The notion of limit stability on Calabi-Yau 3-folds is introduced by the author to construct an approximation of Bridgeland-Douglas stability conditions at the large volume limit. It has also turned out that the wall-crossing phenomena of limit stable objects seem relevant to the rationality conjecture of the generating functions of Pandharipande-Thomas invariants. In this article, we shall make it clear how wall-crossing formula of the counting invariants of limit stable objects solves the above conjecture.
We give a remark on the wall crossing behavior of perverse coherent sheaves on a blow-up and stability condition constructed by Toda. We also explain the wall crossing of twisted stability in terms of stability condition.
We study motivic Donaldson-Thomas invariants in the sense of Behrend-Bryan-Szendroi. A wall-crossing formula under a mutation is proved for a certain class of quivers with potentials.
We develop the technique of weight truncation in the context of wall-crossings in birational cobordisms, parallel to that in [HL15, BFK19]. More precisely, for each such wall-crossing, we embed the bounded above derived category of coherent sheaves of the semistable part as a semi-orthogonal summand of that of the stack in question. Our construction does not require any smoothness assumptions, and exhibits a strong symmetry across the two sides of the wall-crossing. As an application, we show that for wall-crossings satisfying suitable regularity conditions, a certain duality of local cohomolo…
Notes from the report at the Fields institute in Toronto. We introduce the Donaldson-Thomas invariants and describe the wall-crossing formulas for numerical Donaldson-Thomas invariants.
We study the Hilbert scheme of twisted cubics in the three-dimensional projective space by using Bridgeland stability conditions. We use wall-crossing techniques to describe its geometric structure and singularities, which reproves the classical result of Piene and Schlessinger.
We prove, under suitable conditions, that there exist wall-crossing and reduction morphisms for moduli spaces of stable log pairs in all dimensions as one varies the coefficients of the divisor.
We investigate the wall-crossing behavior as Bridgeland moduli spaces for some Simpson moduli spaces of Gieseker-semistable torsion sheaves on $\mathbb{P}^1\times \mathbb{P}^1$ with linear Hilbert polynomial. In particular, we recover some of the birational transformations of these spaces described by Maican and Chung-Moon as wall-crossing maps in certain slices of $Stab(\mathbb{P}^1\times \mathbb{P}^1)$.
For a Fermat quasi-homogeneous polynomial $W$, we study a family of K-theoretic quantum invariants parametrized by a positive rational number $ε$. We prove a wall-crossing formula by showing the generating functions lie on the Lagrangian cone of the permutation-equivariant K-theoretic FJRW theory of $W$.
We prove that the "Thaddeus flips" of $L$-twisted sheaves constructed by Matsuki and Wentworth can be obtained via Bridgeland wall-crossing. Similarly, we realize the change of polarization for moduli spaces of 1-dimensional Gieseker semistable sheaves on a surface by varying a family of stability conditions.
Bridgeland stability condition is preserved under the Fourier-Mukai transform by its definition. We explain the relation with Gieseker stability. By studying the wall-crossing behavior, we reprove that the moduli spaces of stable sheaves on abelian surfaces are birationally equivalent, if the associated Mukai vectors are related by isometries of the Mukai lattice.
We give a new proof of the following theorem: moduli spaces of stable complexes on a complex projective K3 surface, with primitive Mukai vector and with respect to a generic Bridgeland stability condition, are hyperkähler varieties of $\mathrm{K3}^{[n]}$-type of expected dimension. We use derived equivalences, deformations and wall-crossing for Bridgeland stability to reduce to the case of the Hilbert scheme of points.
We compute I-functions and central charges for abelian GLSMs using virtual matrix factorizations of Favero and Kim. In the Calabi-Yau case we provide analytic continuation for the central charges by explicit integral formulas. The integrals in question are called hemisphere partition functions and we call the integral representation Higgs-Coulomb correspondence. We then use it to prove GIT stability wall-crossing for central charges.
For each positive rational number $ε$, we define $K$-theoretic $ε$-stable quasimaps to certain GIT quotients $W\sslash G$. For $ε>1$, this recovers the $K$-theoretic Gromov-Witten theory of $W\sslash G$ introduced in more general context by Givental and Y.-P. Lee. For arbitrary $ε_1$ and $ε_2$ in different stability chambers, these $K$-theoretic quasimap invariants are expected to be related by wall-crossing formulas. We prove wall-crossing formulas for genus zero $K$-theoretic quasimap theory when the target $W\sslash G$ admits a torus action with isolated fixed points and isolated one-dimens…
The moduli space of holomorphic maps from Riemann surfaces to the Grassmannian is known to have two kinds of compactifications: Kontsevich's stable map compactification and Marian-Oprea-Pandharipande's stable quotient compactification. Over a non-singular curve, the latter moduli space is Grothendieck's Quot scheme. In this paper, we give the notion of `$ε$-stable quotients' for a positive real number $ε$, and show that stable maps and stable quotients are related by the wall-crossing phenomena. We will also discuss Gromov-Witten type invariants associated to $ε$-stable quotients, and investig…
In this paper we give a complete description of the Hitchin fibration on all 2-dimensional moduli spaces of rank 2 irregular Higgs bundles with two poles on the projective line. We describe the dependence of the singular fibers of the fibration on the eigenvalues of the Higgs fields, and describe the corresponding motivic wall-crossing phenomenon in the parameter space of parabolic weights.
We show that the Hilbert scheme of curves and Le Potier's moduli space of stable pairs with one dimensional support have a common GIT construction. The two spaces correspond to chambers on either side of a wall in the space of GIT linearisations. We explain why this is not enough to prove the "DT/PT wall crossing" conjecture relating the invariants derived from these moduli spaces when the underlying variety is a 3-fold. We then give a gentle introduction to a small part of Joyce's theory for such wall crossings, and use it to give a short proof of an identity relating the Euler characteristic…
We study the wall-crossing of the moduli spaces $\mathbf{M}^α(d,1)$ of $α$-stable pairs with linear Hilbert polynomial $dm+1$ on the projective plane $\mathbb{P}^2$ as we alter the parameter $α$. When $d$ is 4 and 5, at each wall, the moduli spaces are related by a smooth blow-up morphism followed by a smooth blow-down morphism, where one can describe the blow-up centers geometrically. As a byproduct, we obtain the Poincaré polynomials of the moduli space $\mathbf{M}(d,1)$ of stable sheaves. We also discuss the wall-crossing when the number of stable components in Jordan-Hölder filtrations is …
We remove the global quotient presentation input in the theory of windows in derived categories of smooth Artin stacks of finite type. As an application, we use existing results on flipping of strata for wall-crossing of Gieseker semi-stable torsion-free sheaves of rank two on rational surfaces to produce semi-orthogonal decompositions relating the different moduli stacks. The complementary pieces of these semi-orthogonal decompositions are derived categories of products of Hilbert schemes of points on the surface.
Over the last few years, wall-crossing for Bridgeland stability conditions has led to a large number of results in algebraic geometry, particular on birational geometry of moduli spaces. We illustrate some of the methods behind these result by reproving Lazarsfeld's Brill-Noether theorem for curves on K3 surfaces via wall-crossing. We conclude with a survey of recent applications of stability conditions on surfaces. The intended reader is an algebraic geometer with a limited working knowledge of derived categories. This article is based on the author's talk at the AMS Summer Institute on Algeb…
For a Fermat quasi-homogeneous polynomial, we study the associated weighted Fan-Jarvis-Ruan-Witten theory with narrow insertions. We prove a wall-crossing formula in all genera via localization on a master space, which is constructed by introducing an additional tangent vector to the moduli problem. This is a Landau-Ginzburg theory analogue of the higher-genus quasi-map wall-crossing formula proved by Ciocan-Fontanine and Kim. It generalizes the genus-$0$ result by Ross-Ruan and the genus-$1$ result by Guo-Ross.
The notion of d-critical loci was introduced by Joyce in order to give classical shadows of $(-1)$-shifted symplectic derived schemes. In this paper, we discuss birational geometry for d-critical loci, by introducing notions such as `d-critical flips', `d-critical flops', etc. They are not birational maps of the underlying spaces, but rather should be understood as virtual birational maps. We show that several wall-crossing phenomena of moduli spaces of stable objects on Calabi-Yau 3-folds are described in terms of d-critical birational geometry. Among them, we show that wall-crossing diagrams…
Let $C$ be a curve on a K3 surface $X$ with Picard group $\mathbb{Z}.[C]$. Mukai's program seeks to recover $X$ from $C$ by exhibiting it as a Fourier-Mukai partner to a Brill-Noether locus of vector bundles on $C$. We use wall-crossing in the space of Bridgeland stability conditions to prove this for genus $\ge14$. This paper deals with the case $g-1$ prime left over from Paper I.
We introduce a technique for proving all-genus wall-crossing formulas in the gauged linear sigma model as the stability parameter varies, without assuming factorization properties of the virtual class. We implement this technique explicitly for the hybrid model, which generalizes our previous work to the Landau--Ginzburg phase.
Let $v_d(\mathbb{P}^2)\subset |\mathcal{O}_{\mathbb{P}^2}(d)|$ denote the $d$-uple Veronese surface. After studying some general aspects of the wall-crossing phenomena for stability conditions on surfaces, we are able to describe a sequence of flips of the secant varieties of $v_d(\mathbb{P}^2)$ by embedding the blow-up $\mbox{bl}_{v_d(\mathbb{P}^2)}|\mathcal{O}_{\mathbb{P}^2}(d)|$ into a suitable moduli space of Bridgeland semistable objects on $\mathbb{P}^2$.
Since its introduction in 1995 by Li-Tian and Behrend-Fantechi, the theory of virtual fundamental class has played a key role in algebraic geometry, defining important invariants such as the Gromov-Witten invariant and the Donaldson-Thomas invariant. Quite a few methods for handling the virtual fundamental classes were discovered such as torus localization, degeneration, virtual pullback and cosection localization. Often combining these methods turns out to be quite effective. In this paper, we prove virtual pullback, torus localization and wall crossing formulas for cosection localized virtua…
Motivated by Witten's work (arXiv:hep-th/9312104), we propose the Verlinde/Grassmannian correspondence which relates the GL Verlinde numbers to the K-theoretic quasimap invariants of the Grassmannian. We recover these two types of invariants by imposing different stability conditions on the gauged linear sigma model associated to the Grassmannian. We construct two families of stability conditions connecting the two theories and prove two wall-crossing results. We confirm the Verlinde/Grassmannian correspondence in the rank two case.
For each positive rational number epsilon, the theory of epsilon-stable quasimaps to certain GIT quotients W//G developed in arXiv:1106.3724[math.AG] gives rise to a Cohomological Field Theory. Furthermore, there is an asymptotic theory corresponding to epsilon --> 0. For epsilon >1 one obtains the usual Gromov-Witten theory of W//G, while the other theories are new. However, they are all expected to contain the same information and in particular the numerical invariants should be related by wall-crossing formulas. In this paper we analyze the genus zero picture and find that the wall-crossing…
In previous work (arXiv:1304.7056) we have conjectured wall-crossing formulas for genus zero quasimap invariants of GIT quotients and proved them via localization in many cases. We extend these formulas to higher genus when the target is semi-positive, and prove them for semi-positive toric varieties, in particular for toric local Calabi-Yau targets. The proof also applies to local Calabi-Yau's associated to some non-abelian quotients.
We extend B. Hassett's theory of weighted stable pointed curves ([Has03]) to weighted stable maps. The space of stability conditions is described explicitly, and the wall-crossing phenomenon studied. This can be considered as a non-linear analog of the theory of stability conditions in abelian and triangulated categories. We introduce virtual fundamental classes and thus obtain weighted Gromov-Witten invariants. We show that by including gravitational descendants, one obtains an $\LL$-algebra as introduced in [LM04] as a generalization of a cohomological field theory.
We derive explicit formulae for the generating series of mixed Grothendieck dessins d'enfant/monotone/simple Hurwitz numbers, via the semi-infinite wedge formalism. This reveals the strong piecewise polynomiality in the sense of Goulden-Jackson-Vakil, generalising a result of Johnson, and provides a new explicit proof of the piecewise polynomiality of the mixed case. Moreover, we derive wall-crossing formulae for the mixed case. These statements specialise to any of the three types of Hurwitz numbers, and to the mixed case of any pair.
Following up on the construction of Bridgeland stability condition on $\mathbb{P}^3$ by Macrì, we develop techniques to study concrete wall crossing behavior for the first time on a threefold. In some cases, such as complete intersections of two hypersurfaces of the same degree or twisted cubics, we show that there are two chambers in the stability manifold where the moduli space is given by a smooth projective irreducible variety, respectively the Hilbert scheme. In the case of twisted cubics, we compute all walls and moduli spaces on a path between those two chambers. This allows us to give …
We prove an effective restriction theorem for stable vector bundles $E$ on a smooth projective variety: $E|_D$ is (semi)stable for all irreducible divisors $D \in |kH|$ for all $k$ greater than an explicit constant. As an application, we show that Mercat's conjecture in any rank greater than $2$ fails for curves lying on K3 surfaces. Our technique is to use wall-crossing with respect to (weak) Bridgeland stability conditions which we also use to reprove Camere's result on slope stability of the tangent bundle of $\mathbb{P}^n$ restricted to a K3 surface.
In this paper we study and relate the non-abelian zeta functions introduced by Weng and invariants of the moduli spaces of arbitrary rank stable pairs over curves. We prove a wall-crossing formula for the latter invariants and obtain an explicit formula for these invariants in terms of the motive of a curve. Previously, formulas for these invariants were known only for rank 2 due to Thaddeus and for rank 3 due to Muñoz. Using these results we obtain an explicit formula for the non-abelian zeta functions, we check the uniformity conjecture by Weng for the ranks 2 and 3, and we prove the countin…
Let $C$ be a curve of genus $g \geq 11$ such that $g-1$ is a composite number. Suppose $C$ is on a K3 surface whose Picard group is generated by the curve class $[C]$. We use wall-crossing with respect to Bridgeland stability conditions to generalise Mukai's program to this situation: we show how to reconstruct the K3 surface containing the curve $C$ as a Fourier-Mukai transform of a Brill-Noether locus of vector bundles on $C$.
Consider a smooth projective 3-fold $X$ satisfying the Bogomolov-Gieseker conjecture of Bayer-Macrì-Toda (such as $\mathbb P^3$, the quintic threefold or an abelian threefold). Let $L$ be a line bundle supported on a very positive surface in $X$. If $c_1(L)$ is a primitive cohomology class then we show it has very negative square.
We define rational numbers counting holomorphic disks bounding a complex lagrangian submanifold on a hyperkhaler manifold of real dimension four. We provide a simple a direct proof of Kontsevich-Soibelman Wall Crossing Formula for these rational invariants.
In this paper, we study holomorphic discs in K3 surfaces and defined the open Gromov-Witten invariants. Using this new invariant, we can establish a version of correspondence between tropical discs and holomorphic discs with non-trivial invariants. We give an example of wall-crossing phenomenon of the invariant and expect it satisfies Kontsevich-Soibelman wall-crossing formula.
