Mixed_Hodge_structures Hodge_structure

1 mixed hodge structures

1.1 example of curves
1.2 definition of mixed hodge structure
1.3 mixed hodge structure in cohomology (deligne s theorem)

mixed hodge structures

it noticed jean-pierre serre in 1960s based on weil conjectures singular (possibly reducible) , non-complete algebraic varieties should admit virtual betti numbers . more precisely, 1 should able assign algebraic variety x polynomial px(t), called virtual poincaré polynomial, properties

if x nonsingular , projective (or complete)









p

x


(
t
)
=
∑

rank

(

h

n


(
x
)
)

t

n




{\displaystyle p_{x}(t)=\sum {\text{rank}}(h^{n}(x))t^{n}}






if y closed algebraic subset of x , u = x \ y









p

x


(
t
)
=

p

y


(
t
)
+

p

u


(
t
)


{\displaystyle p_{x}(t)=p_{y}(t)+p_{u}(t)}

the existence of such polynomials follow existence of analogue of hodge structure in cohomologies of general (singular , non-complete) algebraic variety. novel feature nth cohomology of general variety looks if contained pieces of different weights. led alexander grothendieck conjectural theory of motives , motivated search extension of hodge theory, culminated in work of pierre deligne. introduced notion of mixed hodge structure, developed techniques working them, gave construction (based on hironaka s resolution of singularities) , related them weights on l-adic cohomology, proving last part of weil conjectures.

example of curves

to motivate definition, let consider case of reducible complex algebraic curve x consisting of 2 nonsingular components x1 , x2, transversally intersect @ points q1 , q2. further, assume components not compact, can compactified adding points p1, ..., pn. first cohomology group of curve x (with compact support) dual first homology group, easier visualize. there 3 types of one-cycles in group. first, there elements αi representing small loops around punctures pi. there elements βj coming first homology of compactification of 1 of components. lifting of one-cycle in xk, k = 1, 2, cycle in x not canonical: these elements determined modulo span of αi. finally, modulo first 2 types, group generated combinatorial cycle γ goes q1 q2 along path in 1 component x1 , comes along path in other component x2. suggests h(x) admits increasing filtration

0
⊂

w

0


⊂

w

1


⊂

w

2


=

h

1


(
x
)
,



{\displaystyle 0\subset w_{0}\subset w_{1}\subset w_{2}=h^{1}(x),\,}

whose successive quotients wn/wn−1 originate cohomology of smooth complete varieties, hence admit (pure) hodge structures, albeit of different weights. further examples can found in naive guide mixed hodge theory .

definition of mixed hodge structure

a mixed hodge structure on abelian group hz consists of finite decreasing filtration f on complex vector space h (the complexification of hz), called hodge filtration , finite increasing filtration wi on rational vector space hq = hz ⊗z q (obtained extending scalars rational numbers), called weight filtration, subject requirement nth associated graded quotient of hq respect weight filtration, filtration induced f on complexification, pure hodge structure of weight n, integer n. here induced filtration on

gr

n


w


⁡
h
=

w

n


⊗

c


/


w

n
−
1


⊗

c



{\displaystyle \operatorname {gr} _{n}^{w}h=w_{n}\otimes \mathbf {c} /w_{n-1}\otimes \mathbf {c} }

is defined by

f

p



gr

n


w


⁡
h
=
(

f

p


∩

w

n


⊗

c

+

w

n
−
1


⊗

c

)

/


w

n
−
1


⊗

c

.


{\displaystyle f^{p}\operatorname {gr} _{n}^{w}h=(f^{p}\cap w_{n}\otimes \mathbf {c} +w_{n-1}\otimes \mathbf {c} )/w_{n-1}\otimes \mathbf {c} .}

one can define notion of morphism of mixed hodge structures, has compatible filtrations f , w , prove following theorem.

mixed hodge structures form abelian category. kernels , cokernels in category coincide usual kernels , cokernels in category of vector spaces, induced filtrations.

the total cohomology of compact kähler manifold has mixed hodge structure, nth space of weight filtration wn direct sum of cohomology groups (with rational coefficients) of degree less or equal n. therefore, 1 can think of classical hodge theory in compact, complex case providing double grading on complex cohomology group, defines increasing fitration f , decreasing filtration wn compatible in way. in general, total cohomology space still has these 2 filtrations, no longer come direct sum decomposition. in relation third definition of pure hodge structure, 1 can mixed hodge structure cannot described using action of group c*. important insight of deligne in mixed case there more complicated noncommutative proalgebraic group can used same effect using tannakian formalism.

moreover, category of (mixed) hodge structures admits notion of tensor product, corresponding product of varieties, related concepts of inner hom , dual object, making tannakian category. tannaka–krein philosophy, category equivalent category of finite-dimensional representations of group, deligne, milne , et el. has explicitly described, see deligne (1982) , deligne (1994). description of group recast in more geometrical terms kapranov (2012). corresponding (much more involved) analysis rational pure polarizable hodge structures done patrikis (2016).

mixed hodge structure in cohomology (deligne s theorem)

deligne has proved nth cohomology group of arbitrary algebraic variety has canonical mixed hodge structure. structure functorial, , compatible products of varieties (künneth isomorphism) , product in cohomology. complete nonsingular variety x structure pure of weight n, , hodge filtration can defined through hypercohomology of truncated de rham complex.

the proof consists of 2 parts, taking care of noncompactness , singularities. both parts use resolution of singularities (due hironaka) in essential way. in singular case, varieties replaced simplicial schemes, leading more complicated homological algebra, , technical notion of hodge structure on complexes (as opposed cohomology) used.

using theory of motives, possible refine weight filtration on cohomology rational coefficients 1 integral coefficients.