52 research outputs found
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 and , we construct the (reduced scheme underlying the) Hilbert scheme of
points on the Kleinian singularity as a Nakajima quiver
variety for the framed McKay quiver of , 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
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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
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