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Amin Gholampour

Publications and source records attributed to Amin Gholampour.

At least 19 recordsLinked to original sources
arXiv2026

Virtual cycles of 3-term complexes and the Hilbert schemes of surfaces

Given a 3-term perfect complex E over a quasi-projective variety X and a nonnegative integer r, we define two virtual cycles and their refinements supported over the r-th degeneracy loci of E. This is done by modifying the complex E after pulling it back to certain blow ups of X. We establish several Thom-Porteous, comparison, duality and wall-crossing formulas for these virtual cycles. We apply this construction to perfect complexes arising from the universal objects over the Picard variety and the Hilbert schemes of non-singular complex projective surfaces. We recover, reprove and strengthen some of the known results involving the reduced cycles and the virtual cycles of the Hilbert schemes related to the curve counting theory and Vafa-Witten theory, respectively. In the case of elliptic surfaces, we provide an explicit calculation generalizing that of Seiberg-Witten invariants.

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arXiv2021

Stable pairs of 2-dimensional sheaves on 4-folds

We identify Le Potier's moduli spaces of limit stable pairs $(F,s)$, where $F$ is a 2-dimensional sheaf on a nonsingular projective 4-fold $X$ and $s \in H^0(F)$, with the moduli spaces of polynomial stable 2-term complexes in derived category. These stable pairs are 2-dimensional analogs of Pandharipande-Thomas' stable pairs defined for 3-folds. We establish categorical correspondences involving these stable pairs, ideal sheaves of 2-dimensional subschemes of $X$, and 1-dimensional sheaves on $X$. Under some conditions on the Chern character, these lead to Hall algebra correspondences. The generalization of most of these results to higher ranks is also given. In case $X$ is Calabi-Yau, Oh-Thomas' construction gives a new set of invariants of $X$ counting these stable pairs. For certain Chern characters, these are related to the invariants of 2-dimensional stable sheaves. We calculate and study them in some cases and examples such as fibrations by abelian surfaces, local surfaces, and local Fano 3-folds. The last case in particular leads to new invariants of Fano 3-folds counting 2-dimensional stable pairs with reduced supports.

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arXiv2020

Degeneracy loci, virtual cycles and nested Hilbert schemes II

We express nested Hilbert schemes of points and curves on a smooth projective surface as "virtual resolutions" of degeneracy loci of maps of vector bundles on smooth ambient spaces. We show how to modify the resulting obstruction theories to produce the virtual cycles of Vafa-Witten theory and other sheaf-counting problems. The result is an effective way of calculating invariants (VW, SW, local PT and local DT) via Thom-Porteous-like Chern class formulae.

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arXiv2019

Degeneracy loci, virtual cycles and nested Hilbert schemes I

Given a map of vector bundles on a smooth variety, consider the deepest degeneracy locus where its rank is smallest. We show it carries a natural perfect obstruction theory whose virtual cycle can be calculated by the Thom-Porteous formula. We show nested Hilbert schemes of points on surfaces can be expressed as degeneracy loci. We show how to modify the resulting obstruction theories to recover the virtual cycles of Vafa-Witten and reduced local DT theories. The result computes some Vafa-Witten invariants in terms of Carlsson-Okounkov operators. This proves and extends a conjecture of Gholampour-Sheshmani-Yau and generalises a vanishing result of Carlsson-Okounkov.

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arXiv2019

Higher rank sheaves on threefolds and functional equations

We consider the moduli space of stable torsion free sheaves of any rank on a smooth projective threefold. The singularity set of a torsion free sheaf is the locus where the sheaf is not locally free. On a threefold it has dimension $\leq 1$. We consider the open subset of moduli space consisting of sheaves with empty or 0-dimensional singularity set. For fixed Chern classes $c_1,c_2$ and summing over $c_3$, we show that the generating function of topological Euler characteristics of these open subsets equals a power of the MacMahon function times a Laurent polynomial. This Laurent polynomial is invariant under $q \leftrightarrow q^{-1}$ (upon replacing $c_1 \leftrightarrow -c_1$). For some choices of $c_1,c_2$ these open subsets equal the entire moduli space. The proof involves wall-crossing from Quot schemes of a higher rank reflexive sheaf to a sublocus of the space of Pandharipande-Thomas pairs. We interpret this sublocus in terms of the singularities of the reflexive sheaf.

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arXiv2019

Counting stable sheaves on singular curves and surfaces

Given a quasi-projective scheme M over complex numbers equipped with a perfect obstruction theory and a morphism to a nonsingular quasi-projective variety B, we show it is possible to find an affine bundle M'/ M that admits a perfect obstruction theory relative to B. We study the resulting virtual cycles on the fibers of M'/B and relate them to the image of the virtual cycle [M]^vir under refined Gysin homomorphisms. Our main application is when M is a moduli space of stable codimension 1 sheaves with a fixed determinant L on a nonsingular projective surface or Fano threefold, B is the linear system |L|, and the morphism to B is given by taking the divisor associated to a coherent sheaf.

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arXiv2019

Donaldson-Thomas invariants, linear systems and punctual Hilbert schemes

We study certain DT invariants arising from stable coherent sheaves in a nonsingular projective threefold supported on the members of a linear system of a fixed line bundle. When the canonical bundle of the threefold satisfies certain positivity conditions, we relate the DT invariants to Carlsson-Okounkov formulas for the "twisted Euler's number" of the punctual Hilbert schemes of nonsingular surfaces, and conclude they have a modular property.

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arXiv2019

Nested Hilbert schemes on surfaces: Virtual fundamental class

We construct virtual fundamental classes on nested Hilbert schemes of points and curves in complex nonsingular projective surfaces. These classes recover the virtual classes of Seiberg-Witten theory as well as the (reduced) stable theory, and play a crucial role in the reduced Donaldson-Thomas theory of local-surface-threefolds that we study in [GSY17b] (arXiv:1807.05697). We show that certain integrals against the virtual fundamental classes of punctual nested Hilbert schemes are expressed as integrals over the products of the Hilbert scheme of points. We are able to find explicit formulas for some of these integrals by relating them to Carlsson-Okounkov's vertex operator formulas.

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arXiv2019

Sheaves on weighted projective planes and modular forms

We give an explicit description of toric sheaves on the weighted projective plane $\mathbb{P}(a,b,c)$ viewed as a toric Deligne-Mumford stack. The integers $(a,b,c)$ are not necessarily chosen coprime or mutually coprime allowing for gerbe and root stack structures. As an application, we describe the fixed point locus of the moduli scheme of stable rank 1 and 2 torsion free sheaves on $\mathbb{P}(a,b,c)$ with fixed $K$-group class. Summing over all $K$-group classes, we obtain explicit formulae for generating functions of the topological Euler characteristics. In the case of stable rank 2 locally free sheaves on $\mathbb{P}(a,b,c)$ with $a,b,c \leq 2$ the generating functions can be expressed in terms of Hurwitz class numbers and give rise to modular forms of weight $3/2$. This generalizes Klyachko's computation on $\mathbb{P}^2$ and is consistent with $S$-duality predictions from physics.

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arXiv2018

Localized Donaldson-Thomas theory of surfaces

Let $S$ be a projective simply connected complex surface and $\mathcal{L}$ be a line bundle on $S$. We study the moduli space of stable compactly supported 2-dimensional sheaves on the total spaces of $\mathcal{L}$. The moduli space admits a $\mathbb{C}^*$-action induced by scaling the fibers of $\mathcal{L}$. We identify certain components of the fixed locus of the moduli space with the moduli space of torsion free sheaves and the nested Hilbert schemes on $S$. We define the localized Donaldson-Thomas invariants of $\mathcal{L}$ by virtual localization in the case that $\mathcal{L}$ twisted by the anti-canonical bundle of $S$ admits a nonzero global section. When $p_g(S)>0$, in combination with Mochizuki's formulas, we are able to express the localized DT invariants in terms of the invariants of the nested Hilbert schemes defined by the authors in [GSY17a], the Seiberg-Witten invariants of $S$, and the integrals over the products of Hilbert schemes of points on $S$. When $\mathcal{L}$ is the canonical bundle of $S$, the Vafa-Witten invariants defined recently by Tanaka-Thomas, can be extracted from these localized DT invariants. VW invariants are expected to have modular properties as predicted by S-duality.

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arXiv2017

Donaldson-Thomas Invariants of 2-Dimensional sheaves inside threefolds and modular forms

Motivated by the S-duality conjecture, we study the Donaldson-Thomas invariants of the 2 dimensional Gieseker stable sheaves on a threefold. These sheaves are supported on the fibers of a nonsingular threefold X fibered over a nonsingular curve. In the case where X is a K3 fibration, we express these invariants in terms of the Euler characteristic of the Hilbert scheme of points on the K3 fiber and the Noether-Lefschetz numbers of the fibration. We prove that a certain generating function of these invariants is a vector modular form of weight -3/2 as predicted in S-duality.

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arXiv2017

Stable pairs on nodal K3 fibrations

We study Pandharipande-Thomas's stable pair theory on $K3$ fibrations over curves with possibly nodal fibers. We describe stable pair invariants of the fiberwise irreducible curve classes in terms of Kawai-Yoshioka's formula for the Euler characteristics of moduli spaces of stable pairs on $K3$ surfaces and Noether-Lefschetz numbers of the fibration. Moreover, we investigate the relation of these invariants with the perverse (non-commutative) stable pair invariants of the $K3$ fibration. In the case that the $K3$ fibration is a projective Calabi-Yau threefold, by means of wall-crossing techniques, we write the stable pair invariants in terms of the generalized Donaldson-Thomas invariants of 2-dimensional Gieseker semistable sheaves supported on the fibers.

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arXiv2016

Rank 2 sheaves on toric 3-folds: classical and virtual counts

Let $\mathcal{M}$ be the moduli space of rank 2 stable torsion free sheaves with Chern classes $c_i$ on a smooth 3-fold $X$. When $X$ is toric with torus $T$, we describe the $T$-fixed locus of the moduli space. Connected components of $\mathcal{M}^T$ with constant reflexive hulls are isomorphic to products of $\mathbb{P}^1$. We mainly consider such connected components, which typically arise for any $c_1$, "low values" of $c_2$, and arbitrary $c_3$. In the classical part of the paper, we introduce a new type of combinatorics called double box configurations, which can be used to compute the generating function $\mathsf{Z}(q)$ of topological Euler characteristics of $\mathcal{M}$ (summing over all $c_3$). The combinatorics is solved using the double dimer model in a companion paper. This leads to explicit formulae for $\mathsf{Z}(q)$ involving the MacMahon function. In the virtual part of the paper, we define Donaldson-Thomas type invariants of toric Calabi-Yau 3-folds by virtual localization. The contribution to the invariant of an individual connected component of the $T$-fixed locus is in general not equal to its signed Euler characteristic due to $T$-fixed obstructions. Nevertheless, the generating function of all invariants is given by $\mathsf{Z}(q)$ up to signs.

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arXiv2016

Stable reflexive sheaves and localization

We study moduli spaces $\mathcal{N}$ of rank 2 stable reflexive sheaves on $\mathbb{P}^3$. Fixing Chern classes $c_1$, $c_2$, and summing over $c_3$, we consider the generating function $\mathsf{Z}^{\mathrm{refl}}(q)$ of Euler characteristics of such moduli spaces. The action of the torus $T$ on $\mathbb{P}^3$ lifts to $\mathcal{N}$ and we classify all sheaves in $\mathcal{N}^T$. This leads to an explicit expression for $\mathsf{Z}^{\mathrm{refl}}(q)$. Since $c_3$ is bounded below and above, $\mathsf{Z}^{\mathrm{refl}}(q)$ is a polynomial. We find a simple formula for its leading term when $c_1=-1$. Next, we study moduli spaces of rank 2 stable torsion free sheaves on $\mathbb{P}^3$ and consider the generating function of Euler characteristics of such moduli spaces. We give an expression for this generating function in terms of $\mathsf{Z}^{\mathrm{refl}}(q)$ and Euler characteristics of Quot schemes of certain $T$-equivariant reflexive sheaves, which are studied elsewhere. Many techniques of this paper apply to any toric 3-fold. In general, $\mathsf{Z}^{\mathrm{refl}}(q)$ depends on the choice of polarization which leads to wall-crossing phenomena. We briefly illustrate this in the case of $\mathbb{P}^2 \times \mathbb{P}^1$.

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arXiv2016

Generalized Donaldson-Thomas Invariants of 2-Dimensional sheaves on local P^2

Let X be the total space of the canonical bundle of P^2. We study the generalized Donaldson-Thomas invariants, defined in the work of Joyce-Song, of the moduli spaces of the 2-dimensional Gieseker semistable sheaves on X with first Chern class equal to k times the class of the zero section of X. When k=1, 2 or 3, and semistability implies stability, we express the invariants in terms of known modular forms. We prove a combinatorial formula for the invariants when k=2 in the presence of the strictly semistable sheaves, and verify the BPS integrality conjecture of Joyce-Song in some cases.

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arXiv2015

Intersection numbers on the relative Hilbert schemes of points on surfaces

We study certain top intersection products on the Hilbert scheme of points on a nonsingular surface relative to an effective smooth divisor. We find a formula relating these numbers to the corresponding intersection numbers on the non-relative Hilbert schemes. In particular, we obtain a relative version of the explicit formula found by Carlsson-Okounkov for the Euler class of the twisted tangent bundle of the Hilbert schemes.

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arXiv2015

Counting curves on surfaces in Calabi-Yau 3-folds

Motivated by S-duality modularity conjectures in string theory, we define new invariants counting a restricted class of 2-dimensional torsion sheaves, enumerating pairs $Z\subset H$ in a Calabi-Yau threefold X. Here H is a member of a sufficiently positive linear system and Z is a 1-dimensional subscheme of it. The associated sheaf is the ideal sheaf of $Z\subset H$, pushed forward to X and considered as a certain Joyce-Song pair in the derived category of X. We express these invariants in terms of the MNOP invariants of X.

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