The DT/PT correspondence for smooth curves

@article{Ricolfi2018TheDC,
  title={The DT/PT correspondence for smooth curves},
  author={Andrea T. Ricolfi},
  journal={Mathematische Zeitschrift},
  year={2018},
  volume={290},
  pages={699-710}
}
We show a version of the DT/PT correspondence relating local curve counting invariants, encoding the contribution of a fixed smooth curve in a Calabi–Yau threefold. We exploit a local study of the Hilbert–Chow morphism about the cycle of a smooth curve. We compute, via Quot schemes, the global Donaldson–Thomas theory of a general Abel–Jacobi curve of genus 3. 
The Hilbert scheme of hyperelliptic Jacobians and moduli of Picard sheaves
Let $C$ be a hyperelliptic curve embedded in its Jacobian $J$ via an Abel-Jacobi map. We compute the scheme structure of the Hilbert scheme component of $\textrm{Hilb}_J$ containing the Abel-JacobiExpand
Higher rank motivic Donaldson-Thomas invariants of $\mathbb{A}^3$ via wall-crossing, and asymptotics
We compute, via motivic wall-crossing, the generating function of virtual motives of the Quot scheme of points on $\mathbb{A}^3$, generalising to higher rank a result of Behrend, Bryan and Szendrői.Expand
The equivariant Atiyah class
Let $X$ be a complex scheme acted on by an affine algebraic group $G$. We prove that the Atiyah class of a $G$-equivariant perfect complex on $X$, as constructed by Huybrechts and Thomas, isExpand
On the motive of the Quot scheme of finite quotients of a locally free sheaf
Let $X$ be a smooth variety, $E$ a locally free sheaf on $X$. We express the generating function of the motives $[\textrm{Quot}_X(E,n)]$ in terms of the power structure on the Grothendieck ring ofExpand
Unweighted Donaldson-Thomas theory of the banana 3-fold with section classes.
We further the study of the Donaldson-Thomas theory of the banana threefolds which were recently discovered and studied in [Bryan'19]. These are smooth proper Calabi-Yau threefolds which are fibredExpand
The Donaldson-Thomas Theory of the Banana Threefold with Section Classes
We further the study of the Donaldson-Thomas theory of the banana threefolds which were recently discovered and studied in [Bryan'19]. These are smooth proper Calabi-Yau threefolds which are fibredExpand
Virtual classes and virtual motives of Quot schemes on threefolds
For a simple, rigid vector bundle $F$ on a Calabi-Yau $3$-fold $Y$, we construct a symmetric obstruction theory on the Quot scheme $\textrm{Quot}_Y(F,n)$, and we solve the associated enumerativeExpand
Higher rank K-theoretic Donaldson-Thomas Theory of points
Abstract We exploit the critical structure on the Quot scheme $\text {Quot}_{{{\mathbb {A}}}^3}({\mathscr {O}}^{\oplus r}\!,n)$, in particular the associated symmetric obstruction theory, in order toExpand
Framed motivic Donaldson-Thomas invariants of small crepant resolutions
For an arbitrary integer $r\geq 1$, we compute $r$-framed motivic PT and DT invariants of small crepant resolutions of toric Calabi-Yau $3$-folds, establishing a "higher rank" version of the motivicExpand
Framed sheaves on projective space and Quot schemes
We prove that, given integers $m\geq 3$, $r\geq 1$ and $n\geq 0$, the moduli space of torsion free sheaves on $\mathbb P^m$ with Chern character $(r,0,\ldots,0,-n)$ that are trivial along aExpand
...
1
2
...

References

SHOWING 1-10 OF 19 REFERENCES
Curve counting theories via stable objects I. DT/PT correspondence
The Donaldson-Thomas invariant is a curve counting invariant on Calabi-Yau 3-folds via ideal sheaves. Another counting invariant via stable pairs is introduced by Pandharipande and Thomas, whichExpand
Curve counting via stable pairs in the derived category
For a nonsingular projective 3-fold X, we define integer invariants virtually enumerating pairs (C,D) where C⊂X is an embedded curve and D⊂C is a divisor. A virtual class is constructed on theExpand
Hall algebras and curve-counting invariants
We use Joyce's theory of motivic Hall algebras to prove that reduced Donaldson-Thomas curve-counting invariants on Calabi-Yau threefolds coincide with stable pair invariants, and that the generatingExpand
Counting Sheaves on Calabi–Yau and Abelian Threefolds
We survey the foundations for Donaldson–Thomas invariants for stable sheaves on algebraic threefolds with trivial canonical bundle, with emphasis on the case of abelian threefolds.
Zero dimensional Donaldson – Thomas invariants of threefolds
Ever since the pioneer work of Donaldson and Thomas on Yang–Mills theory over Calabi–Yau threefolds [5, 13], people have been searching for their roles in the study of Calabi–Yau geometry and theirExpand
Donaldson-Thomas type invariants via microlocal geometry
We prove that Donaldson-Thomas type invariants are equal to weighted Euler characteristics of their moduli spaces. In particular, such invariants depend only on the scheme structure of the moduliExpand
Curve counting on elliptic Calabi–Yau threefolds via derived categories
We prove the elliptic transformation law of Jacobi forms for the generating series of Pandharipande--Thomas invariants of an elliptic Calabi--Yau 3-fold over a reduced class in the base. This provesExpand
A holomorphic Casson invariant for Calabi-Yau 3-folds, and bundles on K3 fibrations
We briefly review the formal picture in which a Calabi-Yau n-fold is the complex analogue of an oriented real n-manifold, and a Fano with a fixed smooth anticanonical divisor is the analogue of aExpand
Stable pairs and BPS invariants
We define the BPS invariants of Gopakumar-Vafa in the case of irreducible curve classes on Calabi-Yau 3-folds. The main tools are the theory of stable pairs in the derived category and Behrend'sExpand
On reduced stable pair invariants
Let $$X = S \times E$$X=S×E be the product of a K3 surface S and an elliptic curve E. Reduced stable pair invariants of X can be defined via (1) cutting down the reduced virtual class with incidenceExpand
...
1
2
...