Edward Bierstone - Profile on Academia.edu (original) (raw)

Papers by Edward Bierstone

arXiv (Cornell University), Jul 27, 2011

In this sequel to [4], we find the smallest class of singularities in four variables with which w... more In this sequel to [4], we find the smallest class of singularities in four variables with which we necessarily end up if we resolve singularities except for normal crossings. The main new feature is a characterization of singularities in four variables which occur as limits of triple normal crossings singularities, and which cannot be eliminated by a birational morphism that avoids blowing up normal crossings singularities. This result develops the philsophy of [4], that the desingularization invariant together with natural geometric information can be used to compute local normal forms of singularities.

arXiv (Cornell University), Jan 18, 2009

Our aim is to understand the algebraic notion of flatness in explicit geometric terms. Let ϕ : X ... more Our aim is to understand the algebraic notion of flatness in explicit geometric terms. Let ϕ : X → Y be a morphism of complex-analytic spaces, where Y is smooth. We prove that nonflatness of ϕ is equivalent to a severe discontinuity of the fibres-the existence of a vertical component (a local irreducible component at a point of the source whose image is nowhere-dense in Y)-after passage to the n-fold fibred power of ϕ, where n = dim Y. Our main theorem is a more general criterion for flatness over Y of a coherent sheaf of modules F on X. In the case that ϕ is a morphism of complex algebraic varieties, the result implies that the stalk F ξ of F at a point ξ ∈ X is flat over R := O Y,ϕ(ξ) if and only if its n-fold tensor power is a torsion-free R-module (conjecture of Vasconcelos in the case of C-algebras).

arXiv (Cornell University), Nov 27, 2013

The subject is partial resolution of singularities. Given an algebraic variety X (not necessarily... more The subject is partial resolution of singularities. Given an algebraic variety X (not necessarily equidimensional) in characteristic zero (or, more generally, a pair (X, D), where D is a divisor on X), we construct a functorial desingularization of all but stable simple normal crossings (stablesnc) singularities, by smooth blowings-up that preserve such singularities. A variety has stable simple normal crossings at a point if, locally, its irreducible components are smooth and tranverse in some smooth embedding variety. We also show that our main assertion is false for more general simple normal crossings singularities. Contents 1. Introduction 1 2. Presentation of an invariant 6 3. Characterization of stable-snc singularities of a variety with snc divisor 10 4. Cleaning 13 5. Desingularization of a variety preserving stable-snc singularities 15 6. Characterization of stable-snc singularities of a triple 16 7. The Hilbert-Samuel function and stable simple normal crossings 19 8. Algorithm for the main theorem 24 9. Desingularization at the singular locus of X 26 10. The non-reduced case 30 References 31

arXiv (Cornell University), Sep 14, 2011

Let X denote a reduced algebraic variety and D a Weil divisor on X. The pair (X, D) is said to be... more Let X denote a reduced algebraic variety and D a Weil divisor on X. The pair (X, D) is said to be semi-simple normal crossings (semi-snc) at a ∈ X if X is simple normal crossings at a (i.e., a simple normal crossings hypersurface, with respect to a local embedding in a smooth ambient variety), and D is induced by the restriction to X of a hypersurface that is simple normal crossings with respect to X. We construct a composition of blowings-up f : X → X such that the transformed pair (X, D) is everywhere semi-simple normal crossings, and f is an isomorphism over the semi-simple normal crossings locus of (X, D). The result answers a question of Kollár. Contents 1. Introduction 2 2. Characterization of semi-snc points 4 3. Basic notions and structure of the proof 6 4. The Hilbert-Samuel function and semi-simple normal crossings 12 5. Algorithm for the main theorem 20 6. The case of more than 2 components 23 7. The case of two components 31 8. The non-reduced case 37 9. Functoriality 39 References 40

arXiv (Cornell University), Jan 18, 2009

Our aim is to understand the algebraic notion of flatness in explicit geometric terms. Let ϕ : X ... more Our aim is to understand the algebraic notion of flatness in explicit geometric terms. Let ϕ : X → Y be a morphism of complex-analytic spaces, where Y is smooth. We prove that nonflatness of ϕ is equivalent to a severe discontinuity of the fibres-the existence of a vertical component (a local irreducible component at a point of the source whose image is nowhere-dense in Y)-after passage to the n-fold fibred power of ϕ, where n = dim Y. Our main theorem is a more general criterion for flatness over Y of a coherent sheaf of modules F on X. In the case that ϕ is a morphism of complex algebraic varieties, the result implies that the stalk F ξ of F at a point ξ ∈ X is flat over R := O Y,ϕ(ξ) if and only if its n-fold tensor power is a torsion-free R-module (conjecture of Vasconcelos in the case of C-algebras).

arXiv (Cornell University), Apr 27, 2015

The main problem studied here is resolution of singularities of the cotangent sheaf of a complex-... more The main problem studied here is resolution of singularities of the cotangent sheaf of a complex-or real-analytic variety X 0 (or of an algebraic variety X 0 over a field of characteristic zero). Given X 0 , we ask whether there is a global resolution of singularities σ : X → X 0 such that the pulled-back cotangent sheaf of X 0 is generated by differential monomials in suitable coordinates at every point of X ("Hsiang-Pati coordinates"). Desingularization of the cotangent sheaf is equivalent to monomialization of Fitting ideals generated by minors of a given order of the logarithmic Jacobian matrix of σ. We prove resolution of singularities of the cotangent sheaf in dimension up to three. It was previously known for surfaces with isolated singularities (Hsiang-Pati 1985, Pardon-Stern 2001). Consequences include monomialization of the induced Fubini-Study metric on the smooth part of a complex projective variety X 0 ; there have been important applications of the latter to L 2-cohomology.

Bulletin of the American Mathematical Society, 1991

We announce solutions of two fundamental problems in differential analysis and real analytic geom... more We announce solutions of two fundamental problems in differential analysis and real analytic geometry, on composite differentiable functions and on semicoherence of subanalytic sets. Our main theorem asserts that the problems are equivalent and gives several natural necessary and sufficient conditions in terms of semicontinuity of discrete local invariants and metric properties of a closed subanalytic set.

arXiv (Cornell University), Nov 20, 2001

In 1934, Whitney raised the question of how to recognize whether a function f defined on a closed... more In 1934, Whitney raised the question of how to recognize whether a function f defined on a closed subset X of R n is the restriction of a function of class C p. A necessary and sufficient criterion was given in the case n = 1 by Whitney, using limits of finite differences, and in the case p = 1 by Glaeser (1958), using limits of secants. We introduce a necessary geometric criterion, for general n and p, involving limits of finite differences, that we conjecture is sufficient at least if X has a "tame topology". We prove that, if X is a compact subanalytic set, then there exists q = q X (p) such that the criterion of order q implies that f is C p. The result gives a new approach to higher-order tangent bundles (or bundles of differentiable operators) on singular spaces.

arXiv (Cornell University), Aug 29, 2001

We show that a version of the desingularization theorem of Hironaka holds for certain classes of ... more We show that a version of the desingularization theorem of Hironaka holds for certain classes of C ∞ functions (essentially, for subrings that exclude flat functions and are closed under differentiation and the solution of implicit equations). Examples are quasianalytic classes, introduced by E. Borel a century ago and characterized by the Denjoy-Carleman theorem. These classes have been poorly understood in dimension > 1. Resolution of singularities can be used to obtain many new results; for example, topological Noetherianity, Lojasiewicz inequalities, division properties.

arXiv (Cornell University), Jul 27, 2011

The philosophy of this article is that the desingularization invariant together with natural geom... more The philosophy of this article is that the desingularization invariant together with natural geometric information can be used to compute local normal forms of singularities. The idea is used in two related problems: (1) We give a proof of resolution of singularities of a variety or a divisor, except for simple normal crossings (i.e., which avoids blowing up simple normal crossings, and ends up with a variety or a divisor having only simple normal crossings singularities). (2) For more general normal crossings (in a local analytic or formal sense), such a result does not hold. We find the smallest class of singularities (in low dimension or low codimension) with which we necessarily end up if we avoid blowing up normal crossings singularities. Several of the questions studied were raised by Kollár. Contents 21 5. Appendix. Crash course on the desingularization invariant 26 References 40

In this sequel to [4], we find the smallest class of singularities in four variables with which w... more In this sequel to [4], we find the smallest class of singularities in four variables with which we necessarily end up if we resolve singularities except for normal crossings. The main new feature is a characterization of singularities in four variables which occur as limits of triple normal crossings singularities, and which cannot be eliminated by a birational morphism that avoids blowing up normal crossings singularities. This result develops the philsophy of [4], that the desingularization invariant together with natural geometric information can be used to compute local normal forms of singularities.

arXiv (Cornell University), Nov 28, 2022

We address the following question. Given an algebraic (or analytic) variety X in characteristic z... more We address the following question. Given an algebraic (or analytic) variety X in characteristic zero, can we find a finite sequence of blowingsup preserving the normal-crossings locus of X, after which the transform X ′ of X has smooth normalization? More precisely, we ask whether there is such a partial desingularization where X ′ has only singularities from an explicit finite list of minimal singularities, defined using the determinants of circulant matrices. In the case of surfaces, for example, the pinch point or Whitney umbrella is the only singularity needed in addition to normal crossings. We develop techniques for factorization (splitting) of a monic polynomial with regular (or analytic) coefficients, satisfying a generic normal crossings hypothesis, which we use together with resolution of singularities techniques to find local circulant normal forms of singularities. These techniques in their current state are enough for positive answers to the questions above, for dim X ≤ 4, or in arbitrary dimension if we preserve normal crossings only of order at most three. Contents 9 3. Splitting results 11 4. Limits of k-fold normal crossings in k + 1 variables 17 5. Limits of triple normal crossings 24 6. Partial desingularization algorithm 30 References 45

Conference Festschrift Arnol'd (Vladimir Igorevich) on his 60th birthday

We introduce a new point of view towards Glaeser's theorem on composite C^∞ functions [Ann. o... more We introduce a new point of view towards Glaeser's theorem on composite C^∞ functions [Ann. of Math. 1963], with respect to which we can formulate a "C^k composite function property" that is satisfied by all semiproper real analytic mappings. As a consequence, we see that a closed subanalytic set X satisfies the C^∞ composite function property if and only if the ring C^∞ (X) of C^∞ functions on X is the intersection of all finite differentiability classes.

Algebras of composite differentiable functions

[

In this sequel to [4], we find the smallest class of singularities in four variables with which w... more In this sequel to [4], we find the smallest class of singularities in four variables with which we necessarily end up if we resolve singularities except for normal crossings. The main new feature is a characterization of singularities in four variables which occur as limits of triple normal crossings singularities, and which cannot be eliminated by a birational morphism that avoids blowing up normal crossings singularities. This result develops the philsophy of [4], that the desingularization invariant together with natural geometric information can be used to compute local normal forms of singularities.

The structure of orbit spaces and the singularities equivariant mappings

The extension problem and related themes in differential analysis

Proceedings of Symposia in Pure Mathematics, 1983

7:00–9:00 Breakfast 9:00–9:15 Welcome and introduction. SCHEDULE 9:20–10:10 Dan Abramovich, Overview: Resolution of singularities and toroidal geometry

Resolution of singularities, factorization of birational mappings, and

Banach Center Publications, 1988

arXiv (Cornell University), Jul 27, 2011

In this sequel to [4], we find the smallest class of singularities in four variables with which w... more In this sequel to [4], we find the smallest class of singularities in four variables with which we necessarily end up if we resolve singularities except for normal crossings. The main new feature is a characterization of singularities in four variables which occur as limits of triple normal crossings singularities, and which cannot be eliminated by a birational morphism that avoids blowing up normal crossings singularities. This result develops the philsophy of [4], that the desingularization invariant together with natural geometric information can be used to compute local normal forms of singularities.

arXiv (Cornell University), Jan 18, 2009

Our aim is to understand the algebraic notion of flatness in explicit geometric terms. Let ϕ : X ... more Our aim is to understand the algebraic notion of flatness in explicit geometric terms. Let ϕ : X → Y be a morphism of complex-analytic spaces, where Y is smooth. We prove that nonflatness of ϕ is equivalent to a severe discontinuity of the fibres-the existence of a vertical component (a local irreducible component at a point of the source whose image is nowhere-dense in Y)-after passage to the n-fold fibred power of ϕ, where n = dim Y. Our main theorem is a more general criterion for flatness over Y of a coherent sheaf of modules F on X. In the case that ϕ is a morphism of complex algebraic varieties, the result implies that the stalk F ξ of F at a point ξ ∈ X is flat over R := O Y,ϕ(ξ) if and only if its n-fold tensor power is a torsion-free R-module (conjecture of Vasconcelos in the case of C-algebras).

arXiv (Cornell University), Nov 27, 2013

The subject is partial resolution of singularities. Given an algebraic variety X (not necessarily... more The subject is partial resolution of singularities. Given an algebraic variety X (not necessarily equidimensional) in characteristic zero (or, more generally, a pair (X, D), where D is a divisor on X), we construct a functorial desingularization of all but stable simple normal crossings (stablesnc) singularities, by smooth blowings-up that preserve such singularities. A variety has stable simple normal crossings at a point if, locally, its irreducible components are smooth and tranverse in some smooth embedding variety. We also show that our main assertion is false for more general simple normal crossings singularities. Contents 1. Introduction 1 2. Presentation of an invariant 6 3. Characterization of stable-snc singularities of a variety with snc divisor 10 4. Cleaning 13 5. Desingularization of a variety preserving stable-snc singularities 15 6. Characterization of stable-snc singularities of a triple 16 7. The Hilbert-Samuel function and stable simple normal crossings 19 8. Algorithm for the main theorem 24 9. Desingularization at the singular locus of X 26 10. The non-reduced case 30 References 31

arXiv (Cornell University), Sep 14, 2011

Let X denote a reduced algebraic variety and D a Weil divisor on X. The pair (X, D) is said to be... more Let X denote a reduced algebraic variety and D a Weil divisor on X. The pair (X, D) is said to be semi-simple normal crossings (semi-snc) at a ∈ X if X is simple normal crossings at a (i.e., a simple normal crossings hypersurface, with respect to a local embedding in a smooth ambient variety), and D is induced by the restriction to X of a hypersurface that is simple normal crossings with respect to X. We construct a composition of blowings-up f : X → X such that the transformed pair (X, D) is everywhere semi-simple normal crossings, and f is an isomorphism over the semi-simple normal crossings locus of (X, D). The result answers a question of Kollár. Contents 1. Introduction 2 2. Characterization of semi-snc points 4 3. Basic notions and structure of the proof 6 4. The Hilbert-Samuel function and semi-simple normal crossings 12 5. Algorithm for the main theorem 20 6. The case of more than 2 components 23 7. The case of two components 31 8. The non-reduced case 37 9. Functoriality 39 References 40

arXiv (Cornell University), Jan 18, 2009

Our aim is to understand the algebraic notion of flatness in explicit geometric terms. Let ϕ : X ... more Our aim is to understand the algebraic notion of flatness in explicit geometric terms. Let ϕ : X → Y be a morphism of complex-analytic spaces, where Y is smooth. We prove that nonflatness of ϕ is equivalent to a severe discontinuity of the fibres-the existence of a vertical component (a local irreducible component at a point of the source whose image is nowhere-dense in Y)-after passage to the n-fold fibred power of ϕ, where n = dim Y. Our main theorem is a more general criterion for flatness over Y of a coherent sheaf of modules F on X. In the case that ϕ is a morphism of complex algebraic varieties, the result implies that the stalk F ξ of F at a point ξ ∈ X is flat over R := O Y,ϕ(ξ) if and only if its n-fold tensor power is a torsion-free R-module (conjecture of Vasconcelos in the case of C-algebras).

arXiv (Cornell University), Apr 27, 2015

The main problem studied here is resolution of singularities of the cotangent sheaf of a complex-... more The main problem studied here is resolution of singularities of the cotangent sheaf of a complex-or real-analytic variety X 0 (or of an algebraic variety X 0 over a field of characteristic zero). Given X 0 , we ask whether there is a global resolution of singularities σ : X → X 0 such that the pulled-back cotangent sheaf of X 0 is generated by differential monomials in suitable coordinates at every point of X ("Hsiang-Pati coordinates"). Desingularization of the cotangent sheaf is equivalent to monomialization of Fitting ideals generated by minors of a given order of the logarithmic Jacobian matrix of σ. We prove resolution of singularities of the cotangent sheaf in dimension up to three. It was previously known for surfaces with isolated singularities (Hsiang-Pati 1985, Pardon-Stern 2001). Consequences include monomialization of the induced Fubini-Study metric on the smooth part of a complex projective variety X 0 ; there have been important applications of the latter to L 2-cohomology.

Bulletin of the American Mathematical Society, 1991

We announce solutions of two fundamental problems in differential analysis and real analytic geom... more We announce solutions of two fundamental problems in differential analysis and real analytic geometry, on composite differentiable functions and on semicoherence of subanalytic sets. Our main theorem asserts that the problems are equivalent and gives several natural necessary and sufficient conditions in terms of semicontinuity of discrete local invariants and metric properties of a closed subanalytic set.

arXiv (Cornell University), Nov 20, 2001

In 1934, Whitney raised the question of how to recognize whether a function f defined on a closed... more In 1934, Whitney raised the question of how to recognize whether a function f defined on a closed subset X of R n is the restriction of a function of class C p. A necessary and sufficient criterion was given in the case n = 1 by Whitney, using limits of finite differences, and in the case p = 1 by Glaeser (1958), using limits of secants. We introduce a necessary geometric criterion, for general n and p, involving limits of finite differences, that we conjecture is sufficient at least if X has a "tame topology". We prove that, if X is a compact subanalytic set, then there exists q = q X (p) such that the criterion of order q implies that f is C p. The result gives a new approach to higher-order tangent bundles (or bundles of differentiable operators) on singular spaces.

arXiv (Cornell University), Aug 29, 2001

We show that a version of the desingularization theorem of Hironaka holds for certain classes of ... more We show that a version of the desingularization theorem of Hironaka holds for certain classes of C ∞ functions (essentially, for subrings that exclude flat functions and are closed under differentiation and the solution of implicit equations). Examples are quasianalytic classes, introduced by E. Borel a century ago and characterized by the Denjoy-Carleman theorem. These classes have been poorly understood in dimension > 1. Resolution of singularities can be used to obtain many new results; for example, topological Noetherianity, Lojasiewicz inequalities, division properties.

arXiv (Cornell University), Jul 27, 2011

The philosophy of this article is that the desingularization invariant together with natural geom... more The philosophy of this article is that the desingularization invariant together with natural geometric information can be used to compute local normal forms of singularities. The idea is used in two related problems: (1) We give a proof of resolution of singularities of a variety or a divisor, except for simple normal crossings (i.e., which avoids blowing up simple normal crossings, and ends up with a variety or a divisor having only simple normal crossings singularities). (2) For more general normal crossings (in a local analytic or formal sense), such a result does not hold. We find the smallest class of singularities (in low dimension or low codimension) with which we necessarily end up if we avoid blowing up normal crossings singularities. Several of the questions studied were raised by Kollár. Contents 21 5. Appendix. Crash course on the desingularization invariant 26 References 40

In this sequel to [4], we find the smallest class of singularities in four variables with which w... more In this sequel to [4], we find the smallest class of singularities in four variables with which we necessarily end up if we resolve singularities except for normal crossings. The main new feature is a characterization of singularities in four variables which occur as limits of triple normal crossings singularities, and which cannot be eliminated by a birational morphism that avoids blowing up normal crossings singularities. This result develops the philsophy of [4], that the desingularization invariant together with natural geometric information can be used to compute local normal forms of singularities.

arXiv (Cornell University), Nov 28, 2022

We address the following question. Given an algebraic (or analytic) variety X in characteristic z... more We address the following question. Given an algebraic (or analytic) variety X in characteristic zero, can we find a finite sequence of blowingsup preserving the normal-crossings locus of X, after which the transform X ′ of X has smooth normalization? More precisely, we ask whether there is such a partial desingularization where X ′ has only singularities from an explicit finite list of minimal singularities, defined using the determinants of circulant matrices. In the case of surfaces, for example, the pinch point or Whitney umbrella is the only singularity needed in addition to normal crossings. We develop techniques for factorization (splitting) of a monic polynomial with regular (or analytic) coefficients, satisfying a generic normal crossings hypothesis, which we use together with resolution of singularities techniques to find local circulant normal forms of singularities. These techniques in their current state are enough for positive answers to the questions above, for dim X ≤ 4, or in arbitrary dimension if we preserve normal crossings only of order at most three. Contents 9 3. Splitting results 11 4. Limits of k-fold normal crossings in k + 1 variables 17 5. Limits of triple normal crossings 24 6. Partial desingularization algorithm 30 References 45

Conference Festschrift Arnol'd (Vladimir Igorevich) on his 60th birthday

We introduce a new point of view towards Glaeser's theorem on composite C^∞ functions [Ann. o... more We introduce a new point of view towards Glaeser's theorem on composite C^∞ functions [Ann. of Math. 1963], with respect to which we can formulate a "C^k composite function property" that is satisfied by all semiproper real analytic mappings. As a consequence, we see that a closed subanalytic set X satisfies the C^∞ composite function property if and only if the ring C^∞ (X) of C^∞ functions on X is the intersection of all finite differentiability classes.

Algebras of composite differentiable functions

[

In this sequel to [4], we find the smallest class of singularities in four variables with which w... more In this sequel to [4], we find the smallest class of singularities in four variables with which we necessarily end up if we resolve singularities except for normal crossings. The main new feature is a characterization of singularities in four variables which occur as limits of triple normal crossings singularities, and which cannot be eliminated by a birational morphism that avoids blowing up normal crossings singularities. This result develops the philsophy of [4], that the desingularization invariant together with natural geometric information can be used to compute local normal forms of singularities.

The structure of orbit spaces and the singularities equivariant mappings

The extension problem and related themes in differential analysis

Proceedings of Symposia in Pure Mathematics, 1983

7:00–9:00 Breakfast 9:00–9:15 Welcome and introduction. SCHEDULE 9:20–10:10 Dan Abramovich, Overview: Resolution of singularities and toroidal geometry

Resolution of singularities, factorization of birational mappings, and

Banach Center Publications, 1988