374 research outputs found
The 3-fold vertex via stable pairs
The theory of stable pairs in the derived category yields an enumerative
geometry of curves in 3-folds. We evaluate the equivariant vertex for stable
pairs on toric 3-folds in terms of weighted box counting. In the toric
Calabi-Yau case, the result simplifies to a new form of pure box counting. The
conjectural equivalence with the DT vertex predicts remarkable identities. The
equivariant vertex governs primary insertions in the theory of stable pairs for
toric varieties. We consider also the descendent vertex and conjecture the
complete rationality of the descendent theory for stable pairs.Comment: Typos fixed. 40 pages, 8 figure
Suppressor of gamma response 1 modulates the DNA damage response and oxidative stress response in leaves of cadmium-exposed Arabidopsis thaliana
Cadmium (Cd) exposure causes an oxidative challenge and inhibits cell cycle progression, ultimately impacting plant growth. Stress-induced effects on the cell cycle are often a consequence of activation of the DNA damage response (DDR). The main aim of this study was to investigate the role of the transcription factor SUPPRESSOR OF GAMMA RESPONSE 1 (SOG1) and three downstream cyclin-dependent kinase inhibitors of the SIAMESE-RELATED (SMR) family in the Cd-induced DDR and oxidative challenge in leaves of Arabidopsis thaliana. Effects of Cd on plant growth, cell cycle regulation and the expression of DDR genes were highly similar between the wildtype and smr4/5/7 mutant. In contrast, sog1-7 mutant leaves displayed a much lower Cd sensitivity within the experimental time-frame and significantly less pronounced upregulations of DDR-related genes, indicating the involvement of SOG1 in the Cd-induced DDR. Cadmium-induced responses related to the oxidative challenge were disturbed in the sog1-7 mutant, as indicated by delayed Cd-induced increases of hydrogen peroxide and glutathione concentrations and lower upregulations of oxidative stress-related genes. In conclusion, our results attribute a novel role to SOG1 in regulating the oxidative stress response and connect oxidative stress to the DDR in Cd-exposed plants
Moduli of ADHM Sheaves and Local Donaldson-Thomas Theory
The ADHM construction establishes a one-to-one correspondence between framed
torsion free sheaves on the projective plane and stable framed representations
of a quiver with relations in the category of complex vector spaces. This paper
studies the geometry of moduli spaces of representations of the same quiver
with relations in the abelian category of coherent sheaves on a smooth complex
projective curve . In particular it is proven that this moduli space is
virtually smooth and related byrelative Beilinson spectral sequence to the
curve counting construction via stable pairs of Pandharipande and Thomas. This
yields a new conjectural construction for the local Donaldson-Thomas theory of
curves as well as a natural higher rank generalization.Comment: 61 pages AMS Latex; v2: minor corrections, reference added; v3: some
proofs corrected using the GIT construction of the moduli space due to A.
Schmitt; main results unchanged; final version to appear in J. Geom. Phy
Abel-Jacobi maps for hypersurfaces and non commutative Calabi-Yau's
It is well known that the Fano scheme of lines on a cubic 4-fold is a
symplectic variety. We generalize this fact by constructing a closed p-form
with p=2n-4 on the Fano scheme of lines on a (2n-2)-dimensional hypersurface Y
of degree n. We provide several definitions of this form - via the Abel-Jacobi
map, via Hochschild homology, and via the linkage class, and compute it
explicitly for n = 4. In the special case of a Pfaffian hypersurface Y we show
that the Fano scheme is birational to a certain moduli space of sheaves on a
p-dimensional Calabi--Yau variety X arising naturally in the context of
homological projective duality, and that the constructed form is induced by the
holomorphic volume form on X. This remains true for a general non Pfaffian
hypersurface but the dual Calabi-Yau becomes non commutative.Comment: 34 pages; exposition of Hochschild homology expanded; references
added; introduction re-written; some imrecisions, typos and the orbit diagram
in the last section correcte
Curve counting via stable pairs in the derived category
For a nonsingular projective 3-fold , we define integer invariants
virtually enumerating pairs where is an embedded curve and
is a divisor. A virtual class is constructed on the associated
moduli space by viewing a pair as an object in the derived category of . The
resulting invariants are conjecturally equivalent, after universal
transformations, to both the Gromov-Witten and DT theories of . For
Calabi-Yau 3-folds, the latter equivalence should be viewed as a wall-crossing
formula in the derived category.
Several calculations of the new invariants are carried out. In the Fano case,
the local contributions of nonsingular embedded curves are found. In the local
toric Calabi-Yau case, a completely new form of the topological vertex is
described.
The virtual enumeration of pairs is closely related to the geometry
underlying the BPS state counts of Gopakumar and Vafa. We prove that our
integrality predictions for Gromov-Witten invariants agree with the BPS
integrality. Conversely, the BPS geometry imposes strong conditions on the
enumeration of pairs.Comment: Corrected typos and duality error in Proposition 4.6. 47 page
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