to certain Kähler Lie Algebroids with ellipticity.

This short review is the result of a minicourse at the Sapienza University of Rome the author gave about the proof of the $g$-theorem. We review the hard Lefschetz theorem for simplicial spheres, as well as the theory at its core: perturbations of maps, biased Poincaré pairings and a cobordism argument that relates the…

Suppose one is given a discrete group G, a cocompact proper G-manifold M, and a G-self-map f of M. Then we introduce the equivariant Lefschetz class of f, which is globally defined in terms of cellular chain complexes, and the local equivariant Lefschetz class of f, which is locally defined in terms of fixed point data…

In this paper we obtain theorems of Barth-Lefschetz type in Sasakian geometry. As corollaries, this implis connectedness principle and Frankel's type theorem.

We employ a certain labeled finite graph, called a chart, in a closed oriented surface for describing the monodromy of a(n achiral) Lefschetz fibration over the surface. Applying charts and their moves with respect to Wajnryb's presentation of mapping class groups, we first generalize a signature formula for Lefschetz …

Paper proves a Cohen-Dimca-Orlik type theorem for Z-local systems of hyperplane arrangements.

The study shows that symplectic Lefschetz fibrations can have infinitely many sections.

Proves hard Lefschetz theorem and Hodge-Riemann relations for convex valuations.

Using an approach based on the heat kernel we prove an Atiyah-Bott-Lefschetz theorem for the $L^2-$Lefschetz numbers associated to an elliptic complex of cone differential operators over a compact manifold with conical singularities. We then apply our results to the case of the de Rham complex.

Chart descriptions are a graphic method to describe monodromy representations of various topological objects. Here we introduce a chart description for genus-two Lefschetz fibrations, and show that any genus-two Lefschetz fibration can be stabilized by fiber-sum with certain basic Lefschetz fibrations.

A theorem divides hyperplanes evenly with a line through the origin.

In the literature, there are two different versions of Hard Lefschetz theorems for a compact Sasakian manifold. The first version, due to Kacimi-Alaoui, asserts that the basic cohomology of a compact Sasakian manifold satisfies the transverse Lefschetz property. The second version, established far more recently by Capp…

The paper proves a Hard Lefschetz Theorem for push-forwards of polarized twistor modules.

A notion of equivariant spectral flows for families of self-dual elliptic operators on Riemannian manifolds is purposed. As a consequence, a local version of a Lefschetz fix point theorem is proved for Toeplitz operators on odd-dimensional spin manifolds.

We study the transversal hard Lefschetz theorem on a transversely symplectic foliation. This article extends the results of transversally symplectic flows (H.K.~Pak, "Transversal harmonic theory for transversally symplectic flows", J. Aust. Math. Soc. 84 (2008), 233--245) to the general transversely symplectic foliatio…

Proves a limit on hyperplanes in complex manifolds.

Using theorems of Eliashberg and McDuff, Etnyre [Et] proved that the intersection form of a symplectic filling of a contact 3-manifold supported by planar open book is negative definite. In this paper, we prove a signature formula for allowable Lefschetz fibrations over $D^2$ with planar fiber by computing Maslov index…

We establish a Hard Lefschetz Theorem for the de Rham cohomology of compact Vaisman manifolds. A similar result is proved for the basic cohomology with respect to the Lee vector field. Motivated by these results, we introduce the notions of a Lefschetz and of a basic Lefschetz locally conformal symplectic (l.c.s.) mani…

We study the properties of the multiplicative structure on valuations on convex sets. We prove a new version of the hard Lefschetz theorem for even translation invariant continuous valuations, and discuss related problems of integral geometry. Then we formulate a conjectural analogue of this result for odd valuations.

Study $L^{2}$-harmonic forms on almost Kähler manifolds, extending vanishing theorems.

The paper proves conditions for zero Gaussian curvature convex hypersurfaces to be hyperplanes.

We provide a simpler proof of the hard Lefschetz Theorem for face rings of PL spheres: While the algebraic theory remains the same, we replace the geometric constructions by Pachner's Theorem. This simplifies the reasoning for an important special case of the main result of the first author in arxiv:1812.10454, and alr…

Auroux, Donaldson and Katzarkov introduced broken Lefschetz fibrations as a generalization of Lefshcetz fibrations in order to describe near-symplectic 4-manifolds. We first study monodromy representations of higher sides of genus-1 simplified broken Lefschetz fibrations. We then completely classify diffeomorphism type…

We give a general treatment of the somewhat unfamiliar operation on manifolds called Connected Sum at Infinity, or CSI for short. A driving ambition has been to make the geometry behind the well definition and basic properties of CSI as clear and elementary as possible. CSI then yields a very natural and elementary pro…

Symplectic structures simplified for compact manifolds.

Study properties of balanced hyperbolic compact complex manifolds.

Study proves minimality of certain hyperplane intersections in wide cones.

We prove that a hyperplane in a CAT(0) cubical complex X has no self-intersections and separates X into two convex complementary components. These facts were originally proved by Sageev. Our argument shows that his theorem is a corollary of Gromov's link condition. We also give new arguments establishing some combinato…

The paper embeds 4-manifolds into CP^2 x CP^1 using Lefschetz fibrations.

We introduce a Lefschetz filtration for integer cohomology and explore its applications.

We find a family of five dimensional completely solvable compact manifolds that constitute the first examples of $K$-contact manifolds which satisfy the Hard Lefschetz Theorem and have a model of Tievsky type just as Sasakian manifolds but do not admit any Sasakian structure.

Much work has been done on the existence and uniqueness of broken Lefschetz fibrations such as those by Auroux et al., Gay and Kirby, Lekili, Akbulut and Karakurt, Baykur, and Williams, but there has been a lack of explicit examples. A theorem of Gay and Kirby suggests the existence of a broken Lefschetz fibration of S…

We obtain new general results on the structure of the space of translation invariant continuous valuations on convex sets (a version of the hard Lefschetz theorem). Using these and our previous results we obtain explicit characterization of unitarily invariant translation invariant continuous valuations. It implies new…

We prove that a Lefschetz fibration over the disc that, after compactification, has the same singular fibers as an extremal rational elliptic surface can be obtained by deleting a singular fiber and a section from the rational extremal elliptic surface, i.e. such a Lefschetz fibration is determined up to topological eq…

In this paper, we prove a monotonicity formula and some Bernstein type results for translating solitons of hypersurfaces in $\re^{n+1}$, giving some conditions under which a trantranslating soliton is a hyperplane. We also show a gap theorem for the translating soliton of hypersurfaces in $R^{n+k}$, namely, if the $L^n…

We construct two types of non-holomorphic Lefschetz fibrations over $S^2$ with $(-1)$-sections ---hence, they are fiber sum indecomposable--- by giving the corresponding positive relators. One type of the two does not satisfy the slope inequality (a necessary condition for a fibration to be holomorphic) and has a simpl…

The paper proves a theorem linking convex body centroids and category theory.

Formula connects knot invariant to Lefschetz number, proving special case for Seifert solids.