Related papers: Beyond the Elliptic Genus

In this paper we show that for m>n the set of cobordism classes of maps from m-sphere to n-sphere is trivial. The determination of the cobordism homotopy groups of spheres admits applications to the covers for spheres.

For a complete noncompact connected Riemannian manifold with bounded geometry, we prove the existence of isoperimetric regions in a larger space obtained by adding finitely many limit manifolds at infinity. As one of many possible…

For several embedded surfaces with zero self-intersection number in 4-manifolds, we show that an adjunction-type genus bound holds for at least one of the surfaces under certain conditions. For example, we derive certain adjunction…

We introduce Riemannian Lie algebroids as a generalization of Riemannian manifolds and we show that most of the classical tools and results known in Riemannian geometry can be stated in this setting. We give also some new results on the…

We describe the equivariant cobordism ring of smooth toric varieties. This equivariant description is used to compute the ordinary cobordism ring of such varieties.

Any open Riemann surface $R_0$ of finite genus $g$ can be conformally embedded into a closed Riemann surface of the same genus, that is, $R_0$ is realized as a subdomain of a closed Riemann surface of genus $g$. We are concerned with the…

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…

We study the elliptic modular surface attached to the commutator subgroup of the modular group. This has an elliptic curve as base and only one singular fibre. We employ an algebraic approach and then consider some arithmetic questions.

Isoparametric submanifolds and hypersurfaces in space forms are geometric objects that have been studied since E. Cartan. Another important class of geometric objects is the orbits of a polar action on a Riemannian manifold,e.g., the orbits…

Interpreting the number of ramified covering of a Riemann surface by Riemann surfaces as the relative Gromov-Witten invariants and applying a gluing formula, we derive a recursive formula for the number of ramified covering of a Riemann…

We discuss two simple but useful observations that allow the construction of modular forms from given ones using invariant theory. The first one deals with elliptic modular forms and their derivatives, and generalizes the Rankin-Cohen…

K3 surfaces play a prominent role in string theory and algebraic geometry. The properties of their enumerative invariants have important consequences in black hole physics and in number theory. To a K3 surface string theory associates an…

We define the singular orbifold elliptic genus and $E$-function for all normal surfaces without strictly log-canonical singularities, and prove the analogue of the McKay correspondence in this setting. Our invariants generalize the stringy…

It is known that spin cobordism can be determined by Stiefel-Whitney numbers and index theory invariants, namely $KO$-theoretic Pontryagin numbers. In this paper, we show that string cobordism at dimension 24 can be determined by elliptic…

A submanifold of a Riemannian manifold is called a parallel submanifold if its second fundamental form is parallel with respect to the van der Waerden-Bortolotti connection. From submanifold point of view, parallel submanifolds are the…

We introduce a diffeomorphism invariant of $4$-manifolds, the $\mathrm{Pin}^-(2)$-monopole invariant, defined by using the $\mathrm{Pin}^-(2)$-monopole equations. We compute the invariants of several $4$-manifolds, and prove gluing…

We explicitly construct the Kummer variety associated to the Jacobian of a hyperelliptic curve of genus 3 that is defined over a field of characteristic not equal to 2 and has a Weierstra{\ss} point defined over the same field. We also…

We give a K-theory proof of the invariance under cobordism of the family index. We consider elliptic pseudodifferential families on a continuous fibre bundle with smooth fibres over a compact base space B, and define a notion of cobordant…

Borisov and Libgober recently proved a conjecture of Dijkgraaf, Moore, Verlinde, and Verlinde on the elliptic genus of a Hilbert scheme of points on a surface. We show how their result can be used together with our work on complex genera of…

In this article, we study the $(m+2)$-angulations on a Riemann surface, characterized with its boundary components, punctures, and gender. We count the number of arcs in such a surface, and associate a graded quiver with superpotential…