Stable Quotients Research Articles

MAXIMAL STABLE QUOTIENTS OF INVARIANT TYPES IN NIP THEORIES

Abstract For a NIP theory T, a sufficiently saturated model ${\mathfrak C}$ of T, and an invariant (over some small subset of ${\mathfrak C}$ ) global type p, we prove that there exists a finest relatively type-definable over a small set of parameters from ${\mathfrak C}$ equivalence relation on the set of realizations of p which has stable quotient. This is a counterpart for equivalence relations of the main result of [2] on the existence of maximal stable quotients of type-definable groups in NIP theories. Our proof adapts the ideas of the proof of this result, working with relatively type-definable subsets of the group of automorphisms of the monster model as defined in [3].

On Stable Quotients

We solve two problems from a work of Haskel and Pillay concerning maximal stable quotients of groups ∧-definable in NIP theories. The first result says that if G is a ∧-definable group in a distal theory, then Gst=G00 (where Gst is the smallest ∧-definable subgroup with G∕Gst stable, and G00 is the smallest ∧-definable subgroup of bounded index). In order to get it, we prove that distality is preserved under passing from T to the hyperimaginary expansion Theq. The second result is an example of a group G definable in a nondistal NIP theory for which G=G00, but Gst is not an intersection of definable groups. Our example is a saturated extension of (R,+,[0,1]). Moreover, we make some observations on the question whether there is such an example which is a group of finite exponent. We also take the opportunity and give several characterizations of stability of hyperdefinable sets involving continuous logic.

Relative stable categories and birationality

We propose a general method to construct new triangulated categories, relative stable categories, as additive quotients of a given one. This construction enhances results of Beligiannis, particularly in the tensor-triangular setting. We prove a birationality result showing that the original category and its relative stable quotient are equivalent on some open piece of their spectrum.

Relations in the tautological ring of the moduli space of curves

The virtual geometry of the moduli space of stable quotients is used to obtain Chow relations among the κ classes on the moduli space of nonsingular genus g curves. In a series of steps, the stable quotient relations are rewritten in successively simpler forms. The final result is the proof of the Faber-Zagier relations (conjectured in 2000).

Valued fields, metastable groups

We introduce a class of theories called metastable, including the theory of algebraically closed valued fields (\(\mathrm {ACVF}\)) as a motivating example. The key local notion is that of definable types dominated by their stable part. A theory is metastable (over a sort \(\Gamma \)) if every type over a sufficiently rich base structure can be viewed as part of a \(\Gamma \)-parametrized family of stably dominated types. We initiate a study of definable groups in metastable theories of finite rank. Groups with a stably dominated generic type are shown to have a canonical stable quotient. Abelian groups are shown to be decomposable into a part coming from \(\Gamma \), and a definable direct limit system of groups with stably dominated generic. In the case of \(\mathrm {ACVF}\), among definable subgroups of affine algebraic groups, we characterize the groups with stably dominated generics in terms of group schemes over the valuation ring. Finally, we classify all fields definable in \(\mathrm {ACVF}\).

Relative orbifold Donaldson�Thomas theory and the degeneration formula

We generalize the notion of expanded degenerations and pairs for a simple degeneration or smooth pair to the case of smooth Deligne-Mumford stacks. We then define stable quotients on the classifying stacks of expanded degenerations and pairs and prove the properness of their moduli's. On 3-dimensional smooth projective DM stacks this leads to a definition of relative Donaldson-Thomas invariants and the associated degeneration formula.

Stable quotients and the holomorphic anomaly equation

We study the fundamental relationship between stable quotient invariants and the B-model for local P2 in all genera. Our main result is a direct geometric proof of the holomorphic anomaly equation in the precise form predicted by B-model physics. The method yields new holomorphic anomaly equations for an infinite class of twisted theories on projective spaces.An example of such a twisted theory is the formal quintic defined by a hyperplane section of P4 in all genera via the Euler class of a complex. The formal quintic theory is found to satisfy the holomorphic anomaly equations conjectured for the true quintic theory. Therefore, the formal quintic theory and the true quintic theory should be related by transformations which respect the holomorphic anomaly equations.

Endomorphisms, train track maps, and fully irreducible monodromies

Any endomorphism of a finitely generated free group naturally descends to an injective endomorphism of its stable quotient. In this paper, we prove a geometric incarnation of this phenomenon: namely, that every expanding irreducible train track map inducing an endomorphismof the fundamental group gives rise to an expanding irreducible train track representative of the injective endomorphism of the stable quotient. As an application,we prove that the property of having fully irreducible monodromy for a splitting of a hyperbolic free-by-cyclic group depends only on the component of the BNS-invariant containing the associated homomorphism to the integers.

Double and triple Givental’s J-functions for stable quotients invariants

We use mirror formulas for the stable quotients analogue of Givental's J-function for twisted projective invariants obtained in a previous paper to obtain mirror formulas for the analogues of the double and triple Givental's J-functions (with descendants at all marked points) in this setting. We then observe that the genus 0 stable quotients invariants need not satisfy the divisor, string, or dilaton relations of the Gromov-Witten theory, but they do possess the integrality properties of the genus 0 three-point Gromov-Witten invariants of Calabi-Yau manifolds. We also relate the stable quotients invariants to the BPS counts arising in Gromov-Witten theory and obtain mirror formulas for certain twisted Hurwitz numbers.

The geometry of stable quotients in genus one

Stable quotients provide an alternative to stable maps for compactifying spaces of maps. When \(n \ge 2\), the space \(\overline{Q}_{g}({\mathbb {P}}^{n-1},d) = \overline{Q}_{g}(G(1,n),d)\) compactifies the space of degree \(d\) maps of smooth genus \(g\) curves to \({\mathbb {P}}^{n-1}\), while \(\overline{Q}_{g}(G(1,1),d) \simeq \overline{M}_{1, d \cdot \epsilon }/S_d\) is a quotient of a Hassett weighted pointed space. In this paper we study the coarse moduli schemes associated to the smooth proper Deligne–Mumford stacks \(\overline{Q}_{1}({\mathbb {P}}^{n-1},d)\), for all \(n \ge 1\). We show these schemes are projective, unirational, and have Picard number 2. Then we give generators for the Picard group, compute the canonical divisor, the cones of ample divisors, and in the case \(n=1\) the cones of effective divisors. We conclude that \(\overline{Q}_{1}({\mathbb {P}}^{n-1},d)\) is Fano if and only if \(n(d-1)(d+2) < 20\). Moreover, we show that \({\overline{Q}}_{1}({\mathbb {P}}^{n-1},d)\) is a Mori Fiber space for all \(n,d\), hence always minimal in the sense of the minimal model program. In the case \(n=1\), we write in addition a closed formula for the Poincaré polynomial.

Mirror symmetry for stable quotients invariants

The moduli space of stable quotients introduced by Marian-Oprea-Pandharipande provides a natural compactification of the space of morphisms from nonsingular curves to a nonsingular projective variety and carries a natural virtual class. We show that the analogue of Givental's J-function for the resulting twisted projective invariants is described by the same mirror hypergeometric series as the corresponding Gromov-Witten invariants (which arise from the moduli space of stable maps), but without the mirror transform (in the Calabi-Yau case). This implies that the stable quotients and Gromov-Witten twisted invariants agree if there is enough positivity, but not in all cases. As a corollary of the proof, we show that certain twisted Hurwitz numbers arising in the stable quotients theory are also described by a fundamental object associated with this hypergeometric series. We thus completely answer some of the questions posed by Marian-Oprea-Pandharipande concerning their invariants. Our results suggest a deep connection between the stable quotients invariants of complete intersections and the geometry of the mirror families. As in Gromov-Witten theory, computing Givental's J-function (essentially a generating function for genus 0 invariants with 1 marked point) is key to computing stable quotients invariants of higher genus and with more marked points; we exploit this in forthcoming papers.

Stable maps and stable quotients

AbstractWe analyze the relationship between two compactifications of the moduli space of maps from curves to a Grassmannian: the Kontsevich moduli space of stable maps and the Marian–Oprea–Pandharipande moduli space of stable quotients. We construct a moduli space which dominates both the moduli space of stable maps to a Grassmannian and the moduli space of stable quotients, and equip our moduli space with a virtual fundamental class. We relate the virtual fundamental classes of all three moduli spaces using the virtual push-forward formula. This gives a new proof of a theorem of Marian–Oprea–Pandharipande: that enumerative invariants defined as intersection numbers in the stable quotient moduli space coincide with Gromov–Witten invariants.

Automorphisms and connections on Higgs bundles over compact Kähler manifolds

Let (E,φ) be a Higgs vector bundle over a compact connected Kähler manifold X. Fix any filtration of E by coherent analytic subsheaves in which each sheaf is preserved by the Higgs field, and each successive quotient is a torsionfree and stable Higgs sheaf. Denote by G the direct sum of these stable quotients, and let the singular set of G be called S⊂X. We construct a 1-parameter family of filtration preserving C∞ isomorphisms Φt:G|X∖S→E|X∖S, and a Hermitian metric h on G|X\\S, such that as t→+∞, the Chern connection for the Hermitian Higgs bundle (Φt⁎E,Φt⁎φ,h)|X\\S converges, in the C∞ Fréchet topology over any relatively compact open subset of X\\S, to the direct sum of the Yang–Mills–Higgs connections on the direct summands in G.We also prove an analogous result for principal Higgs G-bundles on X.

The limit of the Yang–Mills flow on semi-stable bundles

Abstract By the work of Hong and Tian it is known that given a holomorphic vector bundle E over a compact Kähler manifold X, the Yang–Mills flow converges away from an analytic singular set. If E is semi-stable, then the limiting metric is Hermitian-Einstein and will decompose the limiting bundle E ∞ into a direct sum of stable bundles. Bando and Siu prove E ∞ can be extended to a reflexive sheaf E ^ ∞ ${\hat{E}_\infty }$ on all of X. In this paper, we construct an isomorphism between E ^ ∞ ${\hat{E}_\infty }$ and the double dual of the stable quotients of the graded Seshadri filtration Gr s ( E ) * * ${{\rm Gr}^{\hspace*{0.56905pt}s}(E)^{**}}$ of E.

Virtual push-forwards

Let $p:F\to G$ be a morphism of stacks of positive \emph{virtual} relative dimension $k$ and let $\gamma\in H^k(F)$. We give sufficient conditions for $p_*\gamma\cdot[F]^{virt}$ to be a multiple of $[G]^{virt}$. We apply this result to show an analogue of the conservation of number for virtually smooth families. We show implications to Gromov-Witten invariants and give a new proof of a theorem of Marian, Oprea and Pandharipande which compares the virtual classes of moduli spaces of stable maps and moduli spaces of stable quotients.

The κ ring of the moduli of curves of compact type

The subalgebra of the tautological ring of the moduli of curves of compact type generated by the κ classes is studied in all genera. Relations, constructed via the virtual geometry of the moduli of stable quotients, are used to obtain minimal sets of generators. Bases and Betti numbers of the κ rings are computed. A universality property relating the higher genus κ rings to the genus 0 rings is proven using the virtual geometry of the moduli space of stable maps. The λg-formula for Hodge integrals arises as the simplest consequence.

The moduli space of stable quotients

A moduli space of stable quotients of the rank n trivial sheaf on stable curves is introduced. Over nonsingular curves, the moduli space is Grothendieck's Quot scheme. Over nodal curves, a relative construction is made to keep the torsion of the quotient away from the singularities. New compactifications of classical spaces arise naturally: a nonsingular and irreducible compactification of the moduli of maps from genus 1 curves to projective space is obtained. Localization on the moduli of stable quotients leads to new relations in the tautological ring generalizing Brill-Noether constructions. The moduli space of stable quotients is proven to carry a canonical 2-term obstruction theory and thus a virtual class. The resulting system of descendent invariants is proven to equal the Gromov-Witten theory of the Grassmannian in all genera. Stable quotients can also be used to study Calabi-Yau geometries. The conifold is calculated to agree with stable maps. Several questions about the behavior of stable quotients for arbitrary targets are raised.

Moduli spaces of stable quotients and wall-crossing phenomena

AbstractThe 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 wall-crossing phenomena. We will also discuss Gromov–Witten type invariants associated to ϵ-stable quotients, and investigate them under wall crossing.

A new triangulated category for rational surface singularities

In this paper, we introduce a new triangulated category for rational surface singularities which in the non-Gorenstein case acts as a substitute for the stable category of matrix factorizations. The category is formed as a stable quotient of the Frobenius category of special CM modules, and we classify the relatively projective-injective objects and thus describe the AR quiver of the quotient. Connections to the corresponding reconstruction algebras are also discussed.

Moduli stacks of stable toric quasimaps

We construct new “virtually smooth” modular compactifications of spaces of maps from nonsingular curves to smooth projective toric varieties. They generalize Givental's compactifications, when the complex structure of the curve is allowed to vary and markings are included, and are the toric counterpart of the moduli spaces of stable quotients introduced by Marian, Oprea, and Pandharipande to compactify spaces of maps to Grassmannians. A brief discussion of the resulting invariants and their (conjectural) relation with Gromov–Witten theory is also included.