Related papers: Counterexamples of Lefschetz hyperplane type resul…

We prove a Lefschetz hypersurface theorem for abelian fundamental groups allowing wild ramification along some divisor. In fact, we show that isomorphism holds if the degree of the hypersurface is large relative to the ramification along…

We prove that the cone of mobile divisors and the cone of curves birationally movable in codimension 1 are dual in the $(K+B)$-negative part for a klt pair $(X/Z,B)$. We also prove the structure theorem and the contraction theorem for the…

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.

We establish consequences of the moving form of Schmidt's Subspace Theorem. Indeed, we obtain inequalities that bound the logarithmic greatest common divisor of moving multivariable polynomials evaluated at moving $S$-unit arguments. In…

The Hard Lefschetz theorem is known to hold for the intersection cohomology of the toric variety associated to a rational convex polytope. One can construct the intersection cohomology combinatorially from the polytope, hence it is well…

We prove that in any strictly convex symmetric cone $\Omega$ there exists a non empty locus where the WDVV equation is satisfied (i.e. there exists a hyperplane being a Frobenius manifold). This result holds over any real division algebra…

In this note we relate the notions of Lefschetz type, decomposability, and isomorphism, on Chow motives with the notions of unit type, decomposability, and isomorphism, on noncommutative motives. Examples, counter-examples, and applications…

Let X be a smooth Mori dream space of dimension at least 4. We show that, if X satisfies a suitable GIT condition which we call "small unstable locus", then every smooth ample divisor Y of X is also a Mori dream space. Moreover, the…

We show a Lefschetz theorem for irreducible overconvergent $F$-isocrystals on smooth varieties defined over a finite field. We derive several consequences from it.

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 $\Theta$.

An explanation is given for the initially surprising ubiquity of separating sets in normal complex surface germs. It is shown that they are quite common in higher dimensions too. The relationship between separating sets and the geometry of…

For a given pair of maps f,g:X->M from an arbitrary topological space to an n-manifold, the Lefschetz homomorphism is a certain graded homomorphism L:H(X)->H(M) of degree (-n). We prove a Lefschetz-type coincidence theorem: if the Lefschetz…

We discuss some variants of cone theorem for movable curves in any codimensions.

The concept of separation by hyperplanes is fundamental for convex geometry and its tropical (max-plus) analogue. However, analogous separation results in max-min convex geometry are based on semispaces. This paper answers the question…

The goal of this work is to study positivity of subvarieties with nef normal bundle in the sense of intersection theory. After Ottem's work on ample subschemes, we introduce the notion of a nef subscheme, which generalizes the notion of a…

A foliation $(M,\mathcal{F})$ is said to be $2$--calibrated if it admits a closed 2-form $\omega$ making each leaf symplectic. By using approximately holomorphic techniques, a sequence $W_k$ of $2$--calibrated submanifolds of…

The quantum hyperplane section theorem is explained for nonnegative decomposable concavex bundle spaces over generalized flag manifolds.

The separation of two sets (or more specific of two cones) plays an important role in different fields of mathematics such as variational analysis, convex analysis, convex geometry, optimization. In the paper, we derive some new results for…

We describe the effective and the big cones of a projective symmetric variety. Moreover, we give a necessary and sufficient combinatorial criterion for the bigness of a nef divisor on a projective symmetric variety. When the variety is…

We present a shift theorem for solutions of the Poisson equation in a finite planar cone (and hence also on plane polygons) for Dirichlet, Neumann, and mixed boundary conditions. The range in which the shift theorem holds depends on the…