Differential forms in algebraic geometry —a new perspective in the singular case

Albert-Ludwigs-Universität Freiburg

Annette HuberMathematisches InstitutAlbert-Ludwigs-Universität FreiburgJune 2015

Today

1 Review of differential forms2 Application: discrete invariants3 Problems and solutions in the singular case

June 2015 Annette Huber – Differential forms in algebraic geometry 2 / 1

Differential forms

June 2015 Annette Huber – Differential forms in algebraic geometry 3 / 1

Back to calculus

derivative

f : (a,b)→ R ⇒ ∂ f∂x

: (a,b)→ R

problem: depends on choice of coordinate!Better point of view: differential form

f 7→ df =∂ f∂x

dx

change of coordinate f (x) = g(y(x))

∂ f∂x

dx =∂g∂y

∂y∂x

dx =∂g∂y

dy

June 2015 Annette Huber – Differential forms in algebraic geometry 4 / 1

Back to calculus

M smooth manifold of dimension nlocal coordinates x1, . . . ,xn near P ∈M

Definition

f : M → R ⇒ df =n

∑i=1

∂ f∂xi

dxi

Geometric interpretation:sections of the cotangent bundle T ∗M.

June 2015 Annette Huber – Differential forms in algebraic geometry 5 / 1

Higher derivatives

Definition

ΩqM =

q∧Ω1

M

Differential graded algebraproduct (ω,ω ′) 7→ ω ∧ω ′

ω ∧ω ′ = (−1)degω degω ′ω ′∧ω

differential d : ΩqM → Ωq+1

Md d = 0ω ∧ω ′ 7→ dω ∧ω ′+ (−1)degωω ∧dω ′

in coordinates f dxj1 ∧·· ·∧dxjq 7→ ∑i

∂ f∂xi

dxi ∧dxj1 ∧·· ·∧dxjq

June 2015 Annette Huber – Differential forms in algebraic geometry 6 / 1

Algebraic differential forms

ObservationDerivatives of polynomials are polynomials

Consequence: Differential forms make sense inalgebraic geometry

June 2015 Annette Huber – Differential forms in algebraic geometry 7 / 1

Algebraic varieties

k algebraically closed field, e.g. complex numbers C

zero sets of polynomial equations in An = kn

Definition1 An affine variety is given as

V (f1, . . . , fm) =

x ∈ kn|fi(x) = 0 for all i

for choice of f1, . . . , fm ∈ k [X1, . . . ,Xn]

2 non-singular if(

∂ fi∂xj

)i ,j

has maximal rank,

i.e., submanifold3 General variety: locally affine, algebraic transition

maps.June 2015 Annette Huber – Differential forms in algebraic geometry 8 / 1

Differential forms for alg. varieties

V = V (f1, . . . , fm) with f1, . . . , fm ∈ k [X1, . . . ,Xn]k [V ] = k [X1, . . . ,Xn]/〈f1, . . . , fm〉

Definitionalgebraic differential forms on V

generators dX1,dX2, . . . ,dXn

relations df1, . . . ,dfmΩ1

V = 〈dX1, . . .dXn〉k [V ]/〈df1, . . . ,dfm〉Higher degree: Ωq

V =∧q Ω1

V

June 2015 Annette Huber – Differential forms in algebraic geometry 9 / 1

Differential forms for alg. varieties

Example: (affine plane) A2 = k2, coordinates X ,Y

Ω1A2 = 〈dX ,dY 〉k [X ,Y ]

Example: (hyperbola) G = V (XY −1)∼= k r0coordinates X ,Y = X−1

d(XY −1) = YdX + XdYΩ1

G = 〈dX ,dY 〉k [X ,X−1]/〈YdX + XdY 〉dY =−X−1YdX =−X−2dX in Ω1

G

Ω1G = k [X ,X−1]dX

June 2015 Annette Huber – Differential forms in algebraic geometry 10 / 1

Use of differential forms

Classification of algebraic varieties!

discrete invariantsdeformation theoryperiod maps. . .

June 2015 Annette Huber – Differential forms in algebraic geometry 11 / 1

Application: source of discrete invariants

June 2015 Annette Huber – Differential forms in algebraic geometry 12 / 1

1. invariant: genus

Definitiongenus of non-singular projective curve C

g = dimk Ω1C(C)

Example: k = C, C = V (Y 4−X 3−X −1), g = 3

June 2015 Annette Huber – Differential forms in algebraic geometry 13 / 1

2. invariant: de Rham cohomology

X non-singular affine variety, characteristic 0de Rham complex:

Ω0(X )d0−→ Ω1(X )

d1−→ Ω2(X )

d2−→ . . . , d i d i−1 = 0

Definitionalgebraic de Rham cohomology

H idR(X ) = Ker(d i)/Im(d i−1)

Example: X = C projective curve

dimk H1dR(X ) = 2g

June 2015 Annette Huber – Differential forms in algebraic geometry 14 / 1

2. invariant: de Rham cohomology

Theoremk = C, X non-singular variety over C.

H idR(X )∼= H i

sing(X an,C)

Completely algebraic way of defining these invariantsstory continues with Hodge theory (Deligne)k = Q: can be used to define period numbers(→ number theory, mathematical physics)

June 2015 Annette Huber – Differential forms in algebraic geometry 15 / 1

2. invariant: de Rham cohomology

Example: k = C, hyperbola G = V (XY −1)

Ω∗G =[C[X ,X−1]

d−→ C[X ,X−1]dX]

H0dR(X ) = Kerd = C

H1dR(X ) = C[X ,X−1dX ]/Imd = C

dXX

G = Cr0 homotopy equivalent to S1

June 2015 Annette Huber – Differential forms in algebraic geometry 16 / 1

3. invariant: Kodaira dimension

input: ω = Ωdx for X non-singular variety of dimension

doutput: Kd(X ) ∈ −∞,0, . . . ,d

Example: C non-singular projective curveKd(X ) =−∞⇔ g = 0 (parabolic)Kd(X ) = 0 ⇔ g = 1 (elliptic/flat)Kd(X ) = 1 ⇔ g ≥ 2 (hyperbolic)

For algebraic geometers: s0, . . . ,sN basis of ω⊗n(X )πn : X −→ PN ,

x 7→ [s0(x) : · · · : sN(x)]Definition: Kd(X ) = maxn dimπn(X )

June 2015 Annette Huber – Differential forms in algebraic geometry 17 / 1

Problems and solutions in the singular case

June 2015 Annette Huber – Differential forms in algebraic geometry 18 / 1

Problems in the singular case

Same definitions possible, give wrong answerΩ1

V no longer a vector bundle

RemarkThis is not surprise. A variety is defined to be non-singularif Ω1

V is a vector bundle.

June 2015 Annette Huber – Differential forms in algebraic geometry 19 / 1

Problems in the singular case

Example:

V = V (XY )⊂ k2

ω = XdY =−YdX 6= 0vanishes on complement of (0,0)

dX ∧dY 6= 0

June 2015 Annette Huber – Differential forms in algebraic geometry 20 / 1

Ad hoc replacements in the singular case

Different replacements in useTorsion free differentials: Ωq

X/torsion

Reflexive differentials: OX -double dual of ΩqX

Dubois complex inspired by Hodge theory. . .

All are useful for certain applications,e.g., classification of algebraic varieties

June 2015 Annette Huber – Differential forms in algebraic geometry 21 / 1

New perspective in the singular case

—, C. Jörder: Differential forms in the h-topology.Algebr. Geom. 1 (2014), no. 4, 449–478.

—, S. Kebekus, S. Kelly: Differential forms in positivecharacteristic avoiding resolution of singularities,

preprint 2014, arXiv:1407.5786

Key idea: Change the topology!

June 2015 Annette Huber – Differential forms in algebraic geometry 22 / 1

Grothendieck topologies

Algebraic varieties are topological spaces.Very few open sets, e.g., all open sets dense in An.Grothendieck ca. 1960: generalize the notion oftopologyuse broader class of morphisms V → X in place ofopen subsets

Example: X topological manifold, allow all localhomeomorphisms V → X .

Extremely successful in algebraic geometry!

June 2015 Annette Huber – Differential forms in algebraic geometry 23 / 1

h-topology (Voevodsky 1996)

generated byopen subsetsX → X proper surjective (think: preimages ofcompact sets are compact)

Theorm (Hironaka 1964)

X variety of characteristic 0. Then there is X → X propersurjective with X non-singular.

Consequence: locally in the h-topology, every variety isnon-singular!

June 2015 Annette Huber – Differential forms in algebraic geometry 24 / 1

h-differentials

h-locally given by algebraic differentials

Definition

Let Ωqh be the sheafification of X 7→Ωq(X ) in the h-topology

more concretely:

Ωqh(X ) = Ker

(Ωq(X0)

p∗1−p∗2−−−−→ Ωq(X1)

)

X0→ X h-cover with X0 non-singular,X1→ X0×X X0 h-cover with X1 non-singular

June 2015 Annette Huber – Differential forms in algebraic geometry 25 / 1

Results (characteristic 0) with C. Jörder

nothing changes in the non-singular case,including cohomologyif X has klt-singularities (mild singularities of minimalmodel program):h-differentitals equal reflexive differentialsalways related to complex of Dubois differentialsdefine algebraic de Rham cohomology

Why I like themUnifies ad hoc definitions, simplifies proofs,very natural

June 2015 Annette Huber – Differential forms in algebraic geometry 26 / 1

Results (characteristic p)with S. Kelly, S. Kebekus

New problem: Frobenius x 7→ xp has

dF (x) = dxp = pxp−1dx = 0.

Consequence: Ωqh = 0

Instead: cdh-topology or eh-topology

differentials do not change in the non-singular casebad news: torsion exists, not functorialcohomology: work in progress,e.g. well-definedness of rational singularitiesok under resolution of singularities

June 2015 Annette Huber – Differential forms in algebraic geometry 27 / 1

Earlier work

1 Geißer 20062 Lee 20093 Cortiñas, Haesemeyer, Schlichting, Walker and

Weibel 2008-20134 Beilinson 2012

all concentrate on de Rham cohomologyhomotopy invariant case→ motives

Sales pitchh-topology is very useful for individual Ωq

June 2015 Annette Huber – Differential forms in algebraic geometry 28 / 1

Conclusion

differential forms also work in algebraic geometryvery useful source of invariants in the non-singularcaseh-differentials replace ad hoc definitions in thesingular casework in progress in positive characteristic

Thank you

June 2015 Annette Huber – Differential forms in algebraic geometry 29 / 1