We explain how structures analogous to those appearing in the theory of stability conditions on abelian and triangulated categories arise in geometric invariant theory. This leads to an axiomatic notion …
We explain how structures analogous to those appearing in the theory of stability conditions on abelian and triangulated categories arise in geometric invariant theory. This leads to an axiomatic notion of a central charge on a scheme with a group action, and ultimately to a notion of a stability condition on a stack analogous to that on an abelian category. We use these ideas to introduce an axiomatic notion of a stability condition for polarised schemes, defined in such a way that K-stability is a special case. In the setting of axiomatic geometric invariant theory on a smooth projective variety, we produce an analytic counterpart to stability and explain the role of the Kempf-Ness theorem. This clarifies many of the structures involved in the study of deformed Hermitian Yang-Mills connections, Z-critical connections and Z-critical K\"ahler metrics.
ABSTRACT We explain how structures analogous to those appearing in the theory of stability conditions on abelian and triangulated categories arise in geometric invariant theory. This leads to an axiomatic …
ABSTRACT We explain how structures analogous to those appearing in the theory of stability conditions on abelian and triangulated categories arise in geometric invariant theory. This leads to an axiomatic notion of a central charge on a scheme with a group action and ultimately to a notion of a stability condition on a stack analogous to that on an abelian category. In the appendix by Ibáñez Núñez, it is explained how central charges can be viewed through the graded points of a stack. We use these ideas to introduce an axiomatic notion of a stability condition for polarized schemes, defined in such a way that K-stability is a special case. In the setting of axiomatic geometric invariant theory on a smooth projective variety, we produce an analytic counterpart to stability and explain the role of the Kempf–Ness theorem. This clarifies many of the structures involved in the study of deformed Hermitian Yang–Mills connections, Z-critical connections and Z-critical Kähler metrics.
Abstract This book gives an account of the fundamental results in geometric stability theory, a subject that has grown out of categoricity and classification theory. This approach studies the fine …
Abstract This book gives an account of the fundamental results in geometric stability theory, a subject that has grown out of categoricity and classification theory. This approach studies the fine structure of models of stable theories, using the geometry of forking; this often achieves global results relevant to classification theory. Topics range from Zilber-Cherlin classification of infinite locally finite homogenous geometries, to regular types, their geometries, and their role in superstable theories. The structure and existence of definable groups is featured prominently, as is work by Hrushovski. The book is unique in the range and depth of material covered and will be invaluable to anyone interested in modern model theory.
These notes cover some of the foundational results of geometric stability theory. We focus on the geometry of minimal sets. The main aim is an account of Hrushovski's result that …
These notes cover some of the foundational results of geometric stability theory. We focus on the geometry of minimal sets. The main aim is an account of Hrushovski's result that unimodular (in particular, locally finite or pseudo finite) minimal sets are locally modular; along the way, we discuss the Zilber trichotomy and the group and field confi gurations. We assume the basics of stability theory (forking calculus, U-rank, canonical bases, stable groups and homogeneous spaces), as can be found e.g. in Daniel Palacín's chapter in this volume \[5].
This thesis studies stability concepts in Algebraic Geometry. The notion of stability comes from Geometric Invariant Theory, introduced by D. Mumford, in order to construct moduli spaces. More recently, the …
This thesis studies stability concepts in Algebraic Geometry. The notion of stability comes from Geometric Invariant Theory, introduced by D. Mumford, in order to construct moduli spaces. More recently, the space of stability conditions on a derived category was introduced by T. Bridgeland. In this thesis, we outline preliminaries such as complex manifolds, divisors and line bundles, sheaf theory and category theory. We then study the stability of vector bundles on curves of small genus and give an overview of Bridgeland stability conditions. We conclude by examining stability conditions on the category of coherent systems and the category of holomorphic triples in this framework.
In this chapter we extend the notion of a geometric homology and cohomology (mock bundle) theory by allowing
In this chapter we extend the notion of a geometric homology and cohomology (mock bundle) theory by allowing
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A complete theory of integration as it appears in geometric and physical problems must include integration over oriented r-dimensional domains in n-space; both the integrand and the domain may be …
A complete theory of integration as it appears in geometric and physical problems must include integration over oriented r-dimensional domains in n-space; both the integrand and the domain may be variable. This is the primary subject matter of the present book, designed to bring out the underlying geometric and analytic ideas and to give clear and complete proofs of the basic theorems. Originally published in 1957. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.
We consider interpretable topological spaces and topological groups in a p-adically closed field K. We identify a special class of "admissible topologies" with topological tameness properties like generic continuity, similar …
We consider interpretable topological spaces and topological groups in a p-adically closed field K. We identify a special class of "admissible topologies" with topological tameness properties like generic continuity, similar to the topology on definable subsets of Kn. We show that every interpretable set has at least one admissible topology, and that every interpretable group has a unique admissible group topology. We then consider definable compactness (in the sense of Fornasiero) on interpretable groups. We show that an interpretable group is definably compact if and only if it has finitely satisfiable generics (fsg), generalizing an earlier result on definable groups. As a consequence, we see that fsg is a definable property in definable families of interpretable groups, and that any fsg interpretable group defined over Qp is definably isomorphic to a definable group.
Abstract We define the notion of Euler characteristic for definable quotients in an arbitrary o‐minimal structure and prove some fundamental properties.
Abstract We define the notion of Euler characteristic for definable quotients in an arbitrary o‐minimal structure and prove some fundamental properties.
Fix a weakly minimal (i.e. superstable [Formula: see text]-rank [Formula: see text]) structure [Formula: see text]. Let [Formula: see text] be an expansion by constants for an elementary substructure, and …
Fix a weakly minimal (i.e. superstable [Formula: see text]-rank [Formula: see text]) structure [Formula: see text]. Let [Formula: see text] be an expansion by constants for an elementary substructure, and let [Formula: see text] be an arbitrary subset of the universe [Formula: see text]. We show that all formulas in the expansion [Formula: see text] are equivalent to bounded formulas, and so [Formula: see text] is stable (or NIP) if and only if the [Formula: see text]-induced structure [Formula: see text] on [Formula: see text] is stable (or NIP). We then restrict to the case that [Formula: see text] is a pure abelian group with a weakly minimal theory, and [Formula: see text] is mutually algebraic (equivalently, weakly minimal with trivial forking). This setting encompasses most of the recent research on stable expansions of [Formula: see text]. Using various characterizations of mutual algebraicity, we give new examples of stable structures of the form [Formula: see text]. Most notably, we show that if [Formula: see text] is a weakly minimal additive subgroup of the algebraic numbers, [Formula: see text] is enumerated by a homogeneous linear recurrence relation with algebraic coefficients, and no repeated root of the characteristic polynomial of [Formula: see text] is a root of unity, then [Formula: see text] is superstable for any [Formula: see text].
Abstract We construct the free fusion of two geometric thories over a common ω-categorical strongly minimal reduct. If the two theories are supersimple of rank 1 (and satisfy an additional …
Abstract We construct the free fusion of two geometric thories over a common ω-categorical strongly minimal reduct. If the two theories are supersimple of rank 1 (and satisfy an additional hypothesis true in particular for stable theories or trivial reduct), the completions of the free fusion are supersimple of rank at most ω.
Abstract It is proved that all groups of finite U -rank that have the descending chain condition on definable subgroups are totally transcendental. A corollary is that any stable group …
Abstract It is proved that all groups of finite U -rank that have the descending chain condition on definable subgroups are totally transcendental. A corollary is that any stable group that is definable in an o -minimal structure is totally transcendental of finite Morley rank.
Abstract We prove the NTP 1 property of a geometric theory T is inherited by theories of lovely pairs and H ‐structures associated to T . We also provide a …
Abstract We prove the NTP 1 property of a geometric theory T is inherited by theories of lovely pairs and H ‐structures associated to T . We also provide a class of examples of nonsimple geometric NTP 1 theories.
Abstract Let T be a complete geometric theory and let $$T_P$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>T</mml:mi> <mml:mi>P</mml:mi> </mml:msub> </mml:math> be the theory of dense pairs of models of T . We …
Abstract Let T be a complete geometric theory and let $$T_P$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>T</mml:mi> <mml:mi>P</mml:mi> </mml:msub> </mml:math> be the theory of dense pairs of models of T . We show that if T is superrosy with "Equation missing"<!-- image only, no MathML or LaTex -->-rank 1 then $$T_P$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>T</mml:mi> <mml:mi>P</mml:mi> </mml:msub> </mml:math> is superrosy with "Equation missing"<!-- image only, no MathML or LaTex -->-rank at most $$\omega $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ω</mml:mi> </mml:math> .
Abstract Recall that a group G has finitely satisfiable generics ( fsg ) or definable f -generics ( dfg ) if there is a global type p on G and …
Abstract Recall that a group G has finitely satisfiable generics ( fsg ) or definable f -generics ( dfg ) if there is a global type p on G and a small model $M_0$ such that every left translate of p is finitely satisfiable in $M_0$ or definable over $M_0$ , respectively. We show that any abelian group definable in a p -adically closed field is an extension of a definably compact fsg definable group by a dfg definable group. We discuss an approach which might prove a similar statement for interpretable abelian groups. In the case where G is an abelian group definable in the standard model $\mathbb {Q}_p$ , we show that $G^0 = G^{00}$ , and that G is an open subgroup of an algebraic group, up to finite factors. This latter result can be seen as a rough classification of abelian definable groups in $\mathbb {Q}_p$ .
Abstract We study the class of weakly locally modular geometric theories introduced in [4], a common generalization of the classes of linear SU-rank 1 and linear o-minimal theories. We find …
Abstract We study the class of weakly locally modular geometric theories introduced in [4], a common generalization of the classes of linear SU-rank 1 and linear o-minimal theories. We find new conditions equivalent to weak local modularity: “weak one-basedness”, absence of type definable “almost quasidesigns”, and “generic linearity”. Among other things, we show that weak one-basedness is closed under reducts. We also show that the lovely pair expansion of a non-trivial weakly one-based ω -categorical geometric theory interprets an infinite vector space over a finite field.
Consider the expansion $T_S$ of a theory $T$ by a predicate for a submodel of a reduct $T_0$ of $T$. We present a setup in which this expansion admits a …
Consider the expansion $T_S$ of a theory $T$ by a predicate for a submodel of a reduct $T_0$ of $T$. We present a setup in which this expansion admits a model companion $TS$. We show that the nice features of the theory $T$ transfer to $TS$. In particular, we study conditions for which this expansion preserves the $\NSOP{1}$-ness, the simplicity or the stability of the starting theory $T$. We give concrete examples of new $\NSOP{1}$ not simple theories obtained by this process, among them the expansion of a perfect $\omega$-free $\PAC$ field of positive characteristic by generic additive subgroups, and the expansion of an algebraically closed field of \emph{any} characteristic by a generic multiplicative subgroup.
Consider the expansion [Formula: see text] of a theory [Formula: see text] by a predicate for a submodel of a reduct [Formula: see text] of [Formula: see text]. We present …
Consider the expansion [Formula: see text] of a theory [Formula: see text] by a predicate for a submodel of a reduct [Formula: see text] of [Formula: see text]. We present a setup in which this expansion admits a model companion [Formula: see text]. We show that some of the nice features of the theory [Formula: see text] transfer to [Formula: see text]. In particular, we study conditions for which this expansion preserves the [Formula: see text]-ness, the simplicity or the stability of the starting theory [Formula: see text]. We give concrete examples of new [Formula: see text] not simple theories obtained by this process, among them the expansion of a perfect [Formula: see text]-free [Formula: see text] field of positive characteristic by generic additive subgroups, and the expansion of an algebraically closed field of any characteristic by a generic multiplicative subgroup.
Let $\mathbb{M}$ be the monster model of a complete first-order theory $T$. If $\mathbb{D}$ is a subset of $\mathbb{M}$, following D. Zambella we consider $e(\mathbb{D})=\{\mathbb{D}^\prime\mid (\mathbb{M},\mathbb{D})\equiv (\mathbb{M},\mathbb{D}^\prime)\}$ and $o(\mathbb{D})=\{\mathbb{D}^\prime\mid (\mathbb{M},\mathbb{D})\cong …
Let $\mathbb{M}$ be the monster model of a complete first-order theory $T$. If $\mathbb{D}$ is a subset of $\mathbb{M}$, following D. Zambella we consider $e(\mathbb{D})=\{\mathbb{D}^\prime\mid (\mathbb{M},\mathbb{D})\equiv (\mathbb{M},\mathbb{D}^\prime)\}$ and $o(\mathbb{D})=\{\mathbb{D}^\prime\mid (\mathbb{M},\mathbb{D})\cong (\mathbb{M},\mathbb{D}^\prime)\}$. The general question we ask is when $e(\mathbb{D})=o(\mathbb{D})$ ? The case where $\mathbb{D}$ is $A$-invariant for some small set $A$ is rather straightforward: it just mean that $\mathbb{D}$ is definable. We investigate the case where $\mathbb{D}$ is not invariant over any small subset. If T is geometric and $(\mathbb{M},\mathbb{D})$ is an $H$-structure (in the sense of A. Berenstein and E. Vassiliev) or a lovely pair, we get some answers. In the case of $SU$-rank one, $e(\mathbb{D})$ is always different from $o(\mathbb{D})$. In the o-minimal case, everything can happen, depending on the complexity of the definable closure.
We give two alternative proofs of the fact that a modular, surgical pregeometric theory admits geometric elimination of imaginaries.
We give two alternative proofs of the fact that a modular, surgical pregeometric theory admits geometric elimination of imaginaries.
We prove elimination of field quantifiers for strongly dependent henselian fields in the Denef-Pas language. This is achieved by proving the result for a class of fields generalizing algebraically maximal …
We prove elimination of field quantifiers for strongly dependent henselian fields in the Denef-Pas language. This is achieved by proving the result for a class of fields generalizing algebraically maximal Kaplansky fields. We deduce that if $(K,v)$ is strongly dependent then so is its henselization.
An o-minimal expansion $\mathcal{M}=\langle M, <, +, 0, \dots\rangle$ of an ordered group is called <i>semi-bounded</i> if it does not expand a real closed field. Possibly, it defines a real …
An o-minimal expansion $\mathcal{M}=\langle M, <, +, 0, \dots\rangle$ of an ordered group is called <i>semi-bounded</i> if it does not expand a real closed field. Possibly, it defines a real closed field with bounded domain $I\subseteq M$. Let us call
We study sets and groups definable in tame expansions of o-minimal structures. Let $\mathcal {\widetilde M}= \langle \mathcal M, P\rangle$ be an expansion of an o-minimal $\mathcal L$-structure $\cal M$ …
We study sets and groups definable in tame expansions of o-minimal structures. Let $\mathcal {\widetilde M}= \langle \mathcal M, P\rangle$ be an expansion of an o-minimal $\mathcal L$-structure $\cal M$ by a dense set $P$, such that three tameness conditions hold. We prove a structure theorem for definable sets and functions in analogy with the influential cell decomposition theorem known for o-minimal structures. The structure theorem advances the state-of-the-art in all known examples of $\mathcal {\widetilde M}$, as it achieves a decomposition of definable sets into \emph{unions} of `cones', instead of only boolean combinations of them. We also develop the right dimension theory in the tame setting. Applications include: (i) the dimension of a definable set coincides with a suitable pregeometric dimension, and it is invariant under definable bijections, (ii) every definable map is given by an $\cal L$-definable map off a subset of its domain of smaller dimension, and (iii) around generic elements of a definable group, the group operation is given by an $\cal L$-definable map.
We prove the elimination of field quantifiers for strongly dependent henselian fields in the Denef-Pas language. This is achieved by proving the result for a class of fields generalizing algebraically …
We prove the elimination of field quantifiers for strongly dependent henselian fields in the Denef-Pas language. This is achieved by proving the result for a class of fields generalizing algebraically maximal Kaplansky fields. We deduce that if $(K,v)$ is strongly dependent, then so is its henselization.
Let $T$ be a complete theory of fields, possibly with extra structure. Suppose that model-theoretic algebraic closure agrees with field-theoretic algebraic closure, or more generally that model-theoretic algebraic closure has …
Let $T$ be a complete theory of fields, possibly with extra structure. Suppose that model-theoretic algebraic closure agrees with field-theoretic algebraic closure, or more generally that model-theoretic algebraic closure has the exchange property. Then $T$ has uniform finiteness, or equivalently, it eliminates the quantifier $\exists^\infty$. It follows that very slim fields in the sense of Junker and Koenigsmann are the same thing as geometric fields in the sense of Hrushovski and Pillay. Modulo some fine print, these two concepts are also equivalent to algebraically bounded fields in the sense of van den Dries. From the proof, one gets a one-cardinal theorem for geometric theories of fields: any infinite definable set has the same cardinality as the field. We investigate whether this extends to interpretable sets. We show that positive dimensional interpretable sets must have the same cardinality as the field, but zero-dimensional interpretable sets can have smaller cardinality. As an application, we show that any geometric theory of fields has an uncountable model with only countably many finite algebraic extensions.
We construct a topology on a given algebraically closed field with a distinguished subfield which is also algebraically closed. This topology is finer than Zariski topology and it captures the …
We construct a topology on a given algebraically closed field with a distinguished subfield which is also algebraically closed. This topology is finer than Zariski topology and it captures the sets definable in the pair of algebraically closed fields as above; in the sense that definable sets are exactly the constructible sets in this topology.
Abstract We examine several conditions, either the existence of a rank or a particular property of þ-forking that suggest the existence of a well-behaved independence relation, and determine the consequences …
Abstract We examine several conditions, either the existence of a rank or a particular property of þ-forking that suggest the existence of a well-behaved independence relation, and determine the consequences of each of these conditions towards the rosiness of the theory. In particular we show that the existence of an ordinal valued equivalence relation rank is a (necessary and) sufficient condition for rosiness.
We study the theory of the structure induced by parameter free formulas on a “dense” algebraically independent subset of a model of a geometric theory T . We show that …
We study the theory of the structure induced by parameter free formulas on a “dense” algebraically independent subset of a model of a geometric theory T . We show that while being a trivial geometric theory, inherits most of the model theoretic complexity of T related to stability, simplicity, rosiness, the NIP and the NTP 2 . In particular, we show that T is strongly minimal, supersimple of SU‐rank 1, has the NIP or the NTP 2 exactly when has these properties. We show that if T is superrosy of thorn rank 1, then so is , and that the converse holds if T satisfies acl = dcl.
Let M be an O-minimal structure. We use our understanding, acquired in [KPS], of the structure of definable sets of n -tuples in M , to study definable (in M …
Let M be an O-minimal structure. We use our understanding, acquired in [KPS], of the structure of definable sets of n -tuples in M , to study definable (in M ) equivalence relations on M n . In particular, we show that if E is an A -definable equivalence relation on M n ( A ⊂ M ) then E has only finitely many classes with nonempty interior in M n , each such class being moreover also A -definable. As a consequence, we are able to give some conditions under which an O -minimal theory T eliminates imaginaries (in the sense of Poizat [P]). If L is a first order language and M an L -structure, then by a definable set in M , we mean something of the form X ⊂ M n , n ≥ 1, where X = {( a 1 …, a n ) ∈ M n : M ⊨ ϕ (ā)} for some formula ∈ L ( M ). (Here L ( M ) means L together with names for the elements of M .) If the parameters from come from a subset A of M , we say that X is A-definable . M is said to be O-minimal if M = ( M , <,…), where < is a dense linear order with no first or last element, and every definable set X ⊂ M is a finite union of points, and intervals ( a, b ) (where a, b ∈ M ∪ {± ∞}). (This notion is as in [PS] except here we demand the underlying order be dense.) The complete theory T is said to be O-minimal if every model of T is O-minimal. (Note that in [KPS] it is proved that if M is O-minimal, then T = Th( M ) is O-minimal.) In the remainder of this section and in §2, M will denote a fixed but arbitrary O-minimal structure. A,B,C,… will denote subsets of M .
Abstract This book gives an account of the fundamental results in geometric stability theory, a subject that has grown out of categoricity and classification theory. This approach studies the fine …
Abstract This book gives an account of the fundamental results in geometric stability theory, a subject that has grown out of categoricity and classification theory. This approach studies the fine structure of models of stable theories, using the geometry of forking; this often achieves global results relevant to classification theory. Topics range from Zilber-Cherlin classification of infinite locally finite homogenous geometries, to regular types, their geometries, and their role in superstable theories. The structure and existence of definable groups is featured prominently, as is work by Hrushovski. The book is unique in the range and depth of material covered and will be invaluable to anyone interested in modern model theory.