Related papers: Bloch's conjecture for some numerical Campedelli s…
A well-known conjecture asserts that the mapping class group of a surface (possibly with punctures/boundary) does not virtually surject onto $\Z$ if the genus of the surface is large. We prove that if this conjecture holds for some genus,…
Assuming the finiteness of the Shafarevich-Tate group of elliptic curves over number fields we make several observations on the birational Grotendieck anabelian setion conjecture. We prove that the birational setion conjecture for curves…
We prove the numerical Hodge standard conjecture for the square of a simple abelian variety of prime dimension, and also in some related cases.
We construct a new minimal complex surface of general type with $p_g=0$, $K^2=2$ and $H_1=\mathbb{Z}/4\mathbb{Z}$ (in fact $\pi_1^{\text{alg}}=\mathbb{Z}/4\mathbb{Z}$), which settles the existence question for numerical Campedelli surfaces…
We study the enumerative geometry of algebraic curves on abelian surfaces and threefolds. In the abelian surface case, the theory is parallel to the well-developed study of the reduced Gromov-Witten theory of K3 surfaces. We prove complete…
The Beilinson--Bloch conjecture is a generalization of the Birch and Swinnerton-Dyer conjecture, which relates the ranks of Chow groups of smooth projective varieties over global fields to the order of vanishing of $L$-functions. We prove…
We determine the class of the Hilbert scheme of points on a surface in the Grothendieck group of varieties. As a corollary we obtain its class in the Grothendieck group of motives. We give some applications to moduli spaces of sheaves on…
Motivated by various equivalent versions of the SHGH conjecture for $\mathbb{P}^2$ blown up at very general points, we propose a similar conjecture for Hirzebruch surfaces. We prove that this conjecture is true for the Hirzebruch surface…
We give a new proof of the main theorem in the theory of C(6) small cancellation complexes. We prove the fundamental theorem of cubical small cancellation theory for C(9) cubical small cancellation complexes.
A classical discovery known as Fenchel's conjecture and proved in the 1950s, shows that every co-compact Fuchsian group $F$ has a normal subgroup of finite index isomorphic to the fundamental group of a compact unbordered orientable…
This note provides a formula for the character of the Lie algebra of the fundamental group of a surface, viewed as a module over the symplectic group.
I give a conjectural generating function for the numbers of $\delta$-nodal curves in a linear system of dimension $\delta$ on an algebraic surface. It reproduces the results of Vainsencher for the case $\delta\le 6$ and Kleiman-Piene for…
We take the fundamental group of the complement of the branch curve of a generic projection induced from canonical embedding of a surface. This group is stable on connected components of moduli spaces of surfaces. Since for many classes of…
We propose new conjectures about the relationship between the principal blocks of finite groups for different primes and establish evidence for these conjectures.
We prove Cherlin's conjecture, concerning binary primitive permutation groups, for those groups with socle isomorphic to a sporadic simple group.
We extend a conjecture of Kimberley-Robertson on the abelianizations of certain square complex groups.
The purpose of this article is to provide a version of Bloch-Wigner theorem over the class of rings with many units.
Nahm's conjecture relates $q$-hypergeometric modular functions to torsion elements in the Bloch group. An interesting class of such functions can be (conjecturally) obtained from a pair $(X,X')$ of diagrams, each of which is either a Dynkin…
In this paper, we classify the minimal surfaces of general type with $\chi=5$, $K^{2}=9$ whose canonical map is composed with an involution. We obtain 6 families, whose dimensions in the moduli space are 28, 27, 33, 32, 31 and 32…
We compute the fundamental group of moduli spaces of Lie group valued representations of surface and torus groups.