arXiv subjects

Martijn Kool

Publications and source records attributed to Martijn Kool.

23 records · Page 2Linked to original sources

arXiv2018

Zero-dimensional Donaldson-Thomas invariants of Calabi-Yau 4-folds

We study Hilbert schemes of points on a smooth projective Calabi-Yau 4-fold $X$. We define $\mathrm{DT}_4$ invariants by integrating the Euler class of a tautological vector bundle $L^{[n]}$ against the virtual class. We conjecture a formula for their generating series, which we prove in certain cases when $L$ corresponds to a smooth divisor on $X$. A parallel equivariant conjecture for toric Calabi-Yau 4-folds is proposed. This conjecture is proved for smooth toric divisors and verified for more general toric divisors in many examples. Combining the equivariant conjecture with a vertex calculation, we find explicit positive rational weights, which can be assigned to solid partitions. The weighted generating function of solid partitions is given by $\exp(M(q)-1)$, where $M(q)$ denotes the MacMahon function.

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arXiv2018

Counting zero-dimensional subschemes in higher dimensions

Consider zero-dimensional Donaldson-Thomas invariants of a toric threefold or toric Calabi-Yau fourfold. In the second case, invariants can be defined using a tautological insertion. In both cases, the generating series can be expressed in terms of the MacMahon function. In the first case, this follows from a theorem of Maulik-Nekrasov-Okounkov-Pandharipande. In the second case, this follows from a conjecture of the authors and a (more general $K$-theoretic) conjecture of Nekrasov. In this paper, we consider formal analogues of these invariants in any dimension $d \not \equiv 2 \ \mathrm{mod} \, 4$. The direct analogues of the above-mentioned conjectures fail in general when $d>4$, showing that dimensions 3 and 4 are special. Surprisingly, after appropriate specialization of the equivariant parameters, the conjectures seem to hold in all dimensions.

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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.

math.AG↗

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 $\mathbb{P}^2 \times \mathbb{P}^1$.

math.AG↗

arXiv2014

Euler characteristics of moduli spaces of torsion free sheaves on toric surfaces

Given a smooth toric variety $X$, the action of the torus $T$ lifts to the moduli space $\mathcal{M}$ of stable sheaves on $X$. Using the pioneering work of Klyacho, a fairly explicit combinatorial description of the fixed point locus $\mathcal{M}^T$ can be given (as shown by earlier work of the author). In this paper, we apply this description to the case of torsion free sheaves on a smooth toric surface $S$. A general expression for the generating function of the Euler characteristics of such moduli spaces is obtained. The generating function is expressed in terms of Euler characteristics of certain moduli spaces of stable configurations of linear subspaces appearing in classical GIT. The expression holds for any choice of $S$, polarization, rank, and first Chern class. Specializing to various examples allows us to compute some new as well as known generating functions.

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