LECTURE 3: SHEAVES ON R 2, CONSTRUCTIBLE WITH RESPECT TO THE STRATIFICATION BY THE AXES

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1 LECTURE 3: SHEAVES ON R 2, CONSTRUCTIBLE WITH RESPECT TO THE STRATIICATION BY THE AXES VIVEK SHENDE Today we will do a calculation in some detail of the microsupports of sheaves on R 2 which are constructible with respect to the stratification into the four open quadrants, the four half-axes, and the origin. 1. SHEAVES ON R 2, CONSTRUCTIBLE WITH RESPECT TO THE STRATIICATION BY THE AXES Recall from last time that the star of a stratum in a stratification is the union of all strata whose closures contain it, and we define the structure of a poset on the set of strata by taking σ τ when the star of σ contains the star of τ. In the present, case the structure is as follows, drawn superimposed on the stratification: This is poset; all diagrams commute. In the future we will not draw the red arrows. As we discussed last time, from any sheaf and any stratification, we get a representation of the stratification poset, by using the restriction maps of the sheaf. In this example, for a sheaf on R 2, we would get the following: 1

2 2 VIVEK SHENDE ], 0[ ]0, [ ], 0[ R ], 0[ ], 0[ R ]0, [ R R R ], 0[ ]0, [ ]0, [ ]0, [ R ]0, [ ], 0[ Conversely, consider any diagram of the above form, let us say NW N NE W C E SW S W We would like to define from this data a sheaf, constructible with respect to the above stratification. We more-or-less said how to do this last time: take a base β of the topology consisting of open sets minimally contained in each strata; define a β presheaf assigning these open sets to the values NE, N, NW, etc., and then finally sheafify. The resulting sheaf is locally constant with respect to the stratification, with stalk NE in the northeast quadrant, etc. 2. MICROSUPPORTS Our purpose today is to compute the microsupport of such sheaves An approximate definition. We ll use the following approximate definition of the microsupport. The microsupport away from zero of a sheaf on R n is the closure in T R n = R n R n \ 0 of the locus of points x, ξ 0 such that, for 0 < δ << ɛ << 1, the following map is not an isomorphism: H {z R n x z < ɛ and ξx z < δ} H {z R n x z < ɛ and ξx z < δ} This is only an approximation to the real definition because we have just taken linear functions instead of allowing any function with derivative ξ, and also have failed to impose conditions on points near x. Also we re presently only discussing sheaves rather than complexes of sheaves. But in any case, the definition above is good enough for our purposes today.

3 LECTURE 3: SHEAVES ON R 2, CONSTRUCTIBLE WITH RESPECT TO THE STRATIICATION BY THE AXES Cech warmup. The above definition involves computing cohomology groups, and a morphism between them. Let s first practice with the definition of Cech cohomology. Consider the stratification R =, 1 { 1} 1, 1 {1} 1, rom a sheaf on R, we obtain a diagram of the form: ], 1[ ], 1[ ] 1, 1[ ] 1, [ ]1, [ In case the sheaf is constructible with respect to the stratification, this data suffices to determine it; moreover the Cech complex can be identified with ], 1[ ] 1, [ ] 1, 1[ with the first term in degree zero, and the second in degree one. Example. Consider the sheaf Z [ 1,1], which is the constructible sheaf that gives rise to the diagram 0 Z Z Z 0 Its Cech cohomology is calculated by the complex Z Z Z That is, H 0 R, Z [ 1,1] = Z, and all other cohomology groups vanish. Example. Consider the sheaf Z ] 1,1[, which is the constructible sheaf that gives rise to the diagram 0 0 Z 0 0 Its Cech cohomology is calculated by the complex 0 0 Z That is, H 1 R, Z ] 1,1[ = Z, and all other cohomology groups vanish. Example. Consider more generally the constructible sheaf that gives rise to the diagram A B C D E Its Cech cohomology is calculated by the complex B D C

4 4 VIVEK SHENDE 2.3. Microsupport calculations. ix a sheaf constructible with respect to a stratification by the axes we encode its isomorphism type by a diagram NW N NE W C E SW S SE We wish to calculate its microsupport. This is, evidently from the definition, a conical locus in the cotangent bundle; also we have only defined the microsupport away from zero. Thus we can characterize the microsupport by its image in or restriction to the cosphere bundle S R 2, viewed as the real projectivization of or unit covectors in the cotangent bundle. At the cost of fixing a metric for drawing pictures, we always take the Euclidean metric we can indicate this locus graphically by drawing the tangent vectors identified with the desired cotangent vectors. Away from the axes, the sheaf is locally constant; there can be no nonzero microsupport. Let us look along the y-axis in the upper region: NW N NE We are interested in the cotangent vectors above a point along this axes; as mentioned above we represent them by the tangent vectors using the inner product. irst let us consider vectors x, y 0. To compute the microsupport along such a direction involves comparing cohomology over the following sorts of region:

5 LECTURE 3: SHEAVES ON R 2, CONSTRUCTIBLE WITH RESPECT TO THE STRATIICATION BY THE AXES 5 The cohomology over both those open sets is N, and the restriction canonically induces an isomorphism. On the other hand, consider the vector 1, 0. The corresponding microsupport calculation involves studying the following comparison: This time, the restriction map on cohomologies is N NW. Thus, there is microsupport in the 1, 0 direction if and only if the map N NW is not an isomorphism. Note this map is the one going the opposite way as the above indicated arrow. Similarly, to study the microsupport in the direction 1, 0 we consider the following comparison and determine that there s microsupport in the 1, 0 direction iff the map N NE is nontrivial. Let us turn our attention to the origin. Let us investigate whether there is microsupport in the 1, 1 direction. This means studying the following comparison:

6 6 VIVEK SHENDE The red region is covered by the restriction of the star of the center stratum, so its cohomology is just C. The stratification of the blue region has the same shape as Its cohomology is calculated analogously by the complex S W SW. Thus we see that there is no microsupport in the 1, 1 direction or similarly in the x, y direction for x, y > 0 if and only if the following sequence is exact: 0 C S W SW 0 Exercise. Think carefully about the signs in the above sequence.