Euler characteristics of Hilbert schemes of points on simple surface singularities

We study the geometry and topology of Hilbert schemes of points on the orbifold surface [C^2/G], respectively the singular quotient surface C^2/G, where G is a finite subgroup of SL(2,C) of type A or D. We give a decomposition of the (equivariant) Hilbert scheme of the orbifold into affine space strata indexed by a certain combinatorial set, the set of Young walls. The generating series of Euler characteristics of Hilbert schemes of points of the singular surface of type A or D is computed in terms of an explicit formula involving a specialized character of the basic representation of the corresponding affine Lie algebra; we conjecture that the same result holds also in type E. Our results are consistent with known results in type A, and are new for type D.Comment: 57 pages, final version. To appear in European Journal of Mathematic

Non-commutative Donaldson-Thomas theory and the conifold

Given a quiver algebra A with relations defined by a superpotential, this paper defines a set of invariants of A counting framed cyclic A-modules, analogous to rank-1 Donaldson-Thomas invariants of Calabi-Yau threefolds. For the special case when A is the non-commutative crepant resolution of the threefold ordinary double point, it is proved using torus localization that the invariants count certain pyramid-shaped partition-like configurations, or equivalently infinite dimer configurations in the square dimer model with a fixed boundary condition. The resulting partition function admits an infinite product expansion, which factorizes into the rank-1 Donaldson-Thomas partition functions of the commutative crepant resolution of the singularity and its flop. The different partition functions are speculatively interpreted as counting stable objects in the derived category of A-modules under different stability conditions; their relationship should then be an instance of wall crossing in the space of stability conditions on this triangulated category.Comment: Infinite product form, conjectured in v1, now a theorem of Ben Young. Additional discussion of small-volume expansion related to Eisenstein-like serie

Purity for graded potentials and quantum cluster positivity

Consider a smooth quasi-projective variety XX equipped with a C∗C∗-action, and a regular function f:X→Cf:X→C which is C∗C∗-equivariant with respect to a positive weight action on the base. We prove the purity of the mixed Hodge structure and the hard Lefschetz theorem on the cohomology of the vanishing cycle complex of ff on proper components of the critical locus of ff, generalizing a result of Steenbrink for isolated quasi-homogeneous singularities. Building on work by Kontsevich and Soibelman, Nagao, and Efimov, we use this result to prove the quantum positivity conjecture for cluster mutations for all quivers admitting a positively graded nondegenerate potential. We deduce quantum positivity for all quivers of rank at most 4; quivers with nondegenerate potential admitting a cut; and quivers with potential associated to triangulations of surfaces with marked points and nonempty boundary

Punctual Hilbert schemes for Kleinian singularities as quiver varieties

For a finite subgroup $\Gamma\subset \mathrm{SL}(2,\mathbb{C})$ and $n\geq 1$, we construct the (reduced scheme underlying the) Hilbert scheme of $n$ points on the Kleinian singularity $\mathbb{C}^2/\Gamma$ as a Nakajima quiver variety for the framed McKay quiver of $\Gamma$, taken at a specific non-generic stability parameter. We deduce that this Hilbert scheme is irreducible (a result previously due to Zheng), normal, and admits a unique symplectic resolution. More generally, we introduce a class of algebras obtained from the preprojective algebra of the framed McKay quiver by a process called cornering, and we show that fine moduli spaces of cyclic modules over these new algebras are isomorphic to quiver varieties for the framed McKay quiver and certain non-generic choices of stability parameter.Comment: 24 pages. v3: Definition of cornered algebras simplified. To be published in Algebraic Geometr

Euler Characteristics of Hilbert Schemes of Points On Surfaces with Simple Singularities

This is an announcement of conjectures and results concerning the generating series of Euler characteristics of Hilbert schemes of points on surfaces with simple (Kleinian) singularities. For a quotient surface C^2/G with G a finite subgroup of SL(2, C), we conjecture a formula for this generating series in terms of Lie-theoretic data, which is compatible with existing results for type A singularities. We announce a proof of our conjecture for singularities of type D. The generating series in our conjecture can be seen as a specialized character of the basic representation of the corresponding (extended) affine Lie algebra; we discuss possible representation-theoretic consequences of this fact. Our results, respectively conjectures, imply the modularity of the generating function for surfaces with type A and type D, respectively arbitrary, simple singularities, confirming predictions of S-duality

Algebrai geometria és határterületei = Algebraic geometry and related fields

A kutatás hű maradt a pályázatban vázolt interdiszciplináris jelleghez, és a szigorúan vett algebrai geometriai problematikát és módszereket ötvözte más területek technikáival. Sikerült kiszámolnunk a komplex normális algebrai felület-szingularitások csomójának Heegaard-Floer homológiáját, és bevezettük egy új invariáns, a súlyozott gyökér fogalmát. Előrehaladást értünk el a komplex projektív síkgörbék szingularitásainak topológiai jellemzése terén is. Felállítottunk egy Riemann-Roch típusú formulát kanonikus ciklikus szingularitásokkal rendelkező komplex algebrai 3-sokaságokra, s e formula segítségével sikerült új Calabi-Yau 3-sokaságokat konstruálnunk. Kimutattuk, hogy egy racionálisan összefüggő komplex projektív sokaság hurokterei is racionálisan összefüggők. Aritmetikai geometriai kutatásaink legfontosabb eredménye egy számtest felett definiált 1-motívum, illetve a duális 1-motívum Tate-Safarevics csoportja között fennálló dualitás igazolása. Dualitástételeinknek alkalmazását is adtuk számtestek felett definiált szemi-Abel-varietások főhomogén tereinek racionális pontjaira. Invariánselméleti kutatásaink során meghatároztuk az általános lineáris, az ortogonális, és a szimplektikus csoportok kvantum koordinátagyűrűihez tartozó FRT- bialgebrákban a kokommutatív elemek által alkotta részalgebra explicit generátorait és az azok közötti relációkat, és konstrukciót adtunk új kvantum kvázihomogén terekre. | Our research remained faithful to the interdisciplinary spirit of the proposal, and blended the research area of algebraic geometry with techniques from other fields. We succeeded in computing the Heegaard-Floer homology of the link of complex normal algebraic surface singularities, and defined new invariants for them called weighted roots. We also achieved progress on the topological characterization of complex projective plane curves. We established a Riemann-Roch formula for complex algebraic threefolds with canonical cyclic singularities, and used it to construct new Calabi-Yau threefolds. We showed that loop spaces of a rationally connected compex projective manifold are again rationally connected. In arithmetic geometry our main result is a duality theorem between the Tate-Shafarevich group of a 1-motive defined over a number field and that of the dual 1-motive. We gave also applications of our duality theorems to the study of rational points on principal homogeneous spaces under semi-abelian varieties. In invariant theory we found explicit generators and relations for the subalgebra of cocommutative elements in the FRT bialgebras associated with the quantum coordinate rings of general linear, orthogonal and symplectic groups, and constructed new quantum quasi-homogeneous spaces

Sheaves on fibered threefolds and quiver sheaves

This paper classifies a class of holomorphic D-branes, closely related to framed torsion-free sheaves, on threefolds fibered in resolved ADE surfaces over a general curve C, in terms of representations with relations of a twisted Kronheimer--Nakajima-type quiver in the category Coh(C) of coherent sheaves on C. For the local Calabi--Yau case C\cong\A^1 and special choice of framing, one recovers the N=1 ADE quiver studied by Cachazo--Katz--Vafa.Comment: 13 pages, 2 figures, minor change

Opposite prognostic roles of HIF1alpha and HIF2alpha expressions in bone metastatic clear cell renal cell cancer

BACKGROUND: Prognostic markers of bone metastatic clear cell renal cell cancer (ccRCC) are poorly established. We tested prognostic value of HIF1alpha/HIF2alpha and their selected target genes in primary tumors and corresponding bone metastases. RESULTS: Expression of HIF2alpha was lower in mRCC both at mRNA and protein levels (p/mRNA/=0.011, p/protein/=0.001) while HIF1alpha was similar to nmRCC. At the protein level, CAIX, GAPDH and GLUT1 were increased in mRCC. In all primary RCCs, low HIF2alpha and high HIF1alpha as well as CAIX, GAPDH and GLUT1 expressions correlated with adverse prognosis, while VEGFR2 and EPOR gene expressions were associated with favorable prognosis. Multivariate analysis confirmed high HIF2alpha protein expression as an independent risk factor. Prognostic validation of HIFs, LDH, EPOR and VEGFR2 in RNA-Seq data confirmed higher HIF1alpha gene expression in primary RCC as an adverse (p=0.07), whereas higher HIF2alpha and VEGFR2 expressions as favorable prognostic factors. HIF1alpha/HIF2alpha-index (HIF-index) proved to be an independent prognostic factor in both the discovery and the TCGA cohort. PATIENTS AND METHODS: Expressions of HIF1alpha and HIF2alpha as well as their 7 target genes were analysed on the mRNA and protein level in 59 non-metastatic ccRCCs (nmRCC), 40 bone metastatic primary ccRCCs (mRCC) and 55 corresponding bone metastases. Results were validated in 399 ccRCCs from the TCGA project. CONCLUSIONS: We identified HIF2alpha protein as an independent marker of the metastatic potential of ccRCC, however, unlike HIF1alpha, increased HIF2alpha expression is a favorable prognostic factor. The HIF-index incorporated these two markers into a strong prognostic biomarker of ccRCC

Opposite prognostic roles of HIF1α and HIF2α expressions in bone metastatic clear cell renal cell cancer

BACKGROUND Prognostic markers of bone metastatic clear cell renal cell cancer (ccRCC) are poorly established. We tested prognostic value of HIF1α/HIF2α and their selected target genes in primary tumors and corresponding bone metastases. RESULTS Expression of HIF2α was lower in mRCC both at mRNA and protein levels (p/mRNA/=0.011, p/protein/=0.001) while HIF1α was similar to nmRCC. At the protein level, CAIX, GAPDH and GLUT1 were increased in mRCC. In all primary RCCs, low HIF2α and high HIF1α as well as CAIX, GAPDH and GLUT1 expressions correlated with adverse prognosis, while VEGFR2 and EPOR gene expressions were associated with favorable prognosis. Multivariate analysis confirmed high HIF2α protein expression as an independent risk factor. Prognostic validation of HIFs, LDH, EPOR and VEGFR2 in RNA-Seq data confirmed higher HIF1α gene expression in primary RCC as an adverse (p=0.07), whereas higher HIF2α and VEGFR2 expressions as favorable prognostic factors. HIF1α/HIF2α-index (HIF-index) proved to be an independent prognostic factor in both the discovery and the TCGA cohort. PATIENTS AND METHODS Expressions of HIF1α and HIF2α as well as their 7 target genes were analysed on the mRNA and protein level in 59 non-metastatic ccRCCs (nmRCC), 40 bone metastatic primary ccRCCs (mRCC) and 55 corresponding bone metastases. Results were validated in 399 ccRCCs from the TCGA project. CONCLUSIONS We identified HIF2α protein as an independent marker of the metastatic potential of ccRCC, however, unlike HIF1α, increased HIF2α expression is a favorable prognostic factor. The HIF-index incorporated these two markers into a strong prognostic biomarker of ccRCC