V2: 94 pages. Many typos fixed, and proofs of Theorems 5.2, 5.14, and 5.15 have been clarified (Lemma 5.16 is new). Postscript added (Remark 1.9) to discuss progress in K-moduli since the first version of this paper. To appear in Proceedings of the London Math Society
We construct proper good moduli spaces parametrizing K-polystable $\mathbb{Q}$-Gorenstein smoothable log Fano pairs $(X, cD)$, where $X$ is a Fano variety and $D$ is a rational multiple of the anti-canonical divisor. We then establish a wall-crossing framework of these K-moduli spaces as $c$ varies. The main application in this paper is the case of plane curves of degree $d \geq 4$ as boundary divisors of $\mathbb{P}^2$. In this case, we show that when the coefficient $c$ is small, the K-moduli space of these pairs is isomorphic to the GIT moduli space. We then show that the first wall crossing of these K-moduli spaces are weighted blow-ups of Kirwan type. We also describe all wall crossings for degree 4,5,6, and relate the final K-moduli spaces to Hacking's compactification and the moduli of K3 surfaces.
We conjecture that the Joyce-Song wall-crossing formula for Donaldson-Thomas invariants arises naturally from an asymptotic expansion in the field theoretic work of Gaiotto, Moore and Neitzke. This would also give a new perspective on how the formulae of Joyce-Song and Kontsevich-Soibelman are related. We check the conjecture in many examples.
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 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 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 $Stab(\mathbb{P}^1\times \mathbb{P}^1)$.
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.
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 survey paper, we briefly review various aspects of the SYZ approach to mirror symmetry for non-Calabi-Yau varieties, focusing in particular on Lagrangian fibrations and wall-crossing phenomena in Floer homology. Various examples are presented, some of them new.
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 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 …
This paper solves the global moduli problem for regular holonomic D-modules with normal crossing singularities on a nonsingular complex projective variety. This is done by introducing a level structure (which gives rise to ``pre-D-modules''), and then introducing a notion of (semi-)stability and applying Geometric Invariant Theory to construct a coarse moduli scheme for semistable pre-D-modules. A moduli is constructed also for the corresponding perverse sheaves, and the Riemann-Hilbert correspondence is represented by an analytic morphism between these moduli spaces.
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 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.
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.
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 $W$.
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.
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…
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.
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.
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…
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.
We study two instanton correction problems of Hitchin's moduli spaces along with their wall crossing formulas. The hyperkahler metric of a Hitchin's moduli space can be put into an instanton-corrected form according to physicists Gaiotto, Moore and Neitzke. The problem boils down to the construction of a set of special coordinates which can be constructed as Fock-Goncharov coordinates associated with foliations of quadratic differentials on a Riemann surface. A wall crossing formula of Kontsevich and Soibelman arises both as a crucial consistency condition and an effective computational tool. …
We describe a close relation between wall crossings in the birational geometry of moduli space of Gieseker stable sheaves $M_H(v)$ on $\bb{P}^2$ and mini-wall crossings in the stability manifold $Stab(D^b(\bb{P}^2))$.
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…
In this paper we propose definitions and examples of categorical enhancements of the data involved in the $2d$-$4d$ wall-crossing formulas which generalize both Cecotti-Vafa and Kontsevich-Soibelman motivic wall-crossing formulas.
We prove the equivalence of (a slightly modified version of) the wall-crossing formula of Manschot, Pioline and Sen and the wall-crossing formula of Kontsevich and Soibelman. The former involves abelian analogues of the motivic Donaldson-Thomas type invariants of quivers with stability introduced by Kontsevich and Soibelman, for which we derive positivity and geometricity properties.
We extend the ideas of Friedman and Qin (Flips of moduli spaces and transition formulae for Donaldson polynomial invariants of rational surfaces) to find the wall-crossing formulae for the Donaldson invariants of algebraic surfaces with geometrical genus zero, positive irregularity and anticanonical divisor effective, for any wall $ζ$ with $l_ζ=(ζ\sp{2}-p_1)/4$ being zero or one.
BPS states in N=4 supersymmetric SU(N) gauge theories in four dimensions can be represented as planar string networks with ends lying on D3-branes. We introduce several protected indices which capture information on the spectrum and various quantum numbers of these states, give their wall crossing formula and describe how using the wall crossing formula we can compute all the indices at all points in the moduli space.
We study the wall-crossing phenomena of D4-D2-D0 bound states with two units of D4-brane charge on the resolved conifold. We identify the walls of marginal stability and evaluate the discrete changes of the BPS indices by using the Kontsevich-Soibelman wall-crossing formula. In particular, we find that the field theories on D4-branes in two large radius limits are properly connected by the wall-crossings involving the flop transition of the conifold. We also find that in one of the large radius limits there are stable bound states of two D4-D2-D0 fragments.
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.
For a balanced wall crossing in geometric invariant theory, there exist derived equivalences between the corresponding GIT quotients if certain numerical conditions are satisfied. Given such a wall crossing, I construct a perverse sheaf of categories on a disk, singular at a point, with half-monodromies recovering these equivalences, and with behaviour at the singular point controlled by a GIT quotient stack associated to the wall. Taking complexified Grothendieck groups gives a perverse sheaf of vector spaces: I characterise when this is an intersection cohomology complex of a local system on…
We prove wall-crossing formulas for the motivic invariants of the moduli spaces of framed objects in the ind-constructible abelian categories. Developed techniques are applied in the case of the motivic Donaldson-Thomas invariants of quivers with potentials. Another application is a new proof of the formula for the motivic invariants of smooth models of quiver moduli spaces.
In recent work, the authors derived a tropical interpretation of monotone and strictly monotone double Hurwitz numbers. In this paper, we apply the technique of tropical flows to this interpretation in order to provide a new proof of the piecewise polynomiality of these enumerative invariants. Moreover, we derive new types of wall-crossing formulae.
We study the relation between the instanton counting on ALE spaces and the BPS state counting on a toric Calabi-Yau three-fold. We put a single D4-brane on a divisor isomorphic to A_{N-1}-ALE space in the Calabi-Yau three-fold, and evaluate the discrete changes of BPS partition function of D4-D2-D0 states in the wall-crossing phenomena. In particular, we find that the character of affine SU(N) algebra naturally arises in wall-crossings of D4-D2-D0 states. Our analysis is completely based on the wall-crossing formula for the d=4, N=2 supersymmetric theory obtained by dimensionally reducing the …
