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Equivariant Algebraic Geometry

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Equivariant Algebraic Geometry EQUIVARIANT ALGEBRAIC GEOMETRY TONY FENG BASED ON LECTURES OF RAVI VAKIL CONTENTS Disclaimer 2 1. Examples and motivation 3 Part 1. Some non-equivariant ingredients 6 2. Chow groups 6 3. K-theory 10 4. Intersection theory and Chern classes 13 5. The Riemann-Roch Theorem 17 Part 2. Some equivariant ingredients 31 6. Topological classifying spaces 31 7. Stacks 34 8. Algebraic stacks 43 9. Chow groups of quotient stacks 49 10. Equivariant K -theory 55 Part 3. Equivariant Riemann-Roch 58 11. Equivariant Riemann-Roch 58 12. The geometry of quotient stacks 63 13. Computing Euler characteristics 68 14. Riemann-Roch and Inertial Stacks 75 References 77 1 Equivariant Algebraic Geometry Math 245B DISCLAIMER This document originated from a set of lectures that I “live-TEXed” during a course offered by Ravi Vakil at Stanford University in the winter of 2015. The format of the course was somewhat unusual in that the first two weeks’ worth of lectures were presented by Dan Edidin, as an overview of his theorem on “Riemann-Roch theorem for stacks” via equivariant algebraic geometry. Some background lectures were given outside of class as well, by both Dan and Ravi. Afterwards, Ravi took over the lectures and fleshed out the argument and examples (with a couple of guest lectures by Arnav Tripathy and Dan Litt sprinkled in). While these unusual features of the course worked in the classroom, I felt upon look- ing back at my notes that I had failed to capture an account that would make any sense to an outside reader. Therefore, I have heavily modified the organization and content of the notes. The major addition has been of background material, which was mostly as- sumed (or black-boxed) during the course. My goal was to make everything accessible to a student familiar with algebraic geometry to the extent of a first year of scheme theory - in particular, the guiding principle has been to include background material that I wish I knew at the start of the class. Indeed, a major reason for my revamping of these notes was to solidify my comfort with that background. Aside from that, the organization and presentation have also been significantly re- vised. This inevitably means that there will be many mistakes and typos, which of course should be blamed on me. On the other hand, the mathematical insights should be at- tributed elsewhere - mostly probably to Ravi, but various snippets are drawn from other sources or personal notes. I also apologize for inadequacies in the references. 2 Equivariant Algebraic Geometry Math 245B 1. EXAMPLES AND MOTIVATION The official objective of these notes is to give a proof of a “Riemann-Roch theorem for stacks” via equivariant algebraic geometry, following [Edi13]. However, our real agenda is to use this result as an excuse to meet some of the main characters of modern algebraic geometry - such as Chow groups, intersection theory, and stacks - which lie beyond a first course in algebraic geometry, and to see how they can be marshalled to answer very concrete questions. The goal of this first section is to carefully study some simple examples, to see what such a Riemann-Roch theorem for stacks might say. In particular, we will examine the spaces P(a,b) =: Projk [x,y ] with x in degree a and y in degree b. As schemes, these 1 are of course all isomorphic to P , but we will already see that for these simple examples something more subtle is going on. 1.1. Setup. On P(a,b) we have an action of the group Gm (which we think of as just being ) as follows: acts on x,y sending C∗ Gm C[ ] a x λ− x 7! b y λ− y. 7! (There is no need to restrict our attention to C, but on the other hand we’ll see that there is not much to be gained in aiming for maximum generality.) We’ll write down some explicit line bundles, count the dimension of the space of sections, and compare what we get with what Riemann-Roch tells us. To construct a line bundle, we simply take a free module over C[x,y ] with generator T , and extend the action of to this bundle, say by by the action λ T λ`T . Let us C∗ = call this bundle ξ`. The question that we are interested in, for this and· the upcoming examples, is the following. ` Question. What is the dimension of the space of Gm -invariant sections of ξ on P(a,b)? 0 ` This dimension is denoted, as usual, by h (P(a,b),ξ ). Note that the Gm -invariant sections will be precisely the Gm -invariant elements of the module C[x,y,T ]. 1.2. Examples. Let’s consider first the simplest case: P(1,1). 0 0 When ` = 0, the only invariant sections are the constants, so we see that h (P(1,1),ξ ) = • 1. When ` = 1, the invariant sections are spanned by xT and y T , so we see that 0 1 • h (P(1,1),ξ ) = 2. 2 2 When ` = 2, the invariant sections are spanned by x T,x y T,y T (now you see • why we chose negative weights in defining the original action!). ` 0 ` It is now easy to see that ξ (`), so in particular h (P(1,1),ξ ) = ` + 1. • $ O 3 Equivariant Algebraic Geometry Math 245B 0 ` ` h (P(1,1),ξ ) 0 1 1 2 2 3 ` ` + 1 Let’s move on to the next simplest example: P(1,2). 0 0 When ` = 0, the only invariant sections are again the constants, so h (P(1,2),ξ ) = • 1. 0 1 When ` = 1, the invariant sections are spanned by xT this time, so h (P(1,2),ξ ) = • 1. 2 2 0 2 When ` = 2, the invariant sections are spanned by x T,y T , so h (P(1,2),ξ ) = 2. 3 0 3 • When ` = 3, the invariant sections are spanned by x T,x y T so h (P(1,2),ξ ) = 2. • In general, the dimension we seek is the number of non-negative integral solu- • tions to i + 2j = `, which is easily seen to be `=2 + 1. 0 ` 0 b c ` ` h (P(1,1),ξ ) h (P(1,2),ξ ) 0 1 1 1 2 1 2 3 2 3 4 2 4 5 3 5 6 3 We see that for P(1,2) the dimension of the space of sections grows about half as fast as it did for P(1,1). What’s going on? 0 ` Exercise 1.1. Compute the formula for h (P(a,b),ξ ) for some other values of a,b. Do you have a guess for the general formula? The Riemann-Roch theorem is precisely a statement about the dimension of the global sections of a line bundle. According to Riemann-Roch, 0 1 χ( ) = h ( ) h ( ) = d + 1 g L L − L 1− d χtop. = + 2 With d = `, this agrees well with the column describing the global sections of ξ` on 0 1 h (P (1,1)), as we even noted in that calculation. However, it seems to disagree with the calculation for P(1,2), which grows at about half the rate. In fact, we can easily compute that 0 ` ` 3 ` 1 h (P(1,2),ξ ) = + + ( 1) . (1.1) 2 4 − 4 In general, we predict something like ` h0 a,b ,ξ` something depending only on ` modulo ab . (P( ) ) = ab + ( ) 4 Equivariant Algebraic Geometry Math 245B Let’s see if we can at least heuristically figure out why this might be. What is the Euler 1 characteristic of P(1,2)? Since this is isomorphic to P the first guess would be that it should be 2. However, we can think about this another way. The point [s : t ] P(a,b) corresponds 2 2 to the Gm -orbit of (s,t ) in A = Spec k [x,y ]. Most of the orbits are acted on freely by m , but one is not: the point (0,1) has stabilizer 1,1 =2 because 1 acts on the G ∼= Z t coordinate by its square, which is 1. Therefore, thisf− orbitg is only “half”− as large as the others, and we might imagine it as being only “half” a point in P(a,b). If we count in this way, then P(1,2) only has Euler characteristic 3=2 (we traded a full point for “half a point”). Assuming for simplicity that gcd(a,b) = 1, the same reasoning “shows” that P(a,b) 1 1 should have Euler characteristic 1+ a + b . Plugging this refined Euler characteristic into Riemann-Roch, we guess that ` 1=a + 1=b h0 a,b ,ξ` . (P( ) ) = ab + 2 For a = b = 1, we recover `+1 on the right hand side as before. For a = 1,b = 2 we obtain ` 3 2 + 4 . This is obviously not correct, since we must obtain an integer, but it captures ` 1 everything in (1.1) except for the term ( 1) 4 . This is clearly a sign that something is interesting going on! We refine our prediction− to ` 1=a + 1=b h0 a,b ,ξ` (oscillation term). (P( ) ) = ab + 2 + Exercise 1.2. Work out more examples, and identify the oscillation term. What is this oscillation term? One might be tempted to guess at first that it has some- thing to do with a higher cohomology group, but this doesn’t seem consistent with the fact that the error term even oscillates in sign.
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