3

Question 1

Define the perverse t-structure and its heart

Answer 1

pg 142

Question 2

How does perverse t-structure interact with extensions? Proof?

Answer 2

Stable under extensions

pg 143

Question 3

Describe the perverse t-structure for local systems in terms of the natural t-structure

what is heart?

pf?

Answer 3

Heart is shifted local systems

pg 143

Question 4

Describe how perverse t-structure interacts with functors coming from open and closed embeddings.

Answer 4

3.1.4 - 3.1.6 pg 144

Question 5

Discuss how perverse sheaves on a closed subvariety of X sit inside the perverse sheaves on X

Answer 5

3.1.10

i_* induces an equiv of categories with cat of perverse sheaves supporteed on Z <– a Serre subcategory

pg 146

Question 6

How does Verdier duality functor interact with perverse t-structure?

Answer 6

If k a field, t-exact

pg 147

Question 7

How does RHom interact with perverse t-structure?

Answer 7

pg148

Question 8

How do derived tensor and hom interact with perverse t-structure on local systems?

Answer 8

right/left exact - exact if L locally free

pg 148

Question 9

Discuss how extension of scalars interacts with perverse t-structure

Answer 9

right exact, exact if k’ flat as a k module

pg 149

Question 10

How does external tensor product interact with perverse t-structure?

Answer 10

right exact, exact if k is a field

pg 149

Question 11

Define: intermediate extension functor, intersection cohomology complex, intersection cohomology

Answer 11

intermediate extension - pg 151

intersection cohohomology complex - specific type of intermediate extension - pg 154-155

Question 12

Discuss properties of the intermediate extension functor, alternate characterizations

Answer 12

Lemma 3.3.3 Fully faithful - unique perverse sheaf on X s.t.
-Supported on Y closure
-Restriction to Y is isomorphic to F
-It has no nonzero subobjects or quotient objects supported on Y closure \ Y

Lemma 3.3.4 pullback conditions

Lemma 3.3.5 takes injectives maps to injective maps, surjective maps to surjective maps

Not exact, image not a Serre subcategory

Question 13

Discuss maximal subobjects and quotients of a perverse sheaf supported on a closed subvariety. Related s.e.s?

Answer 13

Lemma 3.3.7, Lemma 3.3.8

Question 14

Discuss alternative descriptions of IC complexes

Answer 14

Lemma 3.3.11 - Support is closure of strata
- Restriction to strata is shifted local system

Lemma 3.3.12

pg 155

Question 15

How does Verdier duality functor interact with IC complex?

Answer 15

Lemma 3.3.13
pg155

Question 16

How does external tensor product interact with IC complex?

Answer 16

pg 156

Question 17

Prove: If X is a smooth, connected variety of dim n, then Loc(X, k)[n] is a Serre subcat of Perv(X,k)

Answer 17

pg 157

Question 18

Prove: Every perverse sheaf admits a finite filtration whose subquotients are IC complexes

Answer 18

Notice here we don’t say simple IC complexes - see Thm 3.4.5
pg 158

Question 19

Discuss steps in showing Perv(X,k) is Noetherian. What if k is a field?

Answer 19

If field, IC simple objects, artinian
pg 158-160

Question 20

Discuss characterization of perverse t-structure using affine open sets

Answer 20

Thm 3.5.3 F in <=0 same as derived global sections on any affine open in <=0 for natural t-structure - i.e. cohomology in degree > 0 vanishes

Thm 3.5.7 F in >= 0 same as derived global sections with compact support on any affine open in >= 0 for natural t-structure - i.e. cohomology with compact support vanishes in degrees < 0

Question 21

Discuss perverse t-exactness of functors for affine morphisms

Answer 21

f_* right t-exact, f_! left exact
In particular, if we are looking at a locally closed embedding that is also an affine morphism, then h_! and h_* are t-exact

pg 167 - 168

Question 22

For a smooth morphism f define the functors f upper and lower dagger. Discuss perverse counterparts. Prove exactness results and adjuntion. Discuss faithfulness

Answer 22

upper is t-exact
lower is left exact

168-169

If f is a smooth surjective morphism, upper dagger is faithful - fully faithful if f has connected fibers

pg 172

Question 23

Discuss smooth descent

Answer 23

Given f:X –> Y and a perverse sheaf on X, how can we tell if it comes from a perverse sheaf on Y? If it is, how can we recover the original sheaf on Y?

Descent data: thm 3.7.4 for smooth surjective morphisms, upper dagger is an equivalence of categories - Perv(Y,k) and Desc(f, k)

pg 173 - 180

Question 24

Define: semismall, small morphisms and their stratified counterparts

Discuss key theorems related to these morphisms

Answer 24

Thm. 3.8.4 If f:X –> Y is a proper, semismall morphism and X is smooth, connected, then the pushforward of any shifted local system of finite type is in Perv(Y,k)

Thm. 3.8.9 For a proper stratified semismall morphism, pushforwad is t-exact for perverse t-structure
pg 181-184

Question 25

What is the Decomposition theorem?

Answer 25

If f is a proper morphism of varieties, then pushforward perserves semisimple complexes

pg 185

Question 26

What is Relative hard Lefschetz theorem? Relationship with decomposition theorem?

Answer 26

pg 185 - 190