Lagrangian intersections as d-critical loci and their perverse sheaves

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1 Lagrangian intersections as d-critical loci and their perverse sheaves Isabell Grosse-Brauckmann Born 15th Mai 1992 in Bonn 13th April 2017 Master's Thesis Mathematics Advisor: Prof. Dr. Huybrechts Mathematical Institute Mathematisch-Naturwissenschaftliche Fakultät der Rheinischen Friedrich-Wilhelms-Universität Bonn

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3 LAGRANGIAN INTERSECTIONS AS d-critical LOCI AND THEIR PERVERSE SHEAVES ISABELL GROSSE-BRAUCKMANN Contents 1. Introduction 1 Part 1. d-critical loci 5 2. Denition 5 3. Symmetric obstruction theories on d-critical loci Perverse sheaves 20 Part 2. Lagrangian intersections Basics on Symplectic Geometry Symmetric obstruction theories on Lagrangian intersections Lagrangian intersections and perverse sheaves Examples of Lagrangian intersections in dimension Comparison with Ext i (O L, O M ) 38 References Introduction We are interested in intersections of Lagrangian submanifolds inside a holomorphic symplectic manifold. Our particular goal is to understand the construction of a perverse sheaf P L,M on such an intersection as dened in [Bu14], [BBDJS] and [KL16]. It is a special case of the construction of a perverse sheaf on a d-critical locus. Therefore, the paper is divided in two parts: Part 1 can be viewed as an introduction to the theory of d-critical loci from [Joy13]. It covers the denition of a d-critical locus (X, s), symmetric obstruction theories as dened in [BF05], and the construction of a perverse sheaf P X,s on (X, s) from [BBDJS]. Part 2 applies the theory to Lagrangian intersections. The starting point is that any such intersection admits the structure of a d-critical locus. We will now describe the main results in more detail: 1

4 2 I. GROSSE-BRAUCKMANN Part 1. d-critical loci. First of all, the local setting we are interested in is as follows. Let S be a smooth scheme and f H 0 (S, O S ). We set X. = crit(f) and generally care about the following constructions on X: (i) Symmetric obstruction theories, i.e. a triple (E D(X), φ: E L X, θ : E [1] E) such that E is locally isomorphic to a complex [E 1 E 0 ] of locally free sheaves, H 1 (φ) is surjective and H 0 (φ) is an isomorphism. Here, L X denotes the (truncated) cotangent complex of X. For instance, a symmetric obstruction theory on X is given by E 2 f = [T S X 2 f Ω S X ] (1.1) ψ df L X = [I XS /I 2 XS d Ω S X ] (Lemma 3.10). (ii) The perverse sheaf of vanishing cycles dened by PV S,f. = φ f C S [dim S 1] Perv(X). (1.2) In general, we deal with schemes that are locally of the form crit(f). These are d-critical loci. According to [Joy13, Denition 2.5], a d-critical locus is either a scheme or an analytic space X together with a section s of a certain sheaf S X. This section satises the property that for every point x X there exists an open neighborhood U X, a smooth scheme S together with a closed embedding U S and f H 0 (S, O S ) such that U = crit(f) and f denes s U. The triple (U, S, f) is called a chart for (X, s). In Section 2.1 we will give the precise denition of the sheaf S X involving what we call the thickened cotangent complex M X. This is a non-coherent version of the truncated cotangent complex L X. More precisely, for any scheme X we construct a morphism D X : O X L X in D(Sh(X)) over the natural dierential d X : O X Ω X. The thickened cotangent complex is the cone of D X and we set S X. = H 1 (M X ). Since the dierential in M X is not O X -linear, S X is in general not a coherent sheaf but only a sheaf of C-algebras. On any d-critical locus (X, s) we can locally dene a symmetric obstruction theory as in (1.1). Unfortunately, these obstruction theories need not glue to a global obstruction theory on X. We discuss this in Section 3. Proposition 1.1 ([Joy13]). There exists a line bundle K X,s on X red such that K X,s Ured det E 2 f Ured = ω 2 S U red for every critical chart (U, S, f).

5 LAGRANGIAN INTERSECTIONS 3 For the precise statement see Proposition The line bundle K X,s is called canonical bundle. In Section 4 we address the question whether the perverse sheaves of vanishing cycles as in (1.2) can be glued to a global perverse sheaf P X,s on (X, s). Proposition 1.2 ([BBDJS]). Assume that there is a square root K 1/2 X,s of the canonical bundle K X,s. Then there exists a perverse sheaf P X,s on X such that P X,s U PV S,f L or U,S for every critical chart (U, S, f). Here, L or U,S that L or U,S C O Ured = (K 1/2 X,s) Ured ω S Ured. For the precise statement see Proposition is a local system on U red such Part 2. Lagrangian intersections. Let (S, σ) be a symplectic manifold of dimension 2n and let X = L M be the intersection of two Lagrangian submanifolds. For instance, let M be a complex manifold and S = Ω M be the total space of its cotangent bundle with the canonical symplectic structure. Then, for any f H 0 (M, O M ) we dene a Lagrangian submanifold by L. = Γ df. Here, Γ df denotes the image of the embedding df : M Ω M, which is given by the section df H 0 (M, Ω M ). The intersection X = L M is the locus where df vanishes, i.e. X = crit(f) M. It turns out that any Lagrangian intersection is locally of this form. Proposition 1.3 ([Bu14]). The intersection of two Lagrangian submanifolds X = L M admits the structure of a d-critical locus with canonical bundle K X,s = ωl Xred ω M Xred. In particular, we can once again locally dene a symmetric obstruction theory on X. On the other hand, we prove in Lemma 6.1 that X carries a global symmetric obstruction theory with E given by [ ] ( res L,res M ) E LM. = Ω S X Ω L X Ω M X. From here we can conclude (Lemma 6.2) that ω X ω X = ωl X ω M X if X is smooth. Moreover, we compare these two obstruction theories in the above situation. Proposition 1.4. Let M be a complex manifold considered as Lagrangian submanifold inside its cotangent bundle S = Ω M. Let f H 0 (M, O M ) and L.. = Γ df S. There is an isomorphism of symmetric obstruction theories Φ: E 2 f E LM, where E 2 f is dened as in (1.1) and E LM as above.

6 4 I. GROSSE-BRAUCKMANN The precise denition of Φ can be found in Proposition 6.4. In the case that a square root of ω L Xred ω M Xred exists the d-critical structure on X also yields the existence of a perverse sheaf P L,M. In Section 7.2 we will describe P L,M explicitly in some special situations. For instance, we have (Lemma 7.6) Lemma 1.5. Assume that the intersection X = L M is smooth and let K 1/2 X,s = ω X. Then P L,M = C X [dim X]. The following Section 8 is dedicated to the study of two concrete examples of Lagrangian intersections in dimension 4. Here, the Hilbert scheme Hilb 2 (Z) of a K3 surface serves as a symplectic manifold. Finally, in the last Section 9 we aim at setting up a connection between dim H (P L,M ) and dim Ext (ω 1/2 L, ω1/2 M ). More precisely, under the assumption that degenerates, we prove E p,q 2 = H p (X, Ext q (ω 1/2 L, ω1/2 M )) Extp+q (ω 1/2 L, ω1/2 M ) Proposition 1.6. Assume that X = L M is smooth and compact. Let ω 1/2 L and ω1/2 M be xed square roots of the canonical bundles of L and M, respectively. Then dim Ext i n O S (ω 1/2 L, ω1/2 M ) = dim Hi (P L,M ). Here, K 1/2 X,s = ω 1/2 L X ω 1/2 M X serves for the computation of P L,M. Acknowledgements. I want to heartily thank my advisor Daniel Huybrechts for all his time and motivation. Many thanks to Johannes Anschütz for continuously answering all my questions and also to Andrey Soldatenkov for the computation of Example Notation. All schemes are of nite type over C. Let X be a scheme. We abbreviate D(Qcoh(X)) by D(X) and call this the derived category of X. We use the notation D + (X), D (X), D b (X) for the derived category of complexes that are bounded from below, from above or from both sides respectivly. Moreover, D b c(x) is the derived category of bounded complexes of constructible sheaves. We denote the category of sheaves of C vector spaces by Sh(X). We do not especially indicate when tensor products are derived tensor products. The theory of d-critical loci works in the analytic and in the Zariski topology. We will not mention this explicitly. All our perverse sheaves are elements of D b c(sh(x)), where constructibility is understood with respect to the analytic topology. In Part 2 we are only interested in the analytic topology. If E is a complex of sheaves, then E [1] is the complex with (E [1]) i = E i+1 and dierential d i E [1] = di+1 E. If φ: E F is a morphism of complexes, then φ[1] is given by φ[1] i = φ i+1.

7 LAGRANGIAN INTERSECTIONS 5 Let X be a scheme. We write N il X for the Nilradical of O X. If X S is a closed embedding with ideal sheaf I, then N il X = I /I, where I is the radical ideal of I dening X red. If X is a scheme and E a vector bundle on X, we denote the total space of E by E = Spec X (Sym (E )). For any section s H 0 (X, E) let Γ s E be the image of the embedding s: X E. If no confusion arises, we write X E for the zero section Γ 0. For projective bundles, we use the convention P(E) = Proj X (Sym (E )). Let M be a scheme and f : M A 1 a morphism, or equivalently, f Γ(M, O M ). Then crit(f) M is the vanishing locus of the section df Γ(M, Ω M ). We could also write crit(f) = M Γ df. Part 1. d-critical loci 2. Definition In this section we will establish the notion of a d-critical locus (X, s). This is a scheme or a compelex analytic space X that is locally of the form crit(f). Any two local reprensentations of the form crit(f) satisfy a certain compatibility condition, which is encoded by a section s H 0 (X, S X ). The sheaf S X is a cohomology sheaf of the thickened cotangent complex that we dene now The thickened cotangent complex. In the following, we let X be any scheme or complex analytic space. We assume that there is a closed embedding X S with ideal sheaf I into some smooth scheme S. Consider the natural dierential d S : O S Ω S. It induces an O X -linear map I /I 2 Ω S X whose cokernel is Ω X and whose kernel carries information on the smoothness of X. The complex L X. = [I /I 2 Ω S X ] (2.1) concentrated in degrees 1 and 0 is called the truncated cotangent complex of X (in S). By construction, L X comes with a natural morphism L X Ω X. In fact, L X is quasi-isomorphic to the truncation τ 1 L X of Illusie's full cotangent complex and therefore only depends on X. Following [HT10], we can also give a direct proof that L X does (up to quasi-isomorphism) not depend on the embedding X S. Assume that X S i with ideal I i for some smooth scheme S i and i {1, 2}. Then the diagonal embedding gives an embedding X S 1 S 2, whose ideal sheaf we denote by I 12. The composition X S 1 S 2 S 1 induces a short exact sequence of two term

8 6 I. GROSSE-BRAUCKMANN complexes [I 1 /I 2 1 Ω S1 X ] [I 12 /I 2 12 Ω S1 X Ω S2 X ] [Ω S2 X Ω S2 X ]. This proves that L X dened via X S 1 and X S 2 are quasi-isomorphic. Another property of the truncated cotangent complex is that for any open subset U X we have L U = LX U in D(U). (2.2) If we do not insist on dealing exclusively with O X -linear morphisms, we can also consider the following C-linear map O S /I 2 Ω S X that is likewise induced by d S. It is well dened because d S being a derivation implies d S (I 2 ) I Ω S. This map will allow us to lift the classical dierential d X : O X Ω X to a morphism D X : O X L X dened in D(Sh(X)) but not in D(X). Consider the inclusion i: I /I 2 O S /I 2, yielding a short exact sequence of two-term complexes [I /I 2 Ω S X ] u [O S /I 2 Ω S X ] [O X 0], (2.3) which we consider as a distinguished triangle in D(Sh(X)). Therefore, after rotation, we obtain a morphism More explicitly, (2.3) gives an isomorphism [ O X [1] = Cone(u) = I /I 2 D X : O X L X in D(Sh(X)). (2.4) (i, d) O S /I 2 Ω S X ] d S +id Ω S X in D(Sh(X)). Here, O X [1] is the complex with O X in degree 1, so that the projection to L X [1] denes (2.4). Given two embeddings X S i for i = 1, 2 we constructed above a quasi-isomorphism between L X dened via S 1 and L X dened via S 2. The construction of D X is compatible with this quasi-isomorphism as we can apply the above proof to (2.3). We conclude that the denition of D X is independent of the embedding X S. Finally, for later use, we will consider the composition N il X O X D X LX. Lemma 2.1 ([Joy13, Proposition 2.3]). The construction of D X is functorial in the following way: For every morphism Φ: X Y there is a commutative

9 LAGRANGIAN INTERSECTIONS 7 diagram Φ 1 Φ O 1 D Y Y Φ 1 L Y O X D X L X. Proof. This follows essentially from functoriality of L X Φ 1. and exactness of Remark 2.2. Let f H 0 (X, O X ). We can give a more explicit description of D X (f) H 0 (X, L X ) as follows: Consider f as a morphism f : X A 1 = Spec C[t]. Then functoriality of D X gives the commutative diagram H 0 (X, f 1 O A 1) H 0 (X, f 1 L A 1) = H 0 (X, f 1 O A 1 dt ) H 0 (X, O X ) H 0 (X, L X ), where t H 0 (X, f 1 O A 1) maps to f H 0 (X, O X ). Hence, D X (f) is the image of the generator dt under the right-hand vertical arrow H 0 (X, f 1 L A 1) = H 0 (X, f 1 O A 1 dt ) H 0 (X, L X ). Denition 2.3. The thickened cotangent complex is M X. = Cone(O X D X LX ) D(Sh(X)) and the reduced thickened contangent complex is M 0 X. = Cone(N il X L X ) D(Sh(X)). We dene two sheaves of C-vector spaces on X by S X. = H 1 (M X ) and S 0 X. = H 1 (M 0 X ). Remark 2.4. This denition of S X is suggested in [Joy13, Remark 2.2(b)]. Remark 2.5. It follows from the spectral sequence that H 0 (X, S X ) = H 1 (X, M X ). E p,q 2 = H p (X, H q (M X )) H p+q (X, M X ) Lemma 2.6. With the above notations we have the following rst properties of the thickened cotangent complex. (i) For any closed embedding X S into a smooth scheme S with ideal sheaf I. The thickened cotangent complex can be represented by M X = [ OS /I 2 Ω S X ] and its reduced version by M 0 [ ] X = I /I 2 [ Ω S X = OS /I 2 ] Ω S X O Xred. (ii) S X and SX 0 are sheaves of C-algebras.

10 8 I. GROSSE-BRAUCKMANN (iii) Let U X be an open subset. Then M U = MX U in D(Sh(U)). In particular, we have S U = S X U. (iv) For all morphisms Φ: X Y there is an induced map Φ 1 M Y M X and thus Φ 1 S Y S X. Proof. (i) follows directly from the denition, (ii) from the Leibniz rule, (iii) from (2.2) and (iv) from the functoriality of D X (cf. Lemma 2.1). Lemma 2.7. The connecting homomorphism δ in (2.5) is the dierential d X : O X Ω X. Therefore, we have a commuting diagram in D(Sh(X)) O X D X L X O X d X Ω X. Proof. We embed X S as above. The Lemma follows from the denitions and the commutative diagram H 1 (L X ) S X O X 0 I /I 2 O S /I 2 O S /I 0 0 Ω S X d Ω S X 0 0 H 0 (L X ) = Ω X. By denition, there is a commutative diagram in D(Sh(X)) with exact rows and columns N il X L X M 0 X O X L X M X O Xred 0 O Xred [1].

11 LAGRANGIAN INTERSECTIONS 9 This induces a commutative diagram of long exact sequences (cf. [Joy13, Theorem 2.1(b)]) 0 H 1 (L X ) S 0 X N il X Ω X H 0 (M 0 X ) 0 0 H 1 (L X ) S X O X δ Ω X H 0 (M X ) 0 and a long exact sequence (2.5) 0 S 0 X S X α O Xred H 0 (M 0 X ) H 0 (M X ) 0. (2.6) Form these sequences we now deduce several properties of the thickened cotangent complex. Corollary 2.8. The zero-th cohomology sheaves of the thickened cotangent complexes are given by H 0 (M X ) = Ω X /d X (O X ) and H 0 (M 0 X ) = Ω X /d X (N il X ). This allows us to give a precise formulation of [Joy13, Corollary 2.14]. Corollary 2.9. We have In particular, S 0 X H 1 (L X ) = SX 0 d ker(o X X Ω X ) N il X = 0. is a coherent sheaf on X in this case. (i) There is a natu- Proposition 2.10 (cf. [Joy13, Theorem 2.1(a),(c)]). ral decomposition S X = C X SX, 0 where the constant sheaf C X is embedded via the isomorphim C X = (C S +I 2 )/I 2. (ii) The sheaf SX 0 is isomorphic to the middle cohomology sheaf of the complex [ ] E. = I 2 d d I Ω S Ω 2 S. Proof. (i) We claim that im(α) = C X = i C Xred in (2.6) with i: X red X being the inclusion. Obviously we have im(α) C X. For the converse inclusion we choose an embedding of X into a smooth scheme S. Then α is given by the upper row in the following commutative diagram S X O S /I 2 i O Xred O X 0 Ω S X Ω X i Ω Xred and thus im(α) ker(i O Xred i Ω Xred ) = i C Xred. Together, this gives 0 S 0 X S X C X 0

12 10 I. GROSSE-BRAUCKMANN and the inclusion C X S X denes a split. (ii) This time, we denote by i the inclusion X S. Consider the commutative diagram with short exact columns I 2 I Ω S Ω 2 S O S Ω S Ω 2 S i (O S /I 2 ) i (Ω S X ) 0. We consider the rows as complexes concentrated in the interval [ 1, 1]. Taking cohomology sheaves yields an exact sequence H 1 (E) = j! C S\X C S i S X H 0 (E) 0, where j : X \ S S is the inclusion. As H 0 (E) is actually supported on X, applying i 1 gives the short exact sequence 0 C X S X H 0 (E) 0. By construction, C X S X is the embedding from part (i), so that applying part (i) nishes the proof. The following example is central to our application of the thickened cotangent complex. Example Let f H 0 (S, O S ) and consider crit(f) = Z(df) S with df ideal I = im(t S O S ). Then, by denition, df I Ω S and thus f H 0 (X, S X ) H 0 (X, O S /I 2 ). Moreover, f H 0 (X, S 0 X ) if and only if f I. In Example 2.12 below, we see that f is not necessarily locally constant on X as one might expect thinking of manifolds. Example 2.12 (cf. [Joy13, Example 2.13]). In general, we can not expect that H 1 (L X ) = SX 0. The following example shows that this is not even true when X = crit(f) for some f H 0 (S, O S ). Let S = Spec C[x, y] and f = x 5 + x 2 y 2 + y 5. Set X = crit(f). Then, X = V (I) with I = (d x f = 5x 4 + 2xy 2, d y f = 2x 2 y + 5y 4 ). In Example 2.11 we already noticed that df X = 0. Write X as the intersection of two reducible curves X = (L 1 C 1 ) (L 2 C 2 ), where L i are lines and C i are cuspidal cubics, i = 1, 2. Hence, the intersection number is 16. Looking more closely, we see that X consists of 5 distinct reduced intersection points of C 1 and C 2 and the point (0, 0) occuring consequently with multiplicity 11. We claim that the local ring at (0, 0) is given by Spec C[x, y]/j, where J = (d x f, d y f, xy 3, y 3 x). This follows, since 2yd x f 5x 2 d y f = xy 3 (4 25xy) and 4 25xy is invertible in the local ring at (0, 0). Similarily for x 3 y. Note that C[x, y]/j really has a basis consisting of 11 elements, namely {1, x, y, x 2, xy, y 2, x 3, x 2 y, xy 2, y 3, x 2 y 2 }.

13 LAGRANGIAN INTERSECTIONS 11 Now, 5f x 2 y 2 = d x f + d y f shows that 0 f C[x, y]/j meaning that f is not locally constant near (0, 0). Moreover, we see f J. Finally, after altering f by its value on the other points of X we nd 0 f N il X ker(o X Ω X ) or, put dierently, f H 0 (X, SX 0 ) but f / H0 (X, H 1 (L X )). We also see that, in this example SX 0 is not an natural O X-module. Otherwise we would have x x 2 y 2 H 0 (X, SX 0 ). However, d(x 3 y 2 ) = xd(x 2 y 2 ) + x 2 y 2 dx = x 2 y 2 dx 0 Ω A 2 X Denition and rst examples. We come now to the denition of a d-critical locus from [Joy13]. A d-critical locus is a scheme that is locally of the form crit(f) with a compatibility condition given by a section of the sheaf SX 0. Depending on the situation, we apply the theory using the analytic or the Zariski topology, so we let X be a scheme or a complex analytic space. Denition 2.13 ([Joy13, Denition 2.5]). A structure of a d-critical locus on X is dened by a section s H 0 (X, SX 0 ) satisfying the condition that for each x X there is an open neighborhood U X of x and a closed embedding U S into a smooth scheme S such that there is f H 0 (S, O S ) with the properties s U = f H 0 (U, O S /I 2 ) and U = crit(f) S. We call (U, S, f) a critical chart around x for the d-critical locus (X, s). An embedding of critical charts ι: (U, S, f) (V, T, g) is a locally closed embedding ι: S T such that g ι = f and ι U : U V X is the inclusion in X. A subchart is an open embedding of critical charts. If (U, S, f) and (V, T, g) are critical charts, then any étale morphism ι: S T such that g ι = f restricts to a morphism ι U : U V. If ι U is the inclusion in X we write ι: (U, S, f) (V, T, g) and call ι an étale morphism of critical charts. A morphism of d-critical loci Φ: (X, s) (Y, t) is a morphism Φ: X Y such that the induced map (cf. Lemma 2.6(iv)) Φ 1 S Y S X maps t to s. With this notion, d-critical loci form a category. Remark The theory also works with s H 0 (X, S X ). However, choosing s H 0 (X, SX 0 ) has the practical advantage that, with the above notations, it enforces f Ured = 0 H 0 (X red, O Xred ). Example Let us apply the denition to the zero-dimensional example X = Spec C[x]/(x n ) A 1. For this, consider the exact sequence of C-vector spaces 0 S X C[x]/(x 2n ) C[x]/(x n ) dx x i ix i 1. We nd SX 0 = (xn+1 ) C[x]/(x 2n ). Note that this is an example of Corollary 2.9 and indeed L X is given by [ ] (x n )/(x 2n d ) C[x]/(x n ) dx d

14 12 I. GROSSE-BRAUCKMANN hence S 0 X = H 1 (L X ). Next, we wonder which f C[x] actually dene a d-critical structure on X. This is the case if and only if there is an open subset U A 1 such that X = V (df) U. Hence, we need that f = a n+1 x n with a n+1 0 and this gives s (x n+1 ) \ (x n+2 ). Example Assume that X is smooth. Then M 0 X = [0 Ω X ] and thus SX 0 = 0. Therefore, X has a unique structure of a d-critical locus given by (X, 0) Critical charts. Let (X, s) be a d-critical locus. In [Joy13, 2.3] Joyce studies how to manipulate critical charts on (X, s). His major results are that any two critical charts can locally be embedded into a third one and that any embedding of critical charts can locally be modied into an étale morphism of a certain standard form. Proposition 2.17 ([Joy13, Theorem 2.20]). Let (U, S, f) and (V, T, g) be two critical charts on (X, s). For every x U V there exist subcharts (U, S, f ) (U, S, f) and (V, T, g ) (V, T, g) around x and another critical chart (W, R, h) together with embeddings of critical charts (U, S, f ) (W, R, h) and (V, T, g ) (W, R, h). Proposition 2.18 ([Joy13, Proposition 2.22]). Let (X, s) be an analytic critical locus and let (U, S, f) (V, T, g) be an embedding of critical charts on (X, s). For every x U there exists an open neighborhood T T and holomorphic maps α: T S, β : T C n, where n = dim T dim S such that α β : (V T, T, g V ) (U, S C n, f z z2 n) is an étale morphism of critical charts, i.e. g T = f α+(z z2 n) β. Furthermore, α β S : S. = S T S C n is the inclusion of S {0}. Remark There is an algebraic analogue of this Proposition ([Joy13, Proposition 2.24]). These propositions are the key for working with d-critical loci. Roughly speaking, when constructing a geometric object on (X, s) we only have to care about (1) étale morphisms of critical charts and (2) the embedding (U, S, f) (U, S C n, f z z2 n). In [KL16] Kiem and Li present another approach to (analytic) d-critical loci. They make use of the above Propositions in order to x a well-behaved set of critical charts from the beginning on. More precisely, they prove the following equivalent denition of d-critical loci. Lemma 2.20 ([KL16, Proposition 1.44]). Let X be a complex analytic space. Then X has a d-critical structure if and only if there exists an open covering X = α X α and complex manifolds V α together with a closed embedding X α V α and holomorphic functions f α : V α C satisfying

15 LAGRANGIAN INTERSECTIONS 13 (1) X α = crit(f α ) V α ; (2) for each pair of indices (α, β) and X αβ = X α X β there is an open neighborhood V αβ (resp. V βα ) of X αβ in V α (resp. V β ) and a biholomorphic map ϕ αβ : V αβ V βα making the following diagram commute X αβ V αβ ϕ αβ V βα f α Vαβ (3) ϕ 1 αβ = ϕ βα and ϕ αα = id Vα. C; f β Vβα In this situation, we nd for all X αβγ. = X α X β X γ an open neighborhood V αβγ in V α such that ϕ αβγ. = ϕ αβ ϕ γα ϕ βγ : V αβγ V α (2.7) is biholomorphic onto its image ([KL16, Remark 1.7]). Then ϕ αβγ is the identity on X αβγ, yet in general not on all of V αβγ. This is the reason why the gluing of objects (as for example the symmetric obstruction theories from the consecutive chapter) over the intersection of critical charts is troublesome. 3. Symmetric obstruction theories on d-critical loci We are going to explain the concept of symmetric obstructions theories as introduced in [BF05]. We will see that a symmetric obstruction theory is naturally dened in the situation of Lagrangian intersections and locally dened on any d-critical locus. For a discussion of the relation between the category of d-critical loci and the category of schemes with (symmetric) obstruction theory we refer to [Joy13, Examples 2.16 and 2.17] Symmetric obstruction theories. Let X be a scheme. We denote by L X the truncated cotangent complex as in (2.1). If E D b (X) is a bounded complex, we can dene E. = RHom(E, O X ) D b (X) as well as E = RHom(E, O X ) D b (X). There is a natural map E E. If E is given by a complex [E 1 α E 0 ] of locally free sheaves, then, following our sign conventions, E = [(E 0 ) such that we get E [1] = [(E 0 ) start with a technical result. α (E 1 ) ] in degree 0 and 1, α (E 1 ) ] again in degree 1 and 0. We Lemma 3.1. Let E D(X) be such that H i (E) = 0 if i / {0, 1}. Then there is an isomorphism H 1 (E ) H 1 (E) = Hom(H 1 (E), O X ).

16 14 I. GROSSE-BRAUCKMANN Proof. We are looking for an isomorphism H 1 (RHom(E, O X )) = Ext 1 (E, O X ) H 1 (E) = Ext 0 (H 1 (E), O X ). Consider the spectral sequence E p,q 2 = Ext p (H q (E), O X ) E p+q = Ext p+q (E, O X ). By assumption, we have E p,q 2 = 0 if p < 0 or q / {0, 1} and hence the only non-zero entry on the diagonal p + q = 1 occurs when p = 0. Thus the spectral sequence degenerates at the E 2 page and yields the desired isomorphism E 1 = Ext 1 (E, O X ) E 0, 1 2 = Ext 0 (H 1 (E), O X ). Corollary 3.2. Let E D(X) be a perfect complex such that H i (E) = 0 if i / { 1, 0}. Then there is an isomorphism H 1 (E) H 1 (E ). Proof. Since E is perfect the natural map E E is an isomorphism (cf. [St17, Tag 07VI]) and thus H 1 (E) H 1 (E ). The corollary follows after postcomposing with the isomorphism H 1 (E ) H 1 (E ) from Lemma 3.1. This result applies in particular to the following type of complexes. Denition 3.3. We say that E D(X) is perfect of amplitude in [ 1, 0] if E is locally isomorphic to a complex [E 1 E 0 ] of locally free sheaves. Denition 3.4 ([BF05, Section 1.3]). A perfect obstruction theory for X is tuple (E, φ), where E is perfect of amplitude in [ 1, 0] and φ: E L X is a morphism in D(X) such that (1) H 1 (φ): H 1 (E) H 1 (L X ) is surjective and (2) H 0 (φ): H 0 (E) H 0 (L X ) = Ω X is an isomorphism. A symmetric obstruction theory is a triple (E, φ, θ), where (E, φ) is a perfect obstruction theory and θ : E [1] E is a non-degenerate symmetric bilinear form, i.e. an isomorphism in D(X) such that θ [1] = θ. An isomorphism of perfect obstruction theories Φ: (E 1, φ 1 ) (E 2, φ 2 ) is given by an isomorphism Φ: E 1 E 2 over L X in D(X). Moreover, Φ is an isomorphism of symmetric obstruction theories (E 1, φ 1, θ 1 ) (E 2, φ 2, θ 2 ) if it is an isometry (E 1, θ 1 ) (E 2, θ 2 ), i.e. if E 1 [1] Φ [1] E 2 [1] commutes. θ 1 θ 2 Φ E 1 E 2

17 LAGRANGIAN INTERSECTIONS 15 Example 3.5. Let S be any scheme containing two smooth and closed subschemes Y, Z with ideal sheaves I Y S, I ZS. Then the intersection X = Y Z carries a perfect obstruction theory [ ] ( res Y,res Z ) E. = Ω S X Ω Y X Ω Z X (3.1) with E φ L X represented by ϕ 1 E = [Ω S X ( res,res) Ω Y X Ω Z X ] F (0,1). = [I Y S /I 2 Y S X Ω Z X ] ϕ 2 L X = [I XZ /I 2 XZ Ω Z X ]. Here, the dierential of F is the composition I Y S /I 2 Y S X I XZ /I 2 XZ d Ω Z X. This can be seen at follows. First of all the exact triangle F ϕ 1 E [Ω Y X = Ω Y X ] implies that ϕ 1 : F E is a quasi-isomorphism. Moreover, we see immediately that H 0 (ϕ 2 ) is an isomorphism and H 1 (ϕ 2 ) is surjective. Let us remark that interchanging the roles of Y and Z yields isomorphic obstruction theories. Remark 3.6. In Proposition 6.1 we will see that if S is a symplectic manifold and Y, Z S are Lagrangian submanifolds, then the above Example 3.5 can be enhanced by an isometry and thus turned into a symmetric obstruction theory. Lemma 3.7 ([BF05, Corollary 1.15]). Let (E, φ, θ) be a symmetric obstruction theory. Then H 1 (E) = Hom OX (Ω X, O X ) =: T X Proof. In Corollary 3.2 we showed that H 1 (E) = H 1 (E ). However, this is all we need to show, since H 1 (E ) = H 0 (E [1]) = H 0 (E) = Ω X by symmetry of the obstruction theory. Remark 3.8. Let E = [E 1 E 0 ] L X be a symmetric obstruction theory on X. Then, if H 1 (E) = Ω X and H0 (E) = Ω X admit a nite resolution by vector bundles their determinants are well-dened and we have det E = (det E 1 ) det E 0 = (det H 1 (E)) det H 0 (E) 3.7 = det(ω X ) det Ω X.

18 16 I. GROSSE-BRAUCKMANN In particular, in this case det E is independent of the obstruction theory. For example, if X is smooth we nd det E = ω 2 X. (3.2) The same is true if X is normal and Gorenstein. In this situation ω X is a line bundle. Moreover, any line bundle on X is uniquely determined by its restriction to X reg, i.e. for every line bundle L Pic(X) the natural morphism L j j L is an isomorphism, where j : X reg X is the inclusion. This can be seen as follows. As X is normal, we have codim(x sing X) 2 and therefore the local cohomology sheaves H k X sing (L) are trivial for k = 0, 1 by [SGA2, VII, Corollaire 1.4]. Here, we use that any line bundle is a coherent Cohen Macauley module. Now, there is an exact sequence 0 H 0 X sing (L) L j j L H 1 X sing (L) 0 (see [SGA2, I, Corollaire 2.11]), which proves the claim. In particular, we have ω X = j ω Xreg and also det E = j j det E = j det(e Xreg ) = j ω 2 X reg = ω 2 X for every symmetric obstruction theory on E on X. We conclude this discussion with Question 3.9. Does det E only depend on X? In other words, if E and E are two symmetric obstruction theories on X. Is there a (canonical) isomorphism det E = det E? 3.2. Symmetric obstruction theories on d-critical loci. The reason why we are interested in symmetric obstruction theories is that every d- critical locus carries locally a symmetric obstruction theory. Let M be a smooth scheme and f : M A 1 a regular function with critical locus X. We consider df H 0 (M, Ω M ) as a morphism T M O M. By denition, its image is the ideal sheaf I XM dening X. Therefore, we obtain df : T M X I XM /I 2 XM. A local computation shows that the composition is just the Hessian 2 f. T M X df I XM /I 2 XM d Ω M X Lemma With the above notations X = crit(f) carries a symmetric obstruction theory (E 2 f, ψ, θ). It is given by [ ]. E 2 f = T M 2 f X Ω M X, the morphism of complexes E 2 f = [T M X 2 f Ω M X ] ψ L X = [I XM /I 2 XM df d Ω M X ]

19 and the isometry θ.. = id E 2 f : E 2 f [1] E 2 f. LAGRANGIAN INTERSECTIONS 17 Proof. Again, we see immediately that (E 2 f, ψ) is a perfect obstruction theory. Moreover, the symmetry of the Hessian yields ( 2 f ) [1] = 2 f so that E 2 f [1] = E 2 f and after this identication the identity serves indeed as a non-degenerate bilinear form. It turns out that these local obstruction theories are in general not compatible and thus do not glue to a global symmetric obstruction theory on (X, s). For instance, in [Joy13, Example 2.17] one nds an example of a smooth scheme X admitting two global critical charts (X, X, 0) and (X, U, f). However, in this example E 0 is not isomorphic to E 2 f. Nevertheless, because X is smooth we have Ext 2 (Ω U, T U ) = 0 for any ane open subscheme U X and therefore, E 2 f U = E0 U. We do not know, whether it is in general (locally) possible to nd a distinguished symmetric obstruction theory on (X, s). At least, if we restrict ourselves to a set of charts as in Lemma 2.20 this can not happen. Remark Let (X α, V α, f α ) as in Lemma Then the biholomorphism ϕ αβ : V αβ V βα is an isomorphism. Moreover it induces dϕ αβ Xαβ : Ω Vα Xαβ = Ω Vαβ Xαβ Ω Vβα Xαβ = Ω Vβ Xαβ so that we can dene an isomorphism of symmetric obstruction theories E 2 f α = [T 2 f α Xαβ Vα Ω Xαβ Vα ] (3.3) 2 f β E 2 f β = [T Xαβ Vβ Ω Xαβ Vβ ]. Here, the vertical arrows are given by (dϕ 1 αβ Xαβ ) and dϕ Xαβ αβ, respectively. Commutativity of (3.3) means dϕ αβ Xαβ 2 f α (dϕ αβ Xαβ ) = 2 f β which follows from f α = f β ϕ αβ. In particular, we see that xing an atlas as in Lemma 2.20 provides us with a distinguished symmetric obstruction theory in the neighborhood of any point x X Canonical bundle. Let (X, s) be a d-critical locus. For each critical chart (U, S, f) we consider the line bundle ω S U. One could hope, that it is possible to glue these in order to obtain a `canonical bundle' associated to (X, s). However, this is not the case (see Example 3.18). What is possible instead, is to glue the square ω 2 S U red = det E 2 f Ured to a line bundle K X,s on X red. As discussed in Section 2.3 we have to dene gluing isomorphisms for every étale morphism of critical charts and for the embedding (U, S, f) (U, S C n, f z z2 n). In the latter situation we will identify ω S C n U with ω S U without mentioning it explicitly. Let ι: (U, S, f) (V, T, g) be an étale morphism of critical charts (Denition 2.13). Recall that this signies among other things that ι U : U V

20 18 I. GROSSE-BRAUCKMANN is just the inclusion in X. The natural morphism dι: ι Ω T an isomorphism ξ 2 ι ΩS induces = det(dι) 2 : det E 2 g U = ω 2 T U det E 2 f U = ω 2 S U. This shall be the gluing datum for K X,s. Proposition 3.12 ([Joy13, Theorem 2.28] or [KL16, Remark 1.18]). There exists a unique line bundle K X,s on X red such that for each critical chart (U, S, f) there is an isomorphism λ (U,S,f) : K X,s Ured det E 2 f Ured satisfying for each étale morphism of critical charts ι: (U, S, f) (V, T, g) the equality λ (U,S,f) = ξ 2 ι Ured λ (V,T,g) Ured. We will not give the proof here, but at least point out its most important step. One has to show that ξι 2 denes a gluing datum. This holds only true on the reduced critical locus X red and relies on the following Lemma 3.13 ([KL16, Lemma 1.16] or [BBDJS, Corollary 3.2]). Let S be a smooth scheme and f : S C a regular function. Set X = crit(f) and assume that there is an étale morphism ι: S S such that the following diagram commutes X S ι S Then f C. f det(dι) Xred : ι ω S Xred = ω S Xred ω S Xred is locally constant with values in {±1}. Proof. Consider the commutative diagram with exact lines 0 T X T S X 2 f (dι) 0 T X T S X 2 f Ω S X dι Ω X 0 Ω S X Ω X 0. (3.4) If T X and Ω X admit a nite resolution by locally free sheaves such that their determinants satisfy det(t X ) = det(ω X ), the Lemma will follow from functoriality of the determinant. For the general proof let x X. We will show that (det ι) 2 (x) = 1 in the residue eld κ(x). For this aim, we choose

21 LAGRANGIAN INTERSECTIONS 19 a decomposition Ω S X (x) = Ω X (x) W so that the commutative diagram of vector spaces T X (x) T S X (x) 2 f (x) T S X (x) Ω X (x) 0 T X (x) (dι(x)) takes the following form T X (x) T X (x) dι(x) Ω S X (x) 2 f (x) Ω S X (x) Ω X (x) 0 Ω X (x) W ( ) 1 0 t A Ω X (x) W ( ) H ( ) H Ω X (x) W ( ) 1 0 A Ω X (x) 0 Ω X (x) W Ω X (x) 0, where H is some invertible matrix. It follows that H = t AHA and thus det A {±1}, which implies the claim. Remark 3.14 ([KL16, Proof of Proposition 1.15]). If one xes charts (X α, V α, f α ), appropriate neighborhoods V αβ and biholomorphisms ϕ αβ : V αβ V βα as in Lemma 2.20, the existence of K X,s follows directly from Lemma Namely, in this case we have to show that the isomorhisms ξ αβ. = det(dϕ αβ ) 2 Xαβ : det(e 2 f α ) Xαβ det(e 2 f β ) Xαβ from Remark 3.11 satisfy which is equivalent to ξ βγ ξ αβ = ξ αγ on (X αβγ ) red ξ γα ξ βγ ξ αβ = det(dϕ γαβ ) 2 = 1 on (X αβγ ) red, where ϕ γαβ is dened as in (2.7). And this follows after applying Lemma 3.13 with S = ϕ γαβ (V γαβ ) V γαβ, φ = ϕ γαβ and f = f γ. Denition An orientation of (X, s) is the choice of a square root line bundle L = K 1/2 X,s on X red together with an isomorphism L L K X,s. We will suppress the choice of the latter in our notation. Question When is it possible to dene K X,s on X and not only on X red? Of course, this is possible, if we are provided with a global symmetric obstruction theory, together with isomorphisms E U E 2 f for every critical chart (U, S, f) satisfying the correct functoriality. On the other hand, as mentioned in the proof of Lemma 3.13, the question also has a positive answer, if det(ω X ) and det(t X ) are well-dened and satisfy det(t X ) = det(ω X ). In this case, we saw in Remark 3.8 that det E = det(ω X ) 2 for every symmetric obstruction theory E on X. Hence, if we apply this to a covering of critical charts, we nd K X,s = det(ωx ) 2. Example Assume that X is smooth. Then we have K X,0 = ω 2 X. In particular, the d-critical locus (X, 0) is canonically oriented.

22 20 I. GROSSE-BRAUCKMANN Example An example of a non-orientable d-critical locus X can be found in [Joy13, Example 2.39] or [KL16, Example 1.19]. In these examples, X is a projective line with a non-reduced point. 4. Perverse sheaves We want to explain the existence of a perverse sheaf P X,s on any d-critical locus (X, s). First, let us review some basics about the perverse sheaf of vanishing cycles following [Di04] Perverse sheaves of vanishing cycles. Let X be a topological space and D b c(sh(x)) the essential image of the category of bounded complexes of constructible sheaves in D(Sh(X)). We denote by Perv(X) the subcategory of perverse sheaves, i.e. (cf. [BBD, Dénition 2.1.2]) for any F D b c(sh(x)) we have F Perv(X) if and only if for each stratication {X α } (1) H k (i αf ) = 0 for all k > dim C X α and (2) H k (i! αf ) = 0 for all k < dim C X α for all α, where i α : X α X denotes the inclusion. By a stratication we mean a nite partition of X into (non-empty) locally closed subsets, called strata. We assume that the closure of each stratum is the union of strata. As an immediate consequence we record Remark 4.1. Let F Perv(X) then H k (F ) = 0 if k / [ dim X, 0]. Proposition 4.2 ([Di04, Theorem ]). Let X be a complex analytic space of pure dimension n which is locally a complete intersection. Then the shifted constant sheaf C X [n] is a perverse sheaf. Remark 4.3. Let P Perv(X) and L a local system on X. Then also P L Perv(X), because tensoring with a local system is exact. In particular, in the situation of the previous proposition, it follows that L[n] is a perverse sheaf. Next, we dene the perverse sheaf of vanishing cycles as in [Di04, Section 4.2]. Let S be a complex analytic variety and f : S C an analytic map. We set S 0. = f 1 (0) with inclusion i: S 0 S and consider the commutative diagram S. = S ρ C p S \ S 0 S (4.1) C C f C, ρ

23 LAGRANGIAN INTERSECTIONS 21 where C is the universal cover of C. More especially, C = C and ρ(z) = exp(2πiz). The nearby cycle functor ψ f : D b c(s) D b c(s 0 ) is dened by sending F D b c(s) to ψ f F. = i Rp p F D b c(s 0 ) and the vanishing cycle functor φ f : D b c(s) D b c(s 0 ) is dened such that there is an exact triangle in D b c(s 0 ) i F ψ f F φ f F i F [1], where the rst arrow comes from the adjunction map F Rp p F. We denote by ψ p f and φp f the shifted functors ψ f [ 1] and φ f [ 1], respectively. We need Proposition 4.4 ([Di04, Theorem ]). The functors ψ p f, φp f : Db c(s) D b c(s 0 ) preserve perversity. In particular, we have induced functors ψ p f, φp f : Perv(S) Perv(S 0). Proposition 4.5. Let Φ: T S be a morphism between two analytic spaces that is smooth of relative dimension d and f : S C. Write g = f Φ and Φ 0 = Φ T0 : T 0 S 0. Then we have natural isomorphisms Φ 0 ψ f ψ g Φ and Φ 0 φ f φ g Φ. Proof. It suces to prove the statement for the nearby cycle functor. Plugging in the denitions, we get Φ 0 ψ f = Φ 0 i S Rp S p S = i T Φ Rp S p S i T Rp T Φ p S = i T Rp T p T Φ = ψ g Φ. Here, we used i S Φ 0 = Φ i T for the rst, Φ = Φ! [2d] and the base change isomorphism for the second and Φ p T = p S Φ for the third equality, where Φ = id C ρ Φ. Lemma 4.6. Let S and f : S C as above. We denote by f n : S C the morphism given by f n (s) = f(s) n. Then there is a natural isomorphism n ψ f nf = ψ f F for every F D b c(s). k=1 Proof. By denition, ψ f n is computed by means of the bre product S p S \ S 0 S C ρ C where ρ(z) = exp(2πiz). We dene µ n, ν n : C C by µ n (z) = nz and ν n (z) = z n and replace S, p by S µn C, µ np, respectively. As µ n is an C, f n

24 22 I. GROSSE-BRAUCKMANN isomorphism this does not aect the vanishing cycle but now, using ρ µ n = ν n ρ, we can express S as the double bre product S C q ρ p n k=1 S π C ζ k f S ν n C, where π = id and ζ is a n-th root of unity. Let i: S 0 : S be the inclusion and j = i and π 0 = π S0. We nd ψ f n = i Rp p = i Rπ Rq q π = Rπ 0 j Rq q π = Rπ 0 ψ ζ k f π = n k=1 ψ f. using i Rπ = Rπ0 j by proper base change. We can compute the stalk cohomology of ψ f or φ f via the Milnor bre (cf. [Di04, Proposition and Example 4.2.6]). Recall that if f vanishes in s S, then this is the bre of the Milnor-Lê bration (cf. [Mi68, Theorem 4.8]) which is given by MF s = B δ (s) f 1 (t) for some small t, δ > 0 and B δ (s) is an open ball of radius δ around s in S dened via embedding the germ (S, s) in an ane space. If S = C n+1 with n 1 and f has an isolated singularity in s, then Milnor proved that MF s has the homotopy type of a bouqet of n-dimensional spheres (cf. [Mi68, Theorem 6.5]). The number of these spheres is ( given by the) Milnor number µ = dim C C[x 0,..., x n ]/J(f), where J(f) = f x 0,..., f x n is the ideal dening crit(f). Finally, we have for all s S 0 a natural isomorphism and if S is smooth f n H k (ψ f C S ) s = H k (MF s, C) (4.2) H k (φ f C S ) s = Hk (MF s, C), (4.3) where H k (MF s, C) is the reduced cohomology of the Milnor bre at s. Now, we let X = crit(f), then f X is locally constant as a map of topological spaces f : X C with nite image. For simplicity, we assume that f X 0 such that X = Sing(S 0 ) is the singular locus of the bre S 0. We dene the perverse sheaf of vanishing cycles PV S,f. = φ p f (C S[dim S]) = φ f C S [dim S 1]. (4.4) It follows from Propositions 4.2 and 4.4 that PV S,f Perv(S 0 ). Moreover, by (4.3) we have H k (PV S,f ) s = H k+dim S 1 (MF s, C) and therefore: Corollary 4.7. The perverse sheaf of vanishing cycles PV S,f is supported on X = crit(f). In particular, we can consider PV S,f as an element of Perv(X). Remark 4.8. More precisely, if i: X S is a closed subspace, then there is an equivalence between the category of perverse sheaves on S supported on X and Perv(X) established by i with inverse Ri.

25 LAGRANGIAN INTERSECTIONS 23 Remark 4.9 ([Di04, Cor ]). If S is a connected complex manifold and f non-constant, then supp(φ f C S ) = Sing(S 0 ). Remark If f has several critical values, we set. = φ p f c (C S[dim S]). PV S,f c f(x) This denes a pervese sheaf on X (cf. [Bu14, Denition 1.7]). We will now compute some examples. Example Let S be any smooth sheme and f = 0. Applying the denition we nd ψ f = 0 and thus φ p f = id D b c(s) yielding PV S,f = C S [dim S]. Example Let S = Spec C[x] and f = x n+1. Combining Remark 4.1 and Corollary 4.7 we nd that PV S,f is a complex concentrated in degree zero and supported on a point. Therefore, PV S,f is determined by H 0 (PV S,f ) 0 = H 0 (MF 0, C). However, MF 0 is given by n + 1 points. Hence, PV S,f = C n 0. Example Let S = Spec C[x 1,..., x n ] and f = n i=1 x2 i. This time, we have to compute H n 1 (MF 0 ) and MF 0 has the homotopy type of a (n 1)- sphere. Hence, = C 0. PV S,f Example Let S = Spec C[x, y] and f = x n y n. Then ψ f C S = n ψ xy C S i=1 by Lemma 4.6. We can compute ψ xy C S as follows. Using the same arguments as above we nd φ xy C S = C0 [ 1] and thus ψ xy C S ts into the exact triangle C X0 ψ xy C S C 0 [ 1], where X 0 = Z(xy) S. This allows to conclude that ψ xy C d S = [CX0 C 0 ] in degrees 0 and 1. Moreover, using (4.2), i.e. H 1 (ψ xy C S ) s = H 1 (MF s, C) we see that d = 0. Hence, ψ xy C S = CX0 C 0 [ 1] yielding PV S,f = C n 1 X 0 [1] C n 0.

26 24 I. GROSSE-BRAUCKMANN 4.2. Local system corresponding to a torsion line bundle. Before stating the construction of the perverse sheaf P X,s we need yet another technical paranthesis. Let X be a scheme and L a line bundle on X such that α: L L O X. Then O X L is an O X -algebra via (a, b)(a, b ) = (aa +α(bb ), ab +a b) and we dene Y. = Spec X (O X L). The structure map f : Y X denes an étale covering of degree two such that f O Y = OX L and, more importantly for our purposes, f C Y = CX L, for some rank one local system L on X. This can be seen as follows. First of all, f C Y is a local system of rank 2 because f is nite and at with 2- dimensional bres. Now, we consider the natural morphism γ : C X f C Y, which is obviously injective. We claim that L. = coker(c X f C Y ) is a local system and satises f C Y = CX L. Indeed, a retract of γ is given explicitly given by mapping a section s H 0 (f 1 (U), C Y ) to the locally constant map X x 1 2 (s(x 1) + s(x 2 )), where x 1 and x 2 are the two preimages of x under f. We call L the local system corresponding to L. If X is an analytic space Y, can either be constructed by gluing disjoint copies of X or as a closed subspace of the total space of the line bundle L. By construction, L satises L L = C X and L = L CX O X. To be more precise, we should say that L is the local system corresponding to (L, α). However, if X is proper and reduced this makes no dierence. Lemma Let X be proper and reduced. Then the local system L depends only on L and not on the isomorphism L L = O X. Proof. Consider the short exact sequence of groups γ 1 Z/2Z C ν 2 C 1, (4.5) where ν 2 is given by ν 2 (z) = z 2. We denote the two-torsion part of H 1 (X, OX ) by H 1 (X, OX )[2] and similarly for H1 (X, C X ). Then (4.5) induces a commutative diagram with exact rows H 0 (X, C X ) ν 2 H 0 (X, C X ) H 1 (X, Z/2Z) H 1 (X, C X )[2] 0 H 0 (X, O X ) ν 2 H 0 (X, O X ) H 1 (X, Z/2Z) H 1 (X, O X )[2] 0. Here, α is given by α(l) = L O X. We want to see that α is an isomoprhism. First of all, we know from the above construction that α is surjective. The morphism ν 2 : H 0 (X, C X ) H0 (X, C X ) is surjective for any analytic space so that H 1 (X, Z/2Z) H 1 (X, C X )[2] is an isomorphism. This implies that the surjectivity of ν 2 : H 0 (X, OX ) H0 (X, OX ) is equivalent to the injectivity of α and thus holds true if X is proper and reduced. Remark The same construction works for line bundles with n-torsion. α

27 LAGRANGIAN INTERSECTIONS 25 Remark The construction is functorial in the following sense. Let L 1 and L 2 be two line bundles with two-torsion given by the isomorphisms α i : L i L i OX. Any morphism ϕ: L 1 L 2 such that α 2 (ϕ ϕ) = α 1 induces a morphism L 1 L 2 between the corresponding local systems. If X is smooth, the RiemannHilbert correspondence ([De70, Théorème 2.17]) provides a second description of the local system L. Namely, there is a unique at connection : L L Ω 1 X such that L = ker. If X is moreover compact, Lemma 4.15 says that is the unique at connection on L such that ker ker = C X. Lemma Let X be a compact Kähler manifold and L, L as above. Then there exists a hermitian metric h on X with corresponding Chern connection h = 1,0 + such that L = ker 1,0. Proof. By Lemma 4.15 is suces to show that there is a hermitian metric h such that ker 1,0 ker 1,0 = CX. Under the RiemannHilbert correspondence, ker 0,1 ker 0,1 corresponds to the pair (L L, 0, ,1 ) and h h is the Chern connection to h h on L L. Therefore, we have to show that there is a hermitian metric h on L such that h h is the constant metric h 0 on O X via the isomorphism α: L L O X. Let g be any hermitian metric on L. Then g g = e f h 0 via α for some real function f on X and h. = e 1/2f g satises h h = e f g g = h 0 as desired Perverse sheaves on d-critical loci. If X S is of the form crit(f) for some f H 0 (S, O S ), we dened in (4.4) the perverse sheaf of vanishing cycles PV S,f on X. Let (X, s) be a d-critical locus. Our goal is to dene a perverse sheaf P X,s on (X, s) that is locally isomorphic to PV U,f. The following example shows that this will be more complicated than naively gluing the perverse sheaves PV U,f over critical charts. Example 4.19 ([BBDJS, Example 5.5]). Let X = Spec C[x, x 1 ]. We will dene two dierent global critical charts (X, S, f) and (X, V, g) on X such that PV S,f and PV T,g are not isomorphic. Let S. = X with f = 0 and T. = Spec[x, x 1, y] with g : T Spec C[t] given by g x k y 2 for some k 0. Then crit(g) = Spec C[x, x 1, y]/(kx k 1 y 2, 2x k y) = Spec C[x, x 1 ] = X. We nd PV S,f = C X [1] by Example However, we claim that PV T,g = { CX [1] if k 0 mod 2 L X [1] if k 1 mod 2. Here, L X is the unique two-torsion local system on X, i.e. it corresponds to the representation π 1 (X) = Z C that is given by 1. This can be seen as follows. First of all, let us consider X and T as topological spaces, i.e. X = C \ {0} and T = C \ {0} C. If k = 2n for some n N, then Φ: T T

28 26 I. GROSSE-BRAUCKMANN given by Φ(x, y) = (x, x n y) denes an isomorphism such that g = Φ h, where h(x, y) = y 2. Moreover, Φ X = id X and thus PV T,g = PVT,h = CX [1]. If k = 2n + 1 we cover T = (U 0 C) (U 1 C), where U 0 = C \ R 0 and U 1 = C \ R 0. For i = 0, 1 the isomorphism allows to compute Φ i : U i C U i C (x, y) (x, x k, y) PV T,g Ui = CUi [1]. Finally, these constant sheaves will be glued as specied by the chart transition of the square root. This proves the claim. In [BBDJS, Theorem 5.4] it is proven that the above example represents the general situation. For any embedding (U, S, f) (V, T, g) of critical charts the perverse sheaves PV S,f and PV T,g dier by a two-torsion local system of rank one. Given the datum of a square root K 1/2 X,s of the canonical bundle K X,s one can control the local systems arising this way and in this case it is actually possible to glue the perverse sheaves of vanishing cycles. We will state this result below. First, let us dene the gluing maps. In Section 2.3 we discussed that it suces to consider étale morphisms of critical charts and the embedding (U, S, f) (U, S C n, f z z2 n). In the latter case the Thom-Sebastini Theorem (cf. [BBDJS, Theorem 2.13]) provides a natural isomoprhism PV S C n,f z z 2 n PV S,f PV C n,z z 2 n = PVS,f. (4.6) Let ι: (U, S, f) (V, T, g) be an étale morphism of critical charts. Then Proposition 4.5 gives an isomorphims of functors ι 0 ψ g ψ f ι and consequently, because dim S = dim T, we obtain an isomorphism ρ ι : PV T,g U PV S,f (4.7) in Perv(U) = Perv(U red ). Let ξ ι : ι ω T ω S be the natural isomorphism coming from ι Ω T Ω S. Then id ξ ι : (K 1/2 X,s) Ured ω T Ured (K 1/2 X,s) Ured ω S Ured is per denition compatible with the trivializations of the squares and therefore induces a morphism ξ ι : L or V,T U L or U,S (4.8) (cf. Remark 4.17). Proposition 4.20 ([BBDJS, Theorem 6.9]). Let (X, s) be a d-critical locus with orientation KX,s. 1/2 Then there exists a perverse sheaf P X,s in Perv(X) such that for each critical chart (U, S, f) on (X, s) there is a natural isomorphism κ (U,S,f) : P X,s U PV S,f CU L or U,S