» MacPherson R. - Intersection homology and perverse sheaves
MacPherson R. - Intersection homology and perverse sheaves
||Intersection homology and perverse sheaves
||This report is about intersection homology and perverse sheaves. Intersex-tion homology is the subject of Chapter 1, which may be read independent!] of the rest. This introduction will concern only perverse sheaves, which an perhaps less well known at present.
The first tbing to know about perverse sheaves is that they are neithei sheaves nor are they perverse '. They have in common with sheaves the Tacl that you can take the cohomology of them, the fact that they form an abeliai category, and the fact that to constrnet one, it is enough to construct it locallj everywhere. The adjective "perverse" is a reference to certain vanishing conditions that their cohomology satisfies, which are unfamiliar looking to someone who is used to ordinary cohomology *.
The second thing to know is that Perverse sheaves are one of the mosi natural and fundamental objects in topology. Their naturality may be seer through their beautiful formal properties. Their fundamental importance ii clear from the list of problems in diverse areas of mathematics that have beer reformulated in terms of perverse sheaves and solved using them.
Although perverse sheaves are geometric objects, it has been difficult foi geometrically minded mathematicians to absorb the theory. There are twe reasons for this. The first reason is a special case of a very general problem of mathematical exposition: Geometry tends to be explained in a way that it algebraically natural, rather than geometrically natural, since algebra is closet to language than geometry is. The second reason is of a more technical nature: Perverse sheaves are defined in terms of the derived category of the categorj of ordinary sheaves. However, perverse sheaves are much simpler and more natural ordinary sheaves, let alone their derived category. We know this frorr their formal properties. Therefore, if we believe in the essential simplicity ol mathematics, the definition starting from ordinary sheaves must not be th< most elegant one.
In these notes, we give a definition of perverse sheaves which relies on Mors» theory as its fundamental tool, rather than on ordinary sheaves or derived categories. Since Morse theory is very geometric in nature, the hope is that i
1 "Lee faisceaux pervers n'etant ni des Ыкет, ni pervers, U taminologie requiert am explication." [BBD] p. 10.
'An «ample of such a vanishing condition ia the (oOoving: Let V bt a perversa sheaf« a complex vector space C". Suppose that the rapport of V la a complex subvariety V о complex dimension ia k, (so It» real dimension ia Ik). Then both the cohomology of V am the compact support cohomology of V are tera outside ranges of degrees of length It, juatssi V had real dimension k. For ordinary constructible sheaves, this is true for the cohomology but not for the cohomology with compact supports. See section ХЛЛ
Actually, the original meaning of perverse was a non-transveraality property of the chaini for intemection homology [GM1], as explained in section 1.1 . This turns out to imply thi vanishing conditions just referred to.