Related papers: Lefschetz Theorem Using Real Morse Theory

While not obvious from its initial motivation in linear algebra, there are many context where iterated traces can be defined. In this paper we prove a very general theorem about iterated 2-categorical traces. We show that many…

The Lefschetz fixed point theorem and its converse have many generalizations. One of these generalizations is to endomorphisms of a space relative to a fixed subspace. In this paper we define relative Lefschetz numbers and Reidemeister…

The classical Lefschetz fixed point theorem states that the number of fixed points, counted with multiplicity $\pm 1$, of a smooth map $f$ from a manifold $M$ to itself can be calculated as the alternating sum $\sum (-1)^k \textrm{ tr }…

The Whitney-Graustein theorem states that regular closed curves in the 2-plane are classified, up to regular homotopy, by their rotation number. Here we give a simple proof based on contact geometry.

In this paper we compute the motivic Chern classes and homology Hirzebruch characteristic classes of (possibly singular) toric varieties, which in the complete case fit nicely with a generalized Hirzebruch-Riemann-Roch theorem. As special…

The Riemann-Roch Theorem is one of the cornerstones of algebraic geometry, connecting algebraic data (sheaf cohomology) with geometric ones (intersection theory). This survey paper provides a self-contained introduction and a complete proof…

We prove the $P=W$ conjecture for $\mathrm{GL}_n$ for all ranks $n$ and curves of arbitrary genus $g\geq 2$. The proof combines a strong perversity result on tautological classes with the curious Hard Lefschetz theorem of Mellit. For the…

In the article the necessary and sufficient conditions for a representation of Lipschitz function of more than two variables as a difference of two convex functions are formulated. An algorithm of this representation is given. The outcome…

Suppose $Y$ is a smooth variety equipped with a top form. We prove a simple theorem giving a sharp lower bound on the geometric genus of a family of subvarieties of $Y$, in terms of the dimension of this family. Two elementary applications…

We show that on a certain hypersurface in P3 there is a (q3 + q2 + q + 1)q+1-symmetric configuration (resp. a ((q3 + 1)(q2 + 1)q+1, (q3 + 1)(q + 1)q2+1)) -configuration) made up of the rational points over Fq (resp. over Fq2) and the lines…

Hurwitz numbers count ramified covers of a Riemann surface with prescribed monodromy. As such, they are purely combinatorial objects. Tautological classes, on the other hand, are distinguished classes in the intersection ring of the moduli…

We formulate and prove a generalization of Zariski-van Kampen theorem on the topological fundamental groups of smooth complex algebraic varieties. As an application, we prove a hyperplane section theorem of Lefschetz-Zariski-van Kampen type…

It is known that in the case of hyperelliptic curves the Shafarevich conjecture can be made effective, i.e., for any number field k and any finite set of places S of k, one can effectively compute the set of isomorphism classes of…

Hurwitz numbers enumerate branched morphisms between Riemannn surfaces with fixed numerical data. They represent important objects in enumerative geometry that are accessible by combinatorial techniques. In the past decade, many variants of…

Using an explicit family of plane quartic curves, we prove the existence of a genus 3 curve over any finite field of characteristic 3 whose number of rational points stays within a fixed distance from the Hasse-Weil-Serre upper bound. We…

We show the existence of toric resolution tower for an irreducible curve singularity which is explicitly described by Tschirnhausen polynomials. We deduce for a smooth affine plane curve from its topology restrictions for its singularity at…

Symplectic Lefschetz fibrations can be described via classifying maps with values in the Deligne-Mumford compactification of the moduli space of curves, by means of constructions relying on symplectic geometry. In this note we prove the…

We introduce invariants of Hurwitz equivalence classes with respect to arbitrary group $G$. The invariants are constructed from any right $G$-modules $M$ and any $G$-invariant bilinear function on $M$, and are of bilinear forms. For…

Using the existence of certain symplectic submanifolds in symplectic 4-manifolds, we prove an estimate from above for the number of singular fibers with separating vanishing cycles in minimal Lefschetz fibrations over surfaces of positive…

We present existence results for certain singular 2-dimensional foliations on 4-manifolds. The singularities can be chosen to be simple, e.g. the same as those that appear in Lefschetz pencils. There seems to be a wealth of such creatures…