A Morse theoretic approach to non-isolated singularities and applications to optimization
AA MORSE THEORETIC APPROACH TO NON-ISOLATEDSINGULARITIES AND APPLICATIONS TO OPTIMIZATION
LAURENTIU G. MAXIM, JOSE ISRAEL RODRIGUEZ, AND BOTONG WANG
Abstract.
Let X be a complex affine variety in C N , and let f : C N → C be a polyno-mial function whose restriction to X is nonconstant. For g : C N → C a general linearfunction, we study the limiting behavior of the critical points of the one-parameterfamily of f t := f − tg as t →
0. Our main result gives an expression of this limit interms of critical sets of the restrictions of g to the singular strata of ( X, f ). We applythis result in the context of optimization problems. For example, we consider nearestpoint problems (e.g., Euclidean distance degrees) for affine varieties and a possiblynongeneric data point. Introduction
The motivation for this work is to study nearest point problems for algebraic modelsand Euclidean distance degrees. For example, given a circle and a point P outside itscenter, there is a unique point on the circle which is closest to P , as seen in Figure 1.However, if P is taken to be the center, then every point on the circle is a closest point.Our aim is to understand such a special (non-generic) behavior on (arbitrary) algebraicvarieties by a limiting procedure on a set of critical points. In terms of applied algebraicgeometry, our results can be understood as describing what happens when genericityassumptions of statements on Euclidean distance degrees are removed (see Section 5.3).In optimization, our results state what happens as we take a regularization term to zero.Before stating the general result, we start with the following simple, but enlighteningexample. Let X = C N , and let f : C N → C be a polynomial function with isolatedcritical points P , . . . , P l . In this case, the singularity behavior of the function f ateach P i is governed by the Milnor number of f at P i (see [23]), which we denote by n i .In particular, f is a holomorphic Morse function (that is, it has only non-degenerateisolated critical points) if and only if each Milnor number n i is 1. Fix a general linearfunction g : C N → C . Then f t := f − tg is a holomorphic Morse function on C N for allbut finitely many t ∈ C . The limit of the critical locus of f t has the following behavior as t goes to 0. In a small neighborhood U i of P i , there are n i non-degenerate critical pointsof f t for nonzero t with small absolute value. As t approaches zero, these critical pointscollide together to P i . This process is the Morsification of f , which is a well-known resultin singularity theory (see [5, Appendix]). Date : February 4, 2020.2010
Mathematics Subject Classification.
Key words and phrases.
Euclidean distance degree, Euler characteristic, local Euler obstruction func-tion, optimal solution, stationary point, maximum likelihood, objective function. a r X i v : . [ m a t h . A T ] F e b LAURENTIU G. MAXIM, JOSE ISRAEL RODRIGUEZ, AND BOTONG WANG
Figure 1.
Each red curves is the set of real points of an algebraic variety X and a purple point on the curve is a critical point of the distance functionwith respect to U . As U moves along the dotted path, the critical pointsmove along the purple arcs. Left : When U is at the origin, every pointon the circle X is equidistant from U . Right : As U approaches the origin,the three critical points move along the cardioid curve X and two of themcome together. The green curve denotes the ED discriminant and when U is in the shaded region there is only one real critical point on the smoothlocus of X .In general, we allow X to be a possibly singular subvariety of C N , and we allow f to be any polynomial whose restriction to X is nonconstant. If g is a general linearfunction, then f t := f − tg is a holomorphic Morse function on the smooth locus X reg of X for all but finitely many t ∈ C . We are interested in the limiting behavior of the set of critical points of f t | X reg as t approaches zero.In order to formulate our main result, let X ⊂ C N be a complex affine variety and let f : C N → C be a polynomial function whose restriction to X is nonconstant. Considera stratification X = (cid:83) i ∈ I X i of X into smooth locally closed subvarieties such that theLagrangian cycles of the perverse vanishing cycle functors p Φ f − c ([ T ∗ X C N ]) are “locallyconstant along X i ” for all values of c ∈ C and all i ∈ I . Such a stratification of X can be obtained explicitly as follows. As it will be explained in Section 2.2, there existsa constructible complex F (cid:5) on C N with support on X , whose characteristic cycle isexactly the conormal space T ∗ X C N . We regard F (cid:5) as a constructible complex on X . Therestriction f | X has only finitely many critical values in the stratified sense (see, e.g., [6,Definition 4.2.7]), and for each such critical value c ∈ C of f | X the (perverse) vanishingcycle functor p Φ f − c : D bc ( X ) → D bc ( X ∩ { f = c } ) is constructible and supported in thestratified singular locus of f (see, e.g., [6, Proposition 4.2.8]). Choose a stratificationof X ∩ { f = c } into smooth locally closed subvarieties with respect to which p Φ f − c ( F (cid:5) ) is constructible. The required stratification of X is then obtained by collecting allthe strata in X ∩ { f = c } for each critical value c of f | X , together with a Whitneystratification of the complement of these critical fibers in X . Once such a stratification X = (cid:83) i ∈ I X i of X is fixed, we have the following equality of Lagrangian cycles(1) (cid:88) c ∈ C p Φ f − c ([ T ∗ X C N ]) = (cid:88) i ∈ I n i [ T ∗ X i C N ]for n i ∈ Z ≥ . Notice that the sum on the left-hand side of (1) is a finite sum, since f | X has only finitely many critical values in the stratified sense, and p Φ f − c ([ T ∗ X C N ]) = 0when c is not a critical value. Moreover, it follows from work of Massey (see Theorem3.5) that the coefficients n i are nonnegative.By the characteristic cycle functor (see (9)), equation (1) amounts to express (up tosigns), for each critical value c of f , the constructible function ϕ f − c (Eu X ) in terms ofthe basis of local Euler obstruction functions Eu X i corresponding to closures of strata in f = c . Here, Eu X denote the local Euler obstruction function introduced by MacPhersonin [16]. In general, an explicit calculation of the coefficients n i is difficult (see Example5.6). However, when f | X has simple singularities, the vanishing cycle on the left-handside of (1) can be computed by hand, as the following examples show. Example 1.1.
Suppose that X is smooth and f | X has isolated critical points P , . . . , P l .Then we can take the stratification X = X \ { P , . . . , P l } and X i = { P i } . The corresponding coefficients n i in (1) are computed directly as n = 0, and n i isequal to the Milnor number of f | X at P i , that is the length of the Artinian algebra O X,P i / (cid:104) ∂f∂z , . . . , ∂f∂z d (cid:105) , where O X,P i is the germ of holomorphic functions on X at P i and z , . . . , z d ∈ O X,P i are the local coordinates. Example 1.2.
Let X ⊂ C N be a possibly singular complex affine variety. Let f : C N → C be a polynomial function whose restriction to X is nonconstant and has onlyisolated critical points P , . . . , P l in the stratified sense. Then formula (1), written inthe language of constructible functions (see Section 2.3), becomes:(2) − (cid:88) c ∈ C Φ f − c (( − dim X Eu X ) = l (cid:88) i =1 n i Eu P i . Fix i ∈ { , . . . , l } and apply the equality of constructible functions in (2) to the point P i to get:(3) n i = ( − dim X − Φ f − f ( P i ) (Eu X )( P i ) . Of course, if X is smooth, then Eu X = 1 X and, via (12), (14) and (18), n i becomes theMilnor number of f at P i , as already mentioned in Example 1.1.Let g : C N → C be a general linear function, and write as above f t := f − tg . Ourmain result is the following: LAURENTIU G. MAXIM, JOSE ISRAEL RODRIGUEZ, AND BOTONG WANG
Theorem 1.3.
The limit of the critical points of f t satisfies (4) lim t → Crit( f t | X reg ) = (cid:88) i ∈ I n i · Crit( g | X i ) where the symbol Crit denotes the set of critical points and the numbers n i are determinedby formula (1) . Remark 1.4.
The limit in (4) is defined in Subsection 2.1. It is always well-defined inour setting, and it does not count the points going to infinity.
Example 1.5.
As in Example 1.1, suppose that X is smooth and f | X has only isolatedcritical points P , . . . , P l . Then Theorem 1.3 specializes to the well-known Morsificationresult that lim t → Crit( f t ) = (cid:88) ≤ i ≤ l n i P i where n i is the Milnor number of f | X at P i .Moreover, consider the situation of Example 1.2 of a possibly singular affine variety X , with f having only isolated stratified singular points P , . . . , P l on X . Theorem 1.3specializes in this case to lim t → Crit( f t ) = (cid:88) ≤ i ≤ l n i P i , with n i computed as in formula (3). Remark 1.6.
The left side of equation (4) does not count the points that go to infinityas t →
0. To be precise, we say that no points of
Crit( f t | X reg ) go to infinity if (cid:91)
0, and 0 < t (cid:28) r , where B r ⊂ C N is the ball of radius r centered at the origin.We can give a topological interpretation of the number of points of Crit( f t | X reg ) goingto infinity at t goes to zero as follows.First, using a result of Seade, Tibˇar and Verjovsky (see [27, Equation (2)]), togetherwith arguments similar to [22, Section 3.3], we have: Theorem 1.7.
Let X be any irreducible subvariety of C N , and let f be any polynomialfunction on C N . For a general linear function g on C N , the number of critical points of ( f − g ) | X reg is equal to ( − dim X χ (Eu X | U ) where U is the complement of the hypersurface { f − g = c } in C N for a general choiceof c ∈ C . Together with Theorem 1.3, this yields the following:
Corollary 1.8.
The number of points of
Crit( f t | X reg ) going to infinity is equal to (5) ( − dim X χ (Eu X | U ) − (cid:88) i ∈ I n i · (cid:12)(cid:12) Crit( g | X i ) (cid:12)(cid:12) where | · | denotes the cardinality of a set. As an immediate application of Corollary 1.8 together with our result from [22, The-orem 1.3], we provide a new formula for the Euclidean distance (ED) degree of an affinevariety. In the previous literature, the Euclidean distance degree of an algebraic varietyis described in terms of a distance function with respect to a generic data point. Thefollowing corollary (with a mild hypothesis regarding critical points at infinity), givesa formula for the ED degree in terms of critical points of a general linear function onstrata X i where f is a distance function with respect to an arbitrary data point. Corollary 1.9.
Fix a data point ( u , . . . , u n ) ∈ C n and an algebraic variety X ⊂ C n .For f = (cid:80) ni =1 ( x i − u i ) , if no points of Crit( f − tg | X reg ) go to infinity as t → , thenthe Euclidean distance degree of X equals (cid:88) i ∈ I n i · | Crit( g | X i ) | . To study the limiting behavior of the set of critical points, we use the work of Ginsburg[9] on characteristic cycles. Another (possibly more direct) approach is to make use ofMassey’s results from [18], which we learnt about as we were in the final stage of writingup this paper. For more details, see Remark 4.12.The paper is organized as follows. In Section 2, we introduce the notion of limit of sets,and recall the necessary background on constructible complexes and their characteristiccycles. In Section 3, we review Ginsburg’s work of pushforward of characteristic cyclesand the characteristic cycle of the nearby cycle functor. Our main result, Theoreom 1.3is proved in Section 4, while Section 5 is devoted to applications.
Acknowledgements
The authors thank J¨org Sch¨urmann for useful discussions andfor bringing the references [18, 19] to their attention. L. Maxim is partially supportedby the Simons Foundation Collaboration Grant
Preliminaries
In this section, we give a precise definition of the limit of sets. We also review thenotion of characteristic cycles, nearby/vanishing cycles, and their relations.
LAURENTIU G. MAXIM, JOSE ISRAEL RODRIGUEZ, AND BOTONG WANG
Limit of sets.
We introduce here the notion of limit for a parametrized family ofsets, which appears in the formulation of our main result, Theorem 1.3.
Definition 2.1.
Throughout this paper, by a set of points , we always mean a finiteset with multiplicity. More precisely, fixing a ground set S , by a set of points M of S ,we mean a function M : S → Z ≥ such that M ( x ) = 0 for all but finitely many x ∈ S .We call M ( x ) the multiplicity of M at x . For two sets of points M and N of S , wewrite M ≥ N , if M ( x ) ≥ N ( x ) for every point x ∈ S .Let φ : S → T be a map of sets, and let M be a set of points in S . Then φ ( M ) is aset of points in T defined by φ ( M )( y ) = (cid:88) x ∈ φ − ( y ) M ( x ) . Example 2.2.
Any finite subset T ⊂ S can be considered as a set of points M T in S ,by setting M T ( x ) = (cid:40) , x ∈ T, , x / ∈ T. Definition 2.3.
Fixing a Hausdorff space S as the ground set, let M t be a family ofsets of points of S , parametrized by t ∈ C ∗ (or more generally a punctured disc centeredat the origin). We define the limit of M t as t →
0, denoted by lim t → M t , to be theset of points given by: (lim t → M t )( x ) := lim ←− U lim t → (cid:88) y ∈ U M t ( y ) , where lim ←− U denotes taking the inverse limit over all open neighborhood of x . Remark 2.4.
The limit lim t → either exists as a finite set with multiplicity, or doesnot exist. If the limit lim t → exists, then for any x ∈ S , and for any sufficiently smallneighborhood U of x , then the limit lim t → (cid:80) y ∈ U M t ( y ) exists as a finite number. Remark 2.5.
From now on, all the limits we work with are of algebraic nature. Moreprecisely, S is an algebraic variety, and there exists a (not necessarily irreducible) alge-braic curve C ∈ S × C ∗ , such that M t = p S ( C ∩ S × { t } ), where p S : S × C ∗ → S isthe projection to the first factor. In this case, it is easy to see that the limit lim t → M t always exists. Lemma 2.6.
Let φ : S → T be a proper continuous map between Hausdorff and locallycompact spaces. Let M t be a family of sets with multiplicity parametrized by t ∈ C ∗ .Then (6) φ (cid:0) lim t → M t (cid:1) = lim t → φ ( M t ) if both limits exist.Proof. The inequality(7) φ (cid:0) lim t → M t (cid:1) ≤ lim t → φ ( M t )is obvious, and does not require any compactness assumption. Now we prove the con-verse. Since the statement is local in T , we may assume that lim t → φ ( M t ) is supported atone point, that is lim t → φ ( M t ) = nQ for some Q ∈ T and n ∈ Z ≥ . To show theconverse of (7), it suffices to show both sides have the same multiplicity at Q .Since T is locally compact, there exists an arbitrarily small compact neighborhood V of Q and (cid:15) V >
0, such that M t ∩ φ − ( V ) consists of n points for any 0 < t < (cid:15) V . Letlim t → M t = (cid:88) i ∈ J m i P i . By (7), we have P i ∈ φ − ( Q ) ⊂ φ − ( V ). By definition, for any x ∈ V , there exists aneighborhood U x of x in V and (cid:15) x >
0, such that M t ∩ U P i consists of m i points, and M t ∩ U x is empty if x / ∈ { P i | i ∈ J } . Since S is Hausdorff, we can also assume that U P i are pairwise disjoint for i ∈ J . Since φ is proper, φ − ( V ) is compact. Hence wecan cover φ − ( V ) by finitely many U x . Let (cid:15) be the smallest (cid:15) x among all x appearingin the index of the above covering. Then for any 0 < t < (cid:15) , the set with multiplicity M t ∩ φ − ( V ) consists of (cid:80) i ∈ J m i points. Thus, (cid:80) i ∈ J m i = n , that is φ (cid:0) lim t → M t (cid:1) andlim t → φ ( M t ) have the same multiplicity at Q . (cid:3) Constructible complexes and characteristic cycles.
A sheaf F of C -vectorspaces on a variety M is constructible if there exists a finite stratification M = (cid:116) j S j of X into locally closed smooth subvarieties (called strata ), such that the restriction of F toeach stratum S j is a C -local system of finite rank. A complex F (cid:5) of sheaves of C -vectorspaces on M is called constructible if its cohomology sheaves H i ( F (cid:5) ) are all constructible.Denote by D bc ( M ) the bounded derived category of constructible complexes (with respectto some stratification) on M , i.e., one identifies constructible complexes containing thesame cohomological information.By associating characteristic cycles to constructible complexes on a smooth variety M (e.g., see [6, Definition 4.3.19] or [15, Chapter IX]), one gets a functor CC : K ( D bc ( M )) −→ LCZ ( T ∗ M )on the Grothendieck group of C -constructible complexes, where LCZ ( T ∗ M ) is the freeabelian group spanned by the irreducible conic Lagrangian cycles in the cotangent bundle T ∗ M . Recall that any element of LCZ ( T ∗ M ) is of the form (cid:80) k n k [ T ∗ Z k M ], for some n k ∈ Z and Z k closed irreducible subvarieties of M . Here, if Z is a closed irreduciblesubvariety of M with smooth locus Z reg , its conormal bundle T ∗ Z M is defined as theclosure in T ∗ M of T ∗ Z reg M . One can then define a group isomorphism T : LCZ ( T ∗ M ) −→ Z ( M )to the group Z ( M ) of algebraic cycles on M by: (cid:88) k n k [ T ∗ Z k M ] (cid:55)−→ (cid:88) k ( − dim Z k n k Z k . Let CF ( M ) be the group of algebraically constructible functions on a complex al-gebraic variety M , i.e., the free abelian group generated by indicator functions 1 Z ofclosed irreducible subvarieties Z ⊂ M . To any constructible complex F (cid:5) ∈ D bc ( M ) one LAURENTIU G. MAXIM, JOSE ISRAEL RODRIGUEZ, AND BOTONG WANG associates a constructible function χ st ( F (cid:5) ) ∈ CF ( M ) by taking stalkwise Euler charac-teristics, i.e., χ st ( F (cid:5) )( x ) := χ ( F (cid:5) x )for any x ∈ X . For example, χ st ( C M ) = 1 M . Another important example of a con-structible function is the local Euler obstruction function Eu M of MacPherson [16], whichis an essential ingredient in the definition of Chern classes for singular varieties. Sincethe Euler characteristic is additive with respect to distinguished triangles, one gets aninduced group homomorphism (in fact, an epimorphism) χ st : K ( D bc ( M )) −→ CF ( M ) . Moreover, since the class map D bc ( M ) → K ( D bc ( M )) is onto, χ st is already an epimor-phism on D bc ( M ).When Z is a closed subvariety of M , we may regard the function Eu Z as being definedon all of M by setting Eu Z ( x ) = 0 for x ∈ M \ Z . In particular, we may consider thegroup homomorphism(8) Eu : Z ( M ) −→ CF ( M )defined on an irreducible cycle Z by the assignment Z (cid:55)→ Eu Z , and then extended by Z -linearity. A well-known result (e.g., see [6, Theorem 4.1.38] and the references therein)states that the homomorphism Eu : Z ( M ) → CF ( M ) is an isomorphism.The Euler obstruction function enters into the formulation of the local index theorem ,which in the above notations and for M smooth asserts the existence of the followingcommutative diagram (e.g., see [25, Section 5.0.3] and the references therein):(9) K ( D bc ( M )) CC (cid:15) (cid:15) χ st (cid:47) (cid:47) CF ( M ) Eu − ∼ = (cid:15) (cid:15) LCZ ( T ∗ M ) T ∼ = (cid:47) (cid:47) Z ( M )In particular, one can associate a characteristic cycle to any constructible function ϕ ∈ CF ( M ) by the formula CC ( ϕ ) := T − ◦ Eu − ( ϕ ) . For example, if Z is a closed irreducible subvariety of M , one has: CC (Eu Z ) = ( − dim Z [ T ∗ Z M ] . Note also that CC ( F (cid:5) ) = CC ( χ st ( F (cid:5) ))for any constructible complex F (cid:5) ∈ D bc ( M ).2.3. Nearby and vanishing cycle functors.
Let M be a complex manifold, and let f : M → ∆ be a holomorphic map to a disc, with i : f − (0) (cid:44) → M the inclusion of thezero-fiber. The canonical fiber M ∞ of f is defined by M ∞ := M × ∆ ∗ (cid:126) , where (cid:126) is the complex upper-half plane (i.e., the universal cover of the punctured discvia the map z (cid:55)→ exp(2 πiz )). Let k : M ∞ (cid:44) → M be the induced map. The nearby cyclefunctor of f , Ψ f : D bc ( M ) → D bc ( f − (0)) is defined by(10) Ψ f ( F (cid:5) ) := i ∗ Rk ∗ k ∗ F (cid:5) . The vanishing cycle functor Φ f : D bc ( M ) → D bc ( f − (0)) is the cone on the comparisonmorphism i ∗ F (cid:5) → Ψ f ( F (cid:5) ), that is, there exists a canonical morphism can : Ψ f ( F (cid:5) ) → Φ f ( F (cid:5) ) such that(11) i ∗ F (cid:5) → Ψ f ( F (cid:5) ) can → Φ f ( F (cid:5) ) [1] → is a distinguished triangle in D bc ( f − (0)).It follows directly from the definition that for x ∈ X ,(12) H j ( M f,x ; Q ) = H j (Ψ f Q X ) x and (cid:101) H j ( M f,x ; Q ) = H j (Φ f Q X ) x , where M f,x denotes the Milnor fiber of f at x .It is also known that the shifted functors p Ψ f := Ψ f [ −
1] and p Φ f := Φ f [ −
1] takeperverse sheaves on M into perverse sheaves on the zero-fiber f − (0) (e.g., see [25,Theorem 6.0.2]).By repeating the above constructions for the function f − c , one gets functorsΨ f − c , Φ f − c : D bc ( M ) → D bc ( f − ( c )) , provided that { f = c } is a nonempty hypersurface.The nearby cycle functor descends to a functor on the category of constructiblefunctions, see, e.g., [29] or [26, Section 4]. In other words, the constructible function χ st (cid:0) Ψ f ( F ) (cid:1) only depends on the function χ st ( F ). Therefore, Ψ f induces a linear map,which we also denote by Ψ f ,(13) Ψ f : CF ( M ) → CF ( f − (0)) , where we regard elements of CF ( f − (0)) as constructible functions om M with supporton f − (0). In fact, the above linear map Ψ f can be defined directly as follows:(14) Ψ f ( α )( x ) = χ ( α · M f,x ) . In particular,(15) Ψ f (1 M ) = µ ∈ CF ( f − (0)) , where µ : f − (0) → Z is the constructible function defined by the rule:(16) µ ( x ) := χ ( M f,x ) , for all x ∈ f − (0). Note that µ = χ st (Ψ f Q X ) . By analogy with (11), one defines a vanishing cycle functor on constructible functions,(17) Φ f : CF ( M ) → CF ( f − (0)) ⊂ CF ( M ) , by setting(18) Φ f ( α ) := Ψ f ( α ) − α | f − (0) . Remark 2.7.
By (9), the characteristic cycle functor CC : CF ( M ) ∼ = → LCZ ( T ∗ M )allows one to regard the nearby and vanishing cycle functors Ψ f , Φ f as functors on conicLagrangian cycles in the cotangent bundle T ∗ M . This will be the way we view nearbyand vanishing cycle functors for the rest of this paper.2.4. Pushforward, Pullback, Attaching triangle.
Let M be a complex manifold asabove, and let f : M → ∆ be a holomorphic map to a disc. Let i : f − (0) (cid:44) → M and j : U = M \ f − (0) (cid:44) → M be the inclusion maps of the zero-fiber and of its complement,respectively. Recall that for any F (cid:5) ∈ D bc ( M ), there is an attaching triangle in D bc ( M ):(19) j ! j ∗ F (cid:5) → F (cid:5) → i ∗ i ∗ F (cid:5) [1] → with i ∗ = i ! . Lemma 2.8.
Under the above notations, we have: (20) χ st ( j ! j ∗ F (cid:5) ) = χ st ( Rj ∗ j ∗ F (cid:5) ) ∈ CF ( M ) . Proof.
Since j ∗ j ! (cid:39) j ∗ Rj ∗ (cid:39) id , we see that the restrictions of the complexes j ! j ∗ F (cid:5) and Rj ∗ j ∗ F (cid:5) to U are quasi-isomorphic, so they have the same stalks over U . At points in f − (0), the complex j ! j ∗ F (cid:5) has zero stalks. So it remains to show that χ (( Rj ∗ j ∗ F (cid:5) ) x ) = 0for all x ∈ f − (0). Next note that for any x ∈ f − (0) and k ∈ Z , we have: H k (( Rj ∗ j ∗ F (cid:5) ) x ) ∼ = H k ( B x ; Rj ∗ j ∗ F (cid:5) ) ∼ = H k ( B x \ f − (0); j ∗ F (cid:5) ) , for B x a small enough ball in M centered at x . Therefore, χ (( Rj ∗ j ∗ F (cid:5) ) x ) = χ ( B x \ f − (0); j ∗ F (cid:5) ) . Finally, using [6, Corollary 4.1.23, Remark 4.1.24] and the implied additivity of Eulercharacteristics from (19), one gets that: χ (( Rj ∗ j ∗ F (cid:5) ) x ) = χ ( B x ; F (cid:5) ) − χ ( B x ∩ f − (0); i ∗ F (cid:5) ) = χ ( F (cid:5) x ) − χ ( F (cid:5) x ) = 0 , thus completing the proof. (cid:3) Remark 2.9.
The above lemma is also a special case of [25, Example 6.0.17(1)], and itcan be deduced from the distinguished triangle j ! j ∗ F (cid:5) → Rj ∗ j ∗ F (cid:5) → i ∗ i ∗ Rj ∗ j ∗ F (cid:5) [1] → (which is obtained by applying (19) to Rj ∗ j ∗ F (cid:5) instead of F (cid:5) , and using j ∗ Rj ∗ (cid:39) id ),by noting that (cf. [25, (6.37)]):[ i ∗ i ∗ Rj ∗ j ∗ F (cid:5) ] = 0 ∈ K ( D bc ( M )) . When coupled with the local index theorem, Lemma 2.8 yields the following.
Corollary 2.10.
In the above notations, we have: (21) CC ( j ! j ∗ F (cid:5) ) = CC ( Rj ∗ j ∗ F (cid:5) ) . It is well known (e.g., see [25, Section 2.3]) that all the usual functors in sheaf theory,which respect the corresponding category of constructible complexes of sheaves, induceby the epimorphism χ st well-defined group homomorphisms on the level of constructiblefunctions. This was already indicated above for the nearby and vanishing cycle functors, and the same applies for the functors i ∗ , i ∗ , j ! , j ∗ , Rj ∗ , which on the level of constructiblefunctions are denoted by i ∗ = i ! , i ∗ , j ! , j ∗ , j ∗ . In particular, by (9), these functors canalso be considered as functors on conic Lagrangian cycles in the cotangent bundle T ∗ M (with support in a certain subvariety, if needed). Proposition 2.11.
In the above notations, let Λ be a conic Lagrangian cycle in T ∗ M .Then p Ψ f (Λ) = j ∗ j ∗ (Λ) + p Φ f (Λ) − Λ ∈ LCZ ( T ∗ M ) . Proof.
Since the characteristic cycle functor CC : K (cid:0) D bc ( M ) (cid:1) → LCZ (cid:0) T ∗ M (cid:1) is surjec-tive, the distinguished triangle (11) implies thatΨ f (Λ) = i ∗ (Λ) + Φ f (Λ) , as an identity of Lagrangian cycles in LCZ ( T ∗ M ), with support in f − (0). In particular,we identify i ∗ (Λ) and i ∗ i ∗ (Λ) in LCZ ( T ∗ M ). Furthermore, the distinguished triangle(19) yields that i ∗ i ∗ (Λ) + j ! j ∗ (Λ) = Λand, by (21), we have j ! j ∗ (Λ) = j ∗ j ∗ (Λ) . Combining the above three equations, we get:Ψ f (Λ) = Λ − j ∗ j ∗ (Λ) + Φ f (Λ) . Notice that as functors of Lagrangian cycles (just as on K ( D bc ( M ))), p Ψ f (Λ) and p Φ f (Λ)are equal to the negative of Ψ f (Λ) and Φ f (Λ), respectively. Thus, the assertion followsfrom the above equation. (cid:3) The characteristic cycle of nearby and vanishing cycles functors
In this section, we review Ginsburg’s work [9] on the pushforward of characteristiccycles and the characteristic cycle of nearby cycle functor.Let M be a complex manifold. Given any holomorphic function f : M → C , let U bethe complement of the hypersurface f − (0) in M . Given any conic Lagrangian cycle Λin T ∗ U , Ginsburg defined the pushforward of Λ by the open inclusion map j : U → M ,denoted by Lim s → Λ s , as follows. For any s ∈ C ∗ , define the non-conic Lagrangian cycleΛ s by(22) Λ s = Λ + s · d log f = (cid:8) ξ + s ( d log f )( x ) | ( x, ξ ) ∈ Λ (cid:9) . The total space of the family Λ s forms a closed subvariety Λ of T ∗ U × C ∗ . We denote itsclosure in T ∗ M × C by Λ . To define the characteristic cycle Lim s → Λ s , one first takesthe scheme-theoretic intersection Λ ∩ ( T ∗ M ×{ } ), and then considers the cycle obtainedby taking the irreducible components of this intersection with the multiplicities given bythe scheme structure. One obtains in this way a conic Lagrangian cycle Lim s → Λ s in T ∗ M . In view of the following result of Ginsburg, one should regard it as the pushforwardof Λ by the open embedding j : U → M . Theorem 3.1. [9, Theorem 3.2]
Let Λ be a conic Lagrangian cycle in T ∗ U . Then (23) Rj ∗ (Λ) = Lim s → Λ s . Ginsburg also computed the characteristic cycle of the nearby cycle of a constructiblecomplex by using a similar construction. Denote the projection T ∗ M × C → T ∗ M by w . Proposition 3.2. [9, Proposition 2.14.1]
Under the above notations, over a neighborhoodof f − (0) ⊂ M ,(1) the set Λ is an analytic variety of dimension dim M + 1 ;(2) if ( ξ x , s ) ∈ Λ and f ( x ) = 0 , then s = 0 ;(3) the restriction w | Λ : Λ → T ∗ M is a closed embedding. Corollary 3.3. [9, Corollary 2.14.2]
For any conic Lagrangian subvariety Λ of T ∗ U , theclosure of w (Λ ) = { ξ x + s ( d log f )( x ) | ξ x ∈ Λ , s ∈ C ∗ } in T ∗ M is equal to w (Λ ) . Denote the pushforward cycle w ∗ (Λ ) by Λ (cid:92) . Denote by Lim f → Λ (cid:92) the specializationof Λ (cid:92) to f − (0). By Corollary 3.3, Λ (cid:92) is equal to the variety w (Λ ) and Lim f → Λ (cid:92) is theschematic restriction of the variety w (Λ ) to T ∗ M | f − (0) . Theorem 3.4. [9, Theorem 5.5]
Let Λ be a conic Lagrangian cycle in T ∗ M . Then, p Ψ f (Λ) = Lim f → (Λ | T ∗ U ) (cid:92) , where p Ψ f is the perverse nearby cycle functor. It follows from Theorem 3.1 and Theorem 3.4 that if Λ is an irreducible conic La-grangian subvariety in T ∗ M , then both j ∗ j ∗ (Λ) and p Ψ f (Λ) are effective. The same istrue for the vanishing cycle functor by the following result. Theorem 3.5. [18, Theorem 2.10]
Let Λ be an irreducible conic Lagrangian subvarietyin T ∗ M . Then p Φ f (Λ) is an effective conic Lagrangian cycle. See Remark 4.12 for a brief discussion around [18, Theorem 2.10].
Remark 3.6.
Theorem 2.10 in [18] is formulated in the language of sheaves. Never-theless, since the characteristic cycle of a bounded constructible complex only dependson the associated constructible function, in view of (9) the argument also works forLagrangian cycles. See also [19, Remark 1.3] for a correction of sign errors.4.
The limit of critical points
Let X ⊂ C N be an irreducible subvariety with smooth locus X reg . Let f : C N → C be a polynomial map. Let g : C N → C be a general linear function. We will study thecritical locus of f t | X reg as t goes to zero, where f t = f − tg .We introduce some notations that we will use throughout this section. We fix astratification X = (cid:70) i ∈ I X i of X into smooth locally closed subvarieties. Let p : T ∗ C N → C N be the natural projection. Give any algebraic 1-form ω on C N , let Γ ω be the image ofthe 1-form ω in T ∗ C N . Lemma 4.1.
Let Γ X reg ,f = T ∗ X reg C N − Γ df = (cid:8) ( x, η ) ∈ T ∗ C N (cid:12)(cid:12) x ∈ X reg , η + df | x ∈ T ∗ X reg C N (cid:9) . Then for any t ∈ C , we have (24) p (Γ X reg ,f ∩ Γ tdg ) = Crit( f t | X reg ) . Proof.
A point x ∈ X reg is a critical point of f t if and only if the cotangent vector df t at x is contained in T ∗ X reg C N . This is equivalent to Γ df − tdg intersects T ∗ X reg C N in the fiberover x . By definition, p (cid:0) Γ df − tdg ∩ T ∗ X reg C N (cid:1) = p (Γ X reg ,f ∩ Γ tdg ) . Therefore, the assertion in the lemma follows. (cid:3)
We fix a general linear function g : C N → C . Let C be the intersection of Γ dg × P and the closure of Λ in T ∗ C N × P , where Λ = T ∗ X C N | T ∗ U = T ∗ X ∩ U U and Λ is theclosed subvariety of T ∗ X C N × C defined in Section 3. Remark 4.2.
The curve C is a lifting of the polar curve (see [28, Definition 7.1.1]) on X to T ∗ C N × P . Lemma 4.3.
The curve C is equal to the closure of Λ ∩ (cid:0) Γ dg × C (cid:1) in T ∗ C N × P .Proof. Notice that Λ is Zariski open and dense in its closure in T ∗ C N × P . Thus, forgeneral a choice of g , the intersection Λ ∩ (cid:0) Γ dg × C (cid:1) is also Zariski open and dense in C . Thus, the lemma follows. (cid:3) Let π : T ∗ C N × P → C N be the composition of the projection to the first factorand the natural cotangent bundle map T ∗ C N → C N . Denote by π C the restrictionof π : T ∗ C N × P → C N to C . By definition, C is closed in Γ dg × P . Clearly, therestriction map π | Γ dg × P : Γ dg × P → C N is a P bundle map. Therefore, the map π C : C → C N is the composition of a closed embedding C → Γ dg × P and a properprojection Γ dg × P → C N . Since a closed embedding is proper and the composition ofproper maps is also proper, we have the following lemma. Lemma 4.4.
The map π C : C → C N is proper. As a subvariety of T ∗ C N × P , we can consider f and s as functions on C by abuse ofnotations:(1) The regular function f is the composition C (cid:44) → T ∗ C N × P π −→ C N f −→ C .(2) The rational function s is the composition of C (cid:44) → T ∗ C N × P → P , wherethe second arrow is the projection to the second factor. Recall that s is thecoordinate of the line C , and hence it extends to a rational function on P . Proposition 4.5.
In a neighborhood of { f = 0 } in C , the rational function s is a regularfunction. In other words, the zero locus of f on C does not intersect the pole locus of s on C . Proof.
Recall that Λ is a closed subvariety of T ∗ C N × C , and by Lemma 4.3, C is theclosure of Λ ∩ (cid:0) Γ dg × C (cid:1) in T ∗ C N × P . Thus, it suffices to show that the intersectionΛ ∩ (cid:0) Γ dg × C (cid:1) is closed in a neighborhood of { f = 0 } in T ∗ C N × P . This is a consequenceof Proposition 3.2 (3), i.e., the restriction of w : T ∗ C N × C → T ∗ C N to Λ is a closedembedding in a neighborhood of { f = 0 } .In fact, since Γ dg is closed in T ∗ C N , Γ dg × C is closed in T ∗ C N × C , and henceΛ ∩ (cid:0) Γ dg × C (cid:1) is closed in Λ . Thus,Λ ∩ (cid:0) Γ dg × C (cid:1) (cid:44) → Λ w −→ T ∗ C N is the composition of a closed embedding and a proper map (Proposition 3.2 (3)) in aneighborhood of { f = 0 } . Therefore, the above composition is a proper map. By thedefinition of C , the above composition factors through the natural inclusion map(25) Λ ∩ (cid:0) Γ dg × C (cid:1) (cid:44) → C . If a composition of maps of algebraic varieties is proper, then the first map must beproper (see e.g. [11, Corollary 4.8 (e)]). Therefore, the open inclusion (25) is proper,and hence an isomorphism in a neighborhood of { f = 0 } . (cid:3) By definition, the pole locus of s as a rational function on C is equal to the complementof Λ ∩ (cid:0) Γ dg × C (cid:1) . Therefore, the above proposition is also equivalent to Λ ∩ (cid:0) Γ dg × C (cid:1) = C in a neighborhood of { f = 0 } . Corollary 4.6.
The map π C : C → C N is injective in a neighborhood of { f = 0 } .Proof. By the arguments preceding the corollary, it suffices to show that the restrictionof π : T ∗ C N × P → C N to Λ ∩ (cid:0) Γ dg × C (cid:1) is injective. The restriction factors throughthe projection w : T ∗ C N × C → T ∗ C N . So the map Λ ∩ (cid:0) Γ dg × C (cid:1) → C N factors as(26) Λ ∩ (cid:0) Γ dg × C (cid:1) → w (Λ ) ∩ Γ dg → C N By Proposition 3.2 (3), the restriction of w to Λ is injective. Hence the first map in(26) is injective. The second map in (26) is injective, because the map Γ dg → C N is anisomorphism. Therefore, the restriction of π to Λ ∩ (cid:0) Γ dg × C (cid:1) , which is equal to thecomposition of (26), is injective. (cid:3) Let (cid:101) C → C be the normalization of C . Then for any nonzero rational function h on C , its pullback to (cid:101) C defines an effective Weil divisors Zero ˜ C ( h ) on (cid:101) C as its zero divisor.We denote the pushforward of Zero ˜ C ( h ) to C by Zero C ( h ). Proposition 4.7.
Under the above notations, as sets with multiplicity (27) π C (cid:0) Zero C ( f /s ) (cid:1) = lim t → Crit( f t | X reg ) in a neighborhood of { f = 0 } . Before proving the proposition, we make the following observation. By abuse of no-tations, we can consider f and s as regular functions on the affine space T ∗ C N × C .Fixing a nonzero complex number t , we have a hypersurface { f = ts } in T ∗ C N × C . Recall that w : T ∗ C N × C → T ∗ C N and p : T ∗ C N → C N are the natural projections,and π : T ∗ C N × C → C N is their composition. Lemma 4.8.
Under the above notations, for any fixed t ∈ C ∗ , we have p (Γ X reg ,f ∩ Γ tdg ) = π (cid:16) Λ ∩ (cid:0) Γ dg × C ∗ (cid:1) ∩ { f = ts } (cid:17) . Proof.
It is straightforward to check one by one that the following conditions are equiv-alent for a point x ∈ X reg .(1) x ∈ π (cid:16) Λ ∩ (cid:0) Γ dg × C ∗ (cid:1) ∩ { f = ts } (cid:17) .(2) The restriction of the 1-form s · dff − dg to X vanishes at x for s = f ( x ) t .(3) The restriction of the 1-form df − tdg to X vanishes at x .(4) x ∈ p (cid:0) Γ X reg ,f ∩ Γ tdg (cid:1) .Thus, the assertion in the lemma follows. (cid:3) Proof of Proposition 4.7.
By Lemma 4.1 and Lemma 4.8, we havelim t → Crit( f t | X reg ) = lim t → p (Γ X reg ,f ∩ Γ tdg )= lim t → π (cid:16) Λ ∩ (cid:0) Γ dg × C ∗ (cid:1) ∩ { f = ts } (cid:17) = lim t → π (cid:16) Λ ∩ (cid:0) Γ dg × C ∗ (cid:1) ∩ { f /s = t } (cid:17) in a neighborhood of { f = 0 } in C N .As before, consider f /s as a rational function on C . By Lemma 4.3, the intersectionΛ ∩ (cid:0) Γ dg × C ∗ (cid:1) is a Zariski open and dense subset of C . Therefore, for all but finitelymany t ∈ C , we haveΛ ∩ (cid:0) Γ dg × C ∗ (cid:1) ∩ { f /s = t } = C ∩ { f /s = t } . Combining the above two equations, we havelim t → Crit( f t | X reg ) = lim t → π C (cid:0) C ∩ { f /s = t } (cid:1) in a neighborhood of { f = 0 } in C N . By Lemma 4.4 and Lemma 2.6, we havelim t → π C (cid:0) C ∩ { f /s = t } (cid:1) = π C (cid:16) lim t → (cid:0) C ∩ { f /s = t } (cid:1)(cid:17) . Clearly, lim t → (cid:0) C ∩ { f /s = t } (cid:1) = Zero C ( f /s ) . Combining the above three equations, we havelim t → Crit( f t | X reg ) = π C (cid:0) Zero C ( f /s ) (cid:1) in a neighborhood of { f = 0 } in C N . (cid:3) Let X ⊂ C N and f : C N → C as before, and let p Ψ f be the perverse nearby cyclefunctor from conic Lagrangian cycles on C N to the ones supported on { f = 0 } . Let U be the complement of { f = 0 } in C N , and let j : U → C N be the inclusion map. We fixa stratification X ∩ { f = 0 } = (cid:116) i ∈ I S i of X ∩ { f = 0 } into locally closed smooth subvarieties such that(28) p Ψ f (cid:0) [ T ∗ X C N ] (cid:1) = (cid:88) i ∈ I m (cid:48) i [ T ∗ S i C N ] , and(29) Rj ∗ (cid:0) [ T ∗ X C N ] | U (cid:1) = [ T ∗ X C N ] + (cid:88) i ∈ I l (cid:48) i [ T ∗ S i C N ] , with m (cid:48) i , l (cid:48) i ∈ Z ≥ for all i ∈ I . Proposition 4.9.
Under the above notations, counting multiplicities yields: (30) π C (cid:0) Zero C ( f ) (cid:1) = (cid:88) i ∈ I m (cid:48) i · Crit( g | S i ) and (31) π C (cid:0) Zero C ( s ) (cid:1) = (cid:88) i ∈ I l (cid:48) i · Crit( g | S i ) in a neighborhood of { f = 0 } of C N .Proof. We will derive the statements in the proposition from Theorem 3.1 and Theorem3.4 using similar arguments as in the proof of Proposition 4.7.Considering f as a regular function on the curve C , we haveZero C ( f ) = lim c → { x ∈ C | f ( x ) = c } , where the limit is taken in C . Equivalently, considering f as a regular function on T ∗ C N × P and { f = c } as a hypersurface of T ∗ C N × P , we haveZero C ( f ) = lim c → C ∩ { f = c } , where the limit is taken in C . By Lemma 4.3, Λ ∩ (cid:0) Γ dg × C (cid:1) is a nonempty Zariskiopen subset of C . Thus,lim c → C ∩ { f = c } = lim c → Λ ∩ (cid:0) Γ dg × C (cid:1) ∩ { f = c } , where both limits are taken in C . Combining the above two equations, we have(32) π C (cid:0) Zero C ( f ) (cid:1) = π (cid:0) lim c → Λ ∩ (cid:0) Γ dg × C (cid:1) ∩ { f = c } (cid:1) . Since the restriction of π : T ∗ C N × P → C N to Γ dg × P is proper, Lemma 2.6 impliesthat(33) π (cid:0) lim c → Λ ∩ (cid:0) Γ dg × C (cid:1) ∩ { f = c } (cid:1) = lim c → π (cid:0) Λ ∩ (cid:0) Γ dg × C (cid:1) ∩ { f = c } (cid:1) where the first limit is taken in T ∗ C N × P and the second limit is taken in Γ dg × P .Recall that in Section 3, w : T ∗ C N × C → T ∗ C N is the natural projection, and Λ (cid:92) isequal to the pushforward w ∗ (Λ ). Therefore,(34) π (cid:0) Λ ∩ (cid:0) Γ dg × C (cid:1) ∩ { f = c } (cid:1) = p (cid:0) Λ (cid:92) ∩ Γ dg ∩ { f = c } (cid:1) where p : T ∗ C N → C N is the cotangent bundle map. Since the restriction of p : T ∗ C N → C N to Γ dg is an isomorphism, in particular proper, by Lemma 2.6, we have(35) p (cid:0) lim c → Λ (cid:92) ∩ Γ dg ∩ { f = c } (cid:1) = lim c → p (cid:0) Λ (cid:92) ∩ Γ dg ∩ { f = c } (cid:1) where the first limit is in T ∗ C N and the second limit is in C N .Recall that lim f → Λ (cid:92) is the schematic restriction of the variety w (Λ ) to T ∗ M | f − (0) .Since Λ = T ∗ X C N , by Theorem 3.4, we haveLim f → Λ (cid:92) = p Ψ f ([ T ∗ X C N ])which by assumption (28) is equal to (cid:80) i ∈ I m (cid:48) i · Crit( g | S i ). Since g : C N → C is a generallinear function, Γ dg intersects T ∗ S i C N transversally and it also intersects Λ (cid:92) ∩ { f = c } transversally for all but finitely many c ∈ C . Therefore,(36) lim c → Λ (cid:92) ∩ Γ dg ∩ { f = c } = (cid:88) i ∈ I m (cid:48) i · T ∗ S i C N ∩ Γ dg as sets with multiplicity, where the limit is taken in T ∗ C N .Finally, equality (30) follows from equations (32), (33), (34), (35) and (36). The proofof equality (31) is similar. The only difference is that, in this case, the term [ T ∗ X C N ]in (29) does not contribute to the right side of (31). In fact, since f is nonconstant on X , the intersection T ∗ X C N ∩ { f = 0 } is of dimension at most N −
1, and hence for ageneral g , the intersection T ∗ X C N ∩ Γ g is empty in a sufficiently small neighborhood of { f = 0 } . (cid:3) Corollary 4.10.
In a neighborhood of { f = 0 } of C , as sets with multiplicity (or Weildivisors), we have Zero C ( f /s ) = Zero C ( f ) − Zero C ( s ) . Proof.
It suffices to show that in a neighborhood of { f = 0 } , the underlying set Zero C ( f )does not contain any pole of s and Zero C ( f ) ≥ Zero C ( s ).The first part follows from Proposition 4.5. Now, we prove the second part. ByProposition 2.11 and Theorem 3.5, we have m (cid:48) i − l (cid:48) i = n (cid:48) i ≥ i ∈ I . Thus, by Proposition 4.9, π C (cid:0) Zero C ( f ) (cid:1) ≥ π C (cid:0) Zero C ( s ) (cid:1) as sets of multiplicity. Since π C is injective in a neighborhood of { f = 0 } (Corollary 4.6),we have Zero C ( f ) ≥ Zero C ( s ). (cid:3) Before proving Theorem 1.3, we prove a local version of the theorem.
Theorem 4.11.
Let X ∩ { f = 0 } = (cid:70) i ∈ I S i be a stratification of X ∩ { f = 0 } asdiscussed in the paragraphs before Example 1.1. In particular, equations (28) and (29) hold. Then (37) lim t → Crit( f t | X reg ) = (cid:88) i ∈ I n (cid:48) i · Crit( g | S i ) in an analytic neighborhood of { f = 0 } in C N . Moreover, the coefficients n (cid:48) i are deter-mined by the following formula, (38) p Φ f ([ T ∗ X C N ]) = (cid:88) i ∈ I n (cid:48) i [ T ∗ S i C N ] . Proof.
By Proposition 2.11, we have m (cid:48) i − l (cid:48) i = n (cid:48) i for very i ∈ I . Now the assertion in the theorem follows from Corollary 4.10, Proposition4.7 and Proposition 4.9. (cid:3) Proof of Theorem 1.3.
To prove Theorem 1.3, it suffices to show the assertions hold ina neighborhood of { f = c } for every c ∈ C . This follows from Theorem 4.11 with f replaced by f − c . (cid:3) Remark 4.12.
As we shall now explain, it is also possible to derive our results from[18] instead of using [9]. A topological interpretation of the vanishing cycle of a conicLagrangian cycle is obtained in [18, Theorem 2.10]. Let Λ be an irreducible conicLagrangian subvariety of T ∗ C N and let f : C N → C be a polynomial function. Blowup T ∗ C N along Γ df , the image of the 1-form df . Let ˜Λ be the strict transformation ofΛ, and let E be the exceptional divisor. The natural isomorphism between Γ df and C N induces an isomorphism between E and the projective bundle Proj( T ∗ C N ). Under thisisomorphism, E ∩ ˜Λ = (cid:88) c ∈ C Proj (cid:0) p Φ f − c (Λ) (cid:1) where the first intersection is considered as a schematic intersection counting multiplic-ities.We are interested in the case when f | X has positive dimensional critical locus, whichcorresponds to a positive dimensional intersection of Γ df and T ∗ X C N . The above approachof Massey is exactly the deformation to normal cone (see, e.g., [8, Chapter 5]), whichis designed to construct intersection cycles when the set-theoretic intersection has morethan expected dimensions. See also [20, Part IV] for some discussion related to Lˆe-Vogelcycles. 5. Applications and examples X is an affine space. The first class of examples we consider are when X = C n , f is a polynomial function, and g is a general linear function. Example 5.1.
The following illustrates a special case of Examples 1.5. Consider ageneral linear function g : C → C and the function f : C → C , f ( x ) = x − x , The function f has a critical point at zero and at three, which we denote by X and X respectively. For general t , the function f t := f − tg has three distinct critical points.We have lim t → Crit( f t | X reg ) = { X , X } and see that as t → X and the other has multiplicity one as shown in Figure 5.1. Figure 2.
For g ( x ) = x , the critical points of f t are the roots of 4 x − x − t . From left to right, the critical points for f , f . , and f are plottedin the complex planes above.A stratification of X = C such that f is equisingular on each strata is given by X = C \ { X , X } and X , X . In the language of Theorem 1.3, we havelim t → Crit( f t ) = (cid:88) i ∈{ , , } n i · Crit( g | X i ) = 2 · Crit( g | X ) + 1 · Crit( g | X ) . Note that the second equality follows as g | X has no critical points. The n i are preciselythe multiplicity as addressed in Example 1.5. Example 5.2.
The next example we consider is f t = f − tg : C → C with X = C , f ( x, y, z ) = xy − ( z − x ) , and g a general linear function. The ideal of the varietyof critical points of f is generated by the thee partial derivatives of f . This idealhas a primary decomposition given by (cid:104) z, y , xy, x (cid:105) and (cid:104) y, x − z (cid:105) . Geometrically,this primary decomposition corresponds to the origin P and a parabola C through theorigin. An equisingular decomposition of X with respect to f is given by X = X \ C , X = C \ { P } and X = { P } . For general t the function f t has three critical points, andas t is taken to zero two of the points go to the origin while the third goes to a differentpoint Q in C . The point Q is the critical point of g | C . In the language of Theorem 1.3,we havelim t → Crit( f t ) = (cid:88) i =0 , , n i · Crit( g | X i ) = 1 · Crit( g | X ) + 2 · Crit( g | X ) = 1 · Q + 2 · P. Semidefinite programming and convex algebraic geometry.
Semidefiniteprogramming (SDP) is a subfield of convex optimization and has been studied throughthe lens of algebraic geometry [4]. The aim of an SDP is to optimize a linear objectivefunction over a convex set called a spectrahedron, which is the intersection of the coneof positive semidefinite symmetric matrices with an affine space.Let S n denote the set of n × n real symmetric matrices and denote the set of n × n positive semidefinite matrices by S n + . A set S ⊂ R m is a spectrahedron if it has the form S = { ( x , . . . , x m ) ∈ R m : A + m (cid:88) i =1 A i x i ∈ S n + } , for some given symmetric matrices A , A , . . . , A m ∈ S n . The algebraic boundary of aspectrahedron S is the complex hypersurface given by ∂S := { ( x , . . . , x m ) ∈ C n : det( A + m (cid:88) i =1 A i x i ) = 0 } . An algebraic approach to SDP is to study the critical points of a linear function on ∂S and to determine the algebraic degree of this optimization problem [10, 24]. Example 5.3 (Elliptic curve algebraic boundary) . Consider the spectrahedron S = { ( x, y ) ∈ R : (cid:104) x +1 0 y − x − y − x − (cid:105) ∈ S } , which has an algebraic boundary defined by the elliptic curve ∂S = { ( x, y ) ∈ C : − x − x − y + x + 3 = 0 } . For an illustration of the real points on the algebraic boundary and a description of thespectrahedron S , see [4, Example 2.7]. In the following, we take X to be the algebraicboundary ∂S , which is smooth. Let g : X → C denote a general linear function and let f : X → C be the projection given by f ( x, y ) = − x . For t = 0, the function f has threecritical points, which are the three points X , X , X of the curve intersected with the x -axis. On the other hand, for a general t the general linear function f t = f − tg : X → C has four critical points.As we take t to zero, Figure 3 suggests one critical point of f t goes to infinity. Toprove this, by Corollary 1.8, it suffices to determine (5) equals one. This follows as χ ( X ) = − n i = 1, (cid:12)(cid:12) Crit( g | X i ) (cid:12)(cid:12) = 1, and1 = − ( χ ( X ) − − (cid:88) i ∈{ , , } n i · (cid:12)(cid:12) Crit( g | X i ) (cid:12)(cid:12) = − − − − (1 + 1 + 1) . In the previous equation we subtract 3 from χ ( X ) because a general linear functionintersects X at three points.5.3. Euclidean distance degree.
The Euclidean distance degree (ED degree) [7] ofan affine algebraic subvariety X of C n is defined as the number of critical points ofthe squared Euclidean distance function d u ( x ) := (cid:80) ni =1 ( x i − u i ) on X reg for generic u = ( u , . . . , u n ). When X ∩ R n is smooth and compact, the closest point will be acritical point and a solution to the nearest point problem. Results on Euclidean distancedegrees have a hypothesis requiring genericity of the data point u [1, 2, 3, 12, 14, 17, 22]or study discriminant loci [13]. Our results allow us to handle situations when the datais not generic. Instances of nongeneric behavior include when the data may be sparseas in Example 5.4 or satisfy some algebraic property like in Example 5.6.With generic noise (cid:15) ∈ C n and arbitrary data u , the data u + (cid:15) is generic. In thecontext of distance geometry, Theorem 1.3 describes what happens to the set of criticalpoints of d u + t(cid:15) on X reg as t is taken to zero. Figure 3.
For t = 1, the critical points for the general linear function f ( x, y ) − tg ( x, y ) = − x − t (1 . x − . y ) are plotted as purple dots onthe elliptic curve X . As t is taken to zero, three of the four critical pointsapproach the x -axis and one goes to infinity.Let X denote an subvariety of C n with a Whitney stratification { S i } i ∈ Λ . For arbitrarydata u ∈ C n , generic (cid:15) ∈ C n , and t ∈ C , consider the squared distance function d u + t(cid:15) ( x ) = n (cid:88) i =1 ( x i − ( u i + t(cid:15) i )) = n (cid:88) i =1 x i − n (cid:88) i =1 u i x i − n (cid:88) i =1 t(cid:15) i x i + n (cid:88) i =1 ( u i + t(cid:15) i ) = n (cid:88) i =1 ( x i − u i ) − n (cid:88) i =1 t(cid:15) i x i + n (cid:88) i =1 ( u i + t(cid:15) i ) − n (cid:88) i =1 u i = d u ( x ) − tg ( x ) + c with(39) g ( x ) = 2 n (cid:88) i =1 (cid:15) i x i and c = (cid:80) ni =1 ( u i + t(cid:15) i ) − (cid:80) ni =1 u i . The set of critical points does not depend on c because c is constant with respect to x . So the critical points of d u + t(cid:15) coincide withthose of d u − tg . Moreover, since (cid:15) is generic, we have g is a generic linear function andTheorem 4 applies to d u − tg . Example 5.4 (Sparse data) . Consider the curve X in C defined by x + y = 1 andthe squared distance function from the point p t = ( t(cid:15) , t(cid:15) ) ∈ C , which is f t ( x ) = ( x − t(cid:15) ) + ( y − t(cid:15) ) . When t is generic f t has two critical points. When t = 0, p is the origin and every pointin the curve is a critical point of f . In terms of Theorem 1.3, we have:lim t → Crit( f t ) = 1 · Crit( g | X ) , with Crit( g | X ) consisting of two points. Example 5.5 (Cardioid Curve) . Let X denote the cardioid curve in Figure 1, whichhas a singular point at the origin P . The function f t ( x, y ) = x + y − t ( (cid:15) x + (cid:15) y ) hasthree critical points for general t and general (cid:15) = ( (cid:15) , (cid:15) ). Moreover, these critical pointscoincide with those of the distance function d t(cid:15) , which are illustrated in Figure 1 with (cid:15) = (3 . , . f : X → C only has two isolated critical points P , P ,and Theorem 1.3 specializes to lim t → Crit( f t ) = 2 P + 1 P . Example 5.6 (Eckart-Young and low rank data) . In this example, we take X to be theeight dimensional singular hypersurface in C × defined by det[ x i,j ] , = 0 consisting of3 × X is known to be three. Moreover, a Whitney stratification of X is given by the rankcondition, i.e., X has a regular stratum consisting of matrices of rank exactly 2 andthe singular locus consists of two strata corresponding to matrices of rank one and zerorespectively.Consider the following four data matrices u = u = u = u = . Each distance function d u i + t(cid:15) exhibits different limiting behavior among the sets of crit-ical points as t → d u + t(cid:15) , the set of three critical points (corresponding to ED degree of X isthree) converges to the set of three distinct critical points on X reg given by (cid:104) (cid:105) , (cid:104) (cid:105) , (cid:104) (cid:105) .The stratified critical locus of d u consists of three isolated points, two of which are inthe singular locus of X . Moreover, the set of three critical points of d u + t(cid:15) converges as t goes to zero to the point (cid:104) (cid:105) on the regular locus X reg and the previously mentionedtwo points (cid:104) (cid:105) , (cid:104) (cid:105) in the rank one stratum of the singular locus of X .The distance function d u has a positive dimensional critical locus given by the unionof an isolated regular point and the quadratic curve Q ⊂ X reg given by the set of matricesof the form (cid:104) a b b − a
00 0 1 (cid:105) with a (2 − a ) = b . The limit set of critical points of d u + t(cid:15) hasthree distinct points in X reg , one given by (cid:104) (cid:105) , and the other two being contained in Q . These two points correspond to Crit( g | Q ) where g is a general linear function givenby (cid:15) as in (39). For d u + t(cid:15) , the limit of the set of critical points consists of one point at the originwith multiplicity one, and another point (cid:104) (cid:105) with multiplicity two in the rank onestratum. These respective multiplicities correspond to coefficients n i in Theorem 1.3. Remark 5.7.
In [21], we studied the number of critical points in the smooth projectivecase by perturbing the squared Euclidean distance function by a general quadratic func-tion. In the projective setting, no points go to infinity, so we have an equality there. Incontrast, in the above examples the emphasis is on perturbing the squared Euclideandistance function with a linear function, and we do not assume the variety to be smooth.
References [1] M. F. Adamer and M. Helmer. Complexity of model testing for dynamical systems with toric steadystates.
Adv. in Appl. Math. , 110:42–75, 2019. 20[2] P. Aluffi and C. Harris. The Euclidean distance degree of smooth complex projective varieties.
Algebra Number Theory , 12(8):2005–2032, 2018. 20[3] J. A. Baaijens and J. Draisma. Euclidean distance degrees of real algebraic groups.
Linear Algebraand its Applications , 467:174 – 187, 2015. 20[4] G. Blekherman, P. A. Parrilo, and R. R. Thomas, editors.
Semidefinite optimization and convexalgebraic geometry , volume 13 of
MOS-SIAM Series on Optimization . Society for Industrial and Ap-plied Mathematics (SIAM), Philadelphia, PA; Mathematical Optimization Society, Philadelphia,PA, 2013. 19, 20[5] E. Brieskorn. Die Monodromie der isolierten Singularit¨aten von Hyperfl¨achen.
Manuscripta Math. ,2:103–161, 1970. 1[6] A. Dimca.
Sheaves in topology . Universitext. Springer-Verlag, Berlin, 2004. 2, 7, 8, 10[7] J. Draisma, E. Horobet¸, G. Ottaviani, B. Sturmfels, and R. R. Thomas. The Euclidean distancedegree of an algebraic variety.
Found. Comput. Math. , 16(1):99–149, 2016. 20[8] W. Fulton.
Intersection theory , volume 2 of
Ergebnisse der Mathematik und ihrer Grenzgebiete. 3.Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas.3rd Series. A Series of Modern Surveys in Mathematics] . Springer-Verlag, Berlin, second edition,1998. 18[9] V. Ginsburg. Characteristic varieties and vanishing cycles.
Invent. Math. , 84(2):327–402, 1986. 5,11, 12, 18[10] H.-C. Graf von Bothmer and K. Ranestad. A general formula for the algebraic degree in semidefiniteprogramming.
Bull. Lond. Math. Soc. , 41(2):193–197, 2009. 20[11] R. Hartshorne.
Algebraic geometry . Springer-Verlag, New York-Heidelberg, 1977. Graduate Textsin Mathematics, No. 52. 14[12] M. Helmer and B. Sturmfels. Nearest points on toric varieties.
Math. Scand. , 122(2):213–238, 2018.20[13] E. Horobet¸. The data singular and the data isotropic loci for affine cones.
Comm. Algebra ,45(3):1177–1186, 2017. 20[14] E. Horobet¸ and M. Weinstein. Offset hypersurfaces and persistent homology of algebraic varieties.
Comput. Aided Geom. Design , 74:101767, 14, 2019. 20[15] M. Kashiwara and P. Schapira.
Sheaves on manifolds , volume 292 of
Grundlehren der Mathematis-chen Wissenschaften [Fundamental Principles of Mathematical Sciences] . Springer-Verlag, Berlin,1990. With a chapter in French by Christian Houzel. 7[16] R. D. MacPherson. Chern classes for singular algebraic varieties.
Ann. of Math. (2) , 100:423–432,1974. 3, 8[17] A. Mart´ın del Campo and J. I. Rodriguez. Critical points via monodromy and local methods.
J.Symbolic Comput. , 79(part 3):559–574, 2017. 20, 22 [18] D. B. Massey. Critical points of functions on singular spaces.
Topology Appl. , 103(1):55–93, 2000.5, 12, 18[19] D. B. Massey. Perverse cohomology and the vanishing index theorem.
Topology Appl. , 125(2):299–313, 2002. 5, 12[20] D. B. Massey. Numerical control over complex analytic singularities.
Mem. Amer. Math. Soc. ,163(778), 2003. 18[21] L. G. Maxim, J. I. Rodriguez, and B. Wang. EDefect of Euclidean distance degree. preprintarXiv:1905.06758 , 2019. 23[22] L. G. Maxim, J. I. Rodriguez, and B. Wang. Euclidean Distance Degree of the Multiview Variety.
SIAM J. Appl. Algebra Geom. , 4(1):28–48, 2020. 4, 5, 20[23] J. Milnor.
Morse theory . Based on lecture notes by M. Spivak and R. Wells. Annals of MathematicsStudies, No. 51. Princeton University Press, Princeton, N.J., 1963. 1[24] J. Nie, K. Ranestad, and B. Sturmfels. The algebraic degree of semidefinite programming.
Math.Program. , 122(2, Ser. A):379–405, 2010. 20[25] J. Sch¨urmann.
Topology of singular spaces and constructible sheaves , volume 63 of
Instytut Matem-atyczny Polskiej Akademii Nauk. Monografie Matematyczne (New Series) Sciences. MathematicalMonographs (New Series)] . Birkh¨auser Verlag, Basel, 2003. 8, 9, 10[26] J. Sch¨urmann. Nearby cycles and characteristic classes of singular spaces. In
Singularities in geom-etry and topology , volume 20 of
IRMA Lect. Math. Theor. Phys. , pages 181–205. Eur. Math. Soc.,Z¨urich, 2012. 9[27] J. Seade, M. Tib˘ar, and A. Verjovsky. Global Euler obstruction and polar invariants.
Math. Ann. ,333(2):393–403, 2005. 4[28] M. Tib˘ar.
Polynomials and vanishing cycles , volume 170 of
Cambridge Tracts in Mathematics .Cambridge University Press, Cambridge, 2007. 13[29] J.-L. Verdier. Sp´ecialisation des classes de Chern. In
The Euler-Poincar´e characteristic (French) ,volume 82 of
Ast´erisque , pages 149–159. Soc. Math. France, Paris, 1981. 9
Department of Mathematics, University of Wisconsin-Madison, 480 Lincoln Drive,Madison WI 53706-1388, USA.
E-mail address : [email protected] URL : Department of Mathematics, University of Wisconsin-Madison, 480 Lincoln Drive,Madison WI 53706-1388, USA.
E-mail address : [email protected] URL : Department of Mathematics, University of Wisconsin-Madison, 480 Lincoln Drive,Madison WI 53706-1388, USA.
E-mail address : [email protected] URL ::