◆ PERVERSE T-STRUCTURE

ARTIN IC SHEAVES // PERVERSE EXACTNESS // VERDIER DUALITY // INTERSECTION COHOMOLOGY

I. T-STRUCTURE ON D^b_c

The perverse t-structure on D^b_c(X) is a non-standard t-structure whose heart is the abelian category Perv(X) of perverse sheaves. Unlike the standard t-structure (which truncates by degree), the perverse t-structure truncates by support dimension.

^pD^{<=0} = { F : dim supp H^{-k}(F) <= k for all k }
^pD^{>=0} = { F : dim supp H^{-k}(D_X F) <= k for all k }

The key insight: the conditions involve BOTH the sheaf AND its Verdier dual D_X F. This self-duality is what makes perverse sheaves so rigid and powerful.

II. IC COMPLEXES

The intersection cohomology complex IC(S, L) is the unique extension of a local system L on a stratum S that satisfies the perverse support and cosupport conditions. It is both the minimal and the maximal such extension — the "Goldilocks" object.

IC(S, L) = im( j_{!*}L[dim S] )

Goresky-MacPherson defined intersection homology to repair Poincare duality for singular spaces. IC complexes are the sheaf-theoretic incarnation. They are simple objects in Perv(X) — the "atoms" of the perverse world.

III. VERDIER DUALITY

Verdier duality D_X is the derived category analog of Poincare duality. For smooth X of dimension n, D_X(F) = RHom(F, omega_X[2n]) where omega_X is the dualizing sheaf. The miraculous fact: IC complexes are self-dual under Verdier duality.

D_X( IC(S, L) ) ~ IC(S, L^*)

This self-duality of IC complexes implies Hard Lefschetz and the decomposition theorem — the most powerful structural results in algebraic geometry. Perverse sheaves on the flag variety encode the entire representation theory of the Lie algebra (Beilinson-Bernstein).

IV. DECOMPOSITION THEOREM

The BBD decomposition theorem states: for a proper map f: X -> Y, the direct image Rf_*(IC_X) decomposes as a direct sum of shifted IC complexes on strata of Y.

Rf_*(IC_X) ~ bigoplus_{S,L,n} IC(S, L)[-n]

No mixing. No extensions. Pure semisimplicity. This is the ultimate structural result — the pushforward of the simplest perverse sheaf remains semisimple in the perverse category.

V. PERVERSE EXACTNESS

Perverse sheaves form an abelian category with remarkable exactness properties. Every short exact sequence in Perv(X) gives a long exact sequence in perverse cohomology. The perverse cohomology functors ^pH^k are the cohomological functors of the perverse t-structure.

0 -> F -> G -> H -> 0 in Perv(X)
... -> ^pH^k(F) -> ^pH^k(G) -> ^pH^k(H) -> ...
C443 // PERVERSE T-STRUCTURE // KASHIWARA-SCHAPIRA D-MODULE FRONTIER