Characteristic Classes in Intersection Theory

Home , Intersection theory

Characteristic classes in Intersection theory

Paolo Aluﬃ

Florida State University

MathemAmplitudes 2019 Intersection Theory & Feynman Integrals. Padova, December 18-20, 2019

1 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory Characteristic classes in Intersection theory

1 Intersection theory in algebraic geometry

2 Characteristic classes of singular/noncompact algebraic varieties

3 cSM classes of graph hypersurfaces

2 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory Intersection theory in algebraic geometry is a deep, well-established ﬁeld, with many facets.

Obvious task: Establish a direct relation with ‘Intersection theory’ as meant in this workshop.

Characteristic classes in Intersection theory

The main message of this talk:

3 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory Obvious task: Establish a direct relation with ‘Intersection theory’ as meant in this workshop.

Characteristic classes in Intersection theory

The main message of this talk: Intersection theory in algebraic geometry is a deep, well-established ﬁeld, with many facets.

3 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory Characteristic classes in Intersection theory

The main message of this talk: Intersection theory in algebraic geometry is a deep, well-established ﬁeld, with many facets.

Obvious task: Establish a direct relation with ‘Intersection theory’ as meant in this workshop.

3 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory Ancient questions, such as Pappus hexagon theorem or the problem of Apollonius (how many circles are tangent to 3 given circles?) (∼2000 years ago); Fast forward: “B´ezout’stheorem” (∼1750) aka intersection theory in projective space, intersection numbers, intersection multiplicities; Fast forward: “Schubert calculus”, (∼1880) aka intersection theory in Grassmannians and beyond, enumerative geometry

Characteristic classes in Intersection theory Intersection theory in algebraic geometry

Intersection theory in algebraic geometry

History: (highly subjective, and a small fraction of the real thing)

4 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory Fast forward: “B´ezout’stheorem” (∼1750) aka intersection theory in projective space, intersection numbers, intersection multiplicities; Fast forward: “Schubert calculus”, (∼1880) aka intersection theory in Grassmannians and beyond, enumerative geometry

Characteristic classes in Intersection theory Intersection theory in algebraic geometry

Intersection theory in algebraic geometry

4 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory Fast forward: “Schubert calculus”, (∼1880) aka intersection theory in Grassmannians and beyond, enumerative geometry

Characteristic classes in Intersection theory Intersection theory in algebraic geometry

Intersection theory in algebraic geometry

History: (highly subjective, and a small fraction of the real thing) Ancient questions, such as Pappus hexagon theorem or the problem of Apollonius (how many circles are tangent to 3 given circles?) (∼2000 years ago); Fast forward: “B´ezout’stheorem” (∼1750) aka intersection theory in projective space, intersection numbers, intersection multiplicities;

4 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory Characteristic classes in Intersection theory Intersection theory in algebraic geometry

Intersection theory in algebraic geometry

4 / 26 Paolo Aluffi Characteristic classes in Intersection theory Maillard (1872): ,. Hilbert’s 15th problem (1900): make such computations rigorous. Many attempts in the XX century, with various degrees of success. A few names: Weil (’40s) for intersection multiplicities; Samuel; Severi; Segre; Chow (’50s) ‘moving lemma’; Serre; Grothendieck (’50s-’60s) Riemann-Roch; etc. Practically a who’s who in algebraic geometry. Modern rigorous, effective theory: Fulton-MacPherson (’70s-’80s); in the language of schemes. In particular: every algebraic scheme has a Chow group. (Think: homology.) The Chow group of a nonsingular algebraic varieties has a well-defined intersection product, making it into a ring. (Think: cohomology.)

Characteristic classes in Intersection theory Intersection theory in algebraic geometry

Typical question in enumerative geometry: how many smooth plane cubic curves are tangent to nine lines in general position?

5 / 26 Paolo Aluffi Characteristic classes in Intersection theory Hilbert’s 15th problem (1900): make such computations rigorous. Many attempts in the XX century, with various degrees of success. A few names: Weil (’40s) for intersection multiplicities; Samuel; Severi; Segre; Chow (’50s) ‘moving lemma’; Serre; Grothendieck (’50s-’60s) Riemann-Roch; etc. Practically a who’s who in algebraic geometry. Modern rigorous, effective theory: Fulton-MacPherson (’70s-’80s); in the language of schemes. In particular: every algebraic scheme has a Chow group. (Think: homology.) The Chow group of a nonsingular algebraic varieties has a well-defined intersection product, making it into a ring. (Think: cohomology.)

Characteristic classes in Intersection theory Intersection theory in algebraic geometry

Typical question in enumerative geometry: how many smooth plane cubic curves are tangent to nine lines in general position? Maillard (1872): ,.

5 / 26 Paolo Aluffi Characteristic classes in Intersection theory Many attempts in the XX century, with various degrees of success. A few names: Weil (’40s) for intersection multiplicities; Samuel; Severi; Segre; Chow (’50s) ‘moving lemma’; Serre; Grothendieck (’50s-’60s) Riemann-Roch; etc. Practically a who’s who in algebraic geometry. Modern rigorous, effective theory: Fulton-MacPherson (’70s-’80s); in the language of schemes. In particular: every algebraic scheme has a Chow group. (Think: homology.) The Chow group of a nonsingular algebraic varieties has a well-defined intersection product, making it into a ring. (Think: cohomology.)

Characteristic classes in Intersection theory Intersection theory in algebraic geometry

5 / 26 Paolo Aluffi Characteristic classes in Intersection theory Weil (’40s) for intersection multiplicities; Samuel; Severi; Segre; Chow (’50s) ‘moving lemma’; Serre; Grothendieck (’50s-’60s) Riemann-Roch; etc. Practically a who’s who in algebraic geometry. Modern rigorous, effective theory: Fulton-MacPherson (’70s-’80s); in the language of schemes. In particular: every algebraic scheme has a Chow group. (Think: homology.) The Chow group of a nonsingular algebraic varieties has a well-defined intersection product, making it into a ring. (Think: cohomology.)

Characteristic classes in Intersection theory Intersection theory in algebraic geometry

5 / 26 Paolo Aluffi Characteristic classes in Intersection theory Practically a who’s who in algebraic geometry. Modern rigorous, effective theory: Fulton-MacPherson (’70s-’80s); in the language of schemes. In particular: every algebraic scheme has a Chow group. (Think: homology.) The Chow group of a nonsingular algebraic varieties has a well-defined intersection product, making it into a ring. (Think: cohomology.)

Characteristic classes in Intersection theory Intersection theory in algebraic geometry

5 / 26 Paolo Aluffi Characteristic classes in Intersection theory Modern rigorous, effective theory: Fulton-MacPherson (’70s-’80s); in the language of schemes. In particular: every algebraic scheme has a Chow group. (Think: homology.) The Chow group of a nonsingular algebraic varieties has a well-defined intersection product, making it into a ring. (Think: cohomology.)

Characteristic classes in Intersection theory Intersection theory in algebraic geometry

5 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory In particular: every algebraic scheme has a Chow group. (Think: homology.) The Chow group of a nonsingular algebraic varieties has a well-deﬁned intersection product, making it into a ring. (Think: cohomology.)

Characteristic classes in Intersection theory Intersection theory in algebraic geometry

Typical question in enumerative geometry: how many smooth plane cubic curves are tangent to nine lines in general position? Maillard (1872): ,. Hilbert’s 15th problem (1900): make such computations rigorous. Many attempts in the XX century, with various degrees of success. A few names: Weil (’40s) for intersection multiplicities; Samuel; Severi; Segre; Chow (’50s) ‘moving lemma’; Serre; Grothendieck (’50s-’60s) Riemann-Roch; etc. Practically a who’s who in algebraic geometry. Modern rigorous, eﬀective theory: Fulton-MacPherson (’70s-’80s); in the language of schemes.

5 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory The Chow group of a nonsingular algebraic varieties has a well-deﬁned intersection product, making it into a ring. (Think: cohomology.)

Characteristic classes in Intersection theory Intersection theory in algebraic geometry

5 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory Characteristic classes in Intersection theory Intersection theory in algebraic geometry

5 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory In a nutshell: Assume X , Y are cycles in V , and X ⊆ V is regularly embedded; so we have a normal bundle NX V .

Deform the embedding X ⊆ V to the zero-section X ⊆ NX V ; and Y ⊆ V to CX ∩Y Y ⊆ NX V . (But don’t ‘move’ anything!) Reduced to intersecting cones with the zero-section in a vector bundle → suitable cocktail of Chern & Segre classes. All expected properties of intersection theory (and more) can be proven from this deﬁnition.

Characteristic classes in Intersection theory Intersection theory in algebraic geometry

Remarks: Diﬀerential forms are not the main tool here. Chow’s ‘moving lemma’ is also not an ingredient. The Chow group is in general much ﬁner than homology.

6 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory Deform the embedding X ⊆ V to the zero-section X ⊆ NX V ; and Y ⊆ V to CX ∩Y Y ⊆ NX V . (But don’t ‘move’ anything!) Reduced to intersecting cones with the zero-section in a vector bundle → suitable cocktail of Chern & Segre classes. All expected properties of intersection theory (and more) can be proven from this deﬁnition.

Characteristic classes in Intersection theory Intersection theory in algebraic geometry

Remarks: Diﬀerential forms are not the main tool here. Chow’s ‘moving lemma’ is also not an ingredient. The Chow group is in general much ﬁner than homology.

In a nutshell: Assume X , Y are cycles in V , and X ⊆ V is regularly embedded; so we have a normal bundle NX V .

6 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory (But don’t ‘move’ anything!) Reduced to intersecting cones with the zero-section in a vector bundle → suitable cocktail of Chern & Segre classes. All expected properties of intersection theory (and more) can be proven from this deﬁnition.

Characteristic classes in Intersection theory Intersection theory in algebraic geometry

Remarks: Diﬀerential forms are not the main tool here. Chow’s ‘moving lemma’ is also not an ingredient. The Chow group is in general much ﬁner than homology.

In a nutshell: Assume X , Y are cycles in V , and X ⊆ V is regularly embedded; so we have a normal bundle NX V .

Deform the embedding X ⊆ V to the zero-section X ⊆ NX V ; and Y ⊆ V to CX ∩Y Y ⊆ NX V .

6 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory → suitable cocktail of Chern & Segre classes. All expected properties of intersection theory (and more) can be proven from this deﬁnition.

Characteristic classes in Intersection theory Intersection theory in algebraic geometry

Remarks: Diﬀerential forms are not the main tool here. Chow’s ‘moving lemma’ is also not an ingredient. The Chow group is in general much ﬁner than homology.

In a nutshell: Assume X , Y are cycles in V , and X ⊆ V is regularly embedded; so we have a normal bundle NX V .

Deform the embedding X ⊆ V to the zero-section X ⊆ NX V ; and Y ⊆ V to CX ∩Y Y ⊆ NX V . (But don’t ‘move’ anything!) Reduced to intersecting cones with the zero-section in a vector bundle

6 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory All expected properties of intersection theory (and more) can be proven from this deﬁnition.

Characteristic classes in Intersection theory Intersection theory in algebraic geometry

Remarks: Diﬀerential forms are not the main tool here. Chow’s ‘moving lemma’ is also not an ingredient. The Chow group is in general much ﬁner than homology.

In a nutshell: Assume X , Y are cycles in V , and X ⊆ V is regularly embedded; so we have a normal bundle NX V .

6 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory Characteristic classes in Intersection theory Intersection theory in algebraic geometry

Remarks: Diﬀerential forms are not the main tool here. Chow’s ‘moving lemma’ is also not an ingredient. The Chow group is in general much ﬁner than homology.

In a nutshell: Assume X , Y are cycles in V , and X ⊆ V is regularly embedded; so we have a normal bundle NX V .

6 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory 1990s: Intersection theory on moduli spaces of curves.

Faber’s conjectures on the tautological ring of Mg,n. (Still partly open!) (and very subtle! Tommasi,...) Witten’s conjectures (Kontsevich’s theorem) on intersection numbers on the moduli space. Keel (’92): Intersection theory of M0,n, TAMS. Manin (’94): Generating functions in algebraic geometry. . . : e.g., Betti numbers for M0,n. ···

Characteristic classes in Intersection theory Intersection theory in algebraic geometry

Example: Rigorous computation of the , cubics (—, Kleiman-Speiser, 1986).

7 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory Faber’s conjectures on the tautological ring of Mg,n. (Still partly open!) (and very subtle! Tommasi,...) Witten’s conjectures (Kontsevich’s theorem) on intersection numbers on the moduli space. Keel (’92): Intersection theory of M0,n, TAMS. Manin (’94): Generating functions in algebraic geometry. . . : e.g., Betti numbers for M0,n. ···

Characteristic classes in Intersection theory Intersection theory in algebraic geometry

Example: Rigorous computation of the , cubics (—, Kleiman-Speiser, 1986).

1990s: Intersection theory on moduli spaces of curves.

7 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory (and very subtle! Tommasi,...) Witten’s conjectures (Kontsevich’s theorem) on intersection numbers on the moduli space. Keel (’92): Intersection theory of M0,n, TAMS. Manin (’94): Generating functions in algebraic geometry. . . : e.g., Betti numbers for M0,n. ···

Characteristic classes in Intersection theory Intersection theory in algebraic geometry

Example: Rigorous computation of the , cubics (—, Kleiman-Speiser, 1986).

1990s: Intersection theory on moduli spaces of curves.

Faber’s conjectures on the tautological ring of Mg,n. (Still partly open!)

7 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory Witten’s conjectures (Kontsevich’s theorem) on intersection numbers on the moduli space. Keel (’92): Intersection theory of M0,n, TAMS. Manin (’94): Generating functions in algebraic geometry. . . : e.g., Betti numbers for M0,n. ···

Characteristic classes in Intersection theory Intersection theory in algebraic geometry

Example: Rigorous computation of the , cubics (—, Kleiman-Speiser, 1986).

1990s: Intersection theory on moduli spaces of curves.

Faber’s conjectures on the tautological ring of Mg,n. (Still partly open!) (and very subtle! Tommasi,...)

7 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory Keel (’92): Intersection theory of M0,n, TAMS. Manin (’94): Generating functions in algebraic geometry. . . : e.g., Betti numbers for M0,n. ···

Characteristic classes in Intersection theory Intersection theory in algebraic geometry

Example: Rigorous computation of the , cubics (—, Kleiman-Speiser, 1986).

1990s: Intersection theory on moduli spaces of curves.

7 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory Manin (’94): Generating functions in algebraic geometry. . . : e.g., Betti numbers for M0,n. ···

Characteristic classes in Intersection theory Intersection theory in algebraic geometry

Example: Rigorous computation of the , cubics (—, Kleiman-Speiser, 1986).

1990s: Intersection theory on moduli spaces of curves.

7 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory Characteristic classes in Intersection theory Intersection theory in algebraic geometry

Example: Rigorous computation of the , cubics (—, Kleiman-Speiser, 1986).

1990s: Intersection theory on moduli spaces of curves.

7 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory Fulton, Pandharipande, Graber, Okounkov, Behrend, Fantechi, ... Intersection theory on stacks Equivariant intersection theory Donaldson-Thomas invariants Modern ‘Schubert calculus’, combinatorial algebraic geometry Characteristic classes for singular varieties ··· Intersection theory in algebraic geometry: item 14C17 in the math reviews; about 1000 papers, ∼30 pending.

Characteristic classes in Intersection theory Intersection theory in algebraic geometry

Much more: Candelas-de la Ossa-Green-Parkes quantum cohomology (still in the ’90s!)

8 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory Intersection theory on stacks Equivariant intersection theory Donaldson-Thomas invariants Modern ‘Schubert calculus’, combinatorial algebraic geometry Characteristic classes for singular varieties ··· Intersection theory in algebraic geometry: item 14C17 in the math reviews; about 1000 papers, ∼30 pending.

Characteristic classes in Intersection theory Intersection theory in algebraic geometry

Much more: Candelas-de la Ossa-Green-Parkes quantum cohomology (still in the ’90s!) Fulton, Pandharipande, Graber, Okounkov, Behrend, Fantechi, ...

8 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory Equivariant intersection theory Donaldson-Thomas invariants Modern ‘Schubert calculus’, combinatorial algebraic geometry Characteristic classes for singular varieties ··· Intersection theory in algebraic geometry: item 14C17 in the math reviews; about 1000 papers, ∼30 pending.

Characteristic classes in Intersection theory Intersection theory in algebraic geometry

Much more: Candelas-de la Ossa-Green-Parkes quantum cohomology (still in the ’90s!) Fulton, Pandharipande, Graber, Okounkov, Behrend, Fantechi, ... Intersection theory on stacks

8 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory Donaldson-Thomas invariants Modern ‘Schubert calculus’, combinatorial algebraic geometry Characteristic classes for singular varieties ··· Intersection theory in algebraic geometry: item 14C17 in the math reviews; about 1000 papers, ∼30 pending.

Characteristic classes in Intersection theory Intersection theory in algebraic geometry

8 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory Modern ‘Schubert calculus’, combinatorial algebraic geometry Characteristic classes for singular varieties ··· Intersection theory in algebraic geometry: item 14C17 in the math reviews; about 1000 papers, ∼30 pending.

Characteristic classes in Intersection theory Intersection theory in algebraic geometry

8 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory Characteristic classes for singular varieties ··· Intersection theory in algebraic geometry: item 14C17 in the math reviews; about 1000 papers, ∼30 pending.

Characteristic classes in Intersection theory Intersection theory in algebraic geometry

8 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory Intersection theory in algebraic geometry: item 14C17 in the math reviews; about 1000 papers, ∼30 pending.

Characteristic classes in Intersection theory Intersection theory in algebraic geometry

Much more: Candelas-de la Ossa-Green-Parkes quantum cohomology (still in the ’90s!) Fulton, Pandharipande, Graber, Okounkov, Behrend, Fantechi, ... Intersection theory on stacks Equivariant intersection theory Donaldson-Thomas invariants Modern ‘Schubert calculus’, combinatorial algebraic geometry Characteristic classes for singular varieties ···

8 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory about 1000 papers, ∼30 pending.

Characteristic classes in Intersection theory Intersection theory in algebraic geometry

8 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory Characteristic classes in Intersection theory Intersection theory in algebraic geometry

8 / 26 Paolo Aluffi Characteristic classes in Intersection theory Some names: Deligne-Grothendieck, Marie-HélèneSchwartz (’60s); Mather, MacPherson, Fulton, Johnson, Brasselet, Yokura, . . . (’70s-’80s); (—), Parusinski, Pragacz, Schürmann;Ohmoto, Maxim, . . . (’90s →) Prototype: Vector bundles have Chern classes;

ci (E) = obstruction to finding rk E + 1 − i linearly independent global sections. V smooth compact ‘(total) Chern class of V ’, c(TV ) ∩ [V ]. Components of c(TV ) ∩ [V ]: obstructions to defining global frames of vector fields on V . Dually (up to sign): obstruction to defining linearly independent global differential forms.

Characteristic classes in Intersection theory Characteristic classes of singular/noncompact algebraic varieties

Characteristic classes

One topic among many: Characteristic classes.

9 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory Mather, MacPherson, Fulton, Johnson, Brasselet, Yokura, . . . (’70s-’80s); (—), Parusinski, Pragacz, Sch¨urmann;Ohmoto, Maxim, . . . (’90s →) Prototype: Vector bundles have Chern classes;

Characteristic classes in Intersection theory Characteristic classes of singular/noncompact algebraic varieties

Characteristic classes

One topic among many: Characteristic classes. Some names: Deligne-Grothendieck, Marie-H´el`eneSchwartz (’60s);

9 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory (—), Parusinski, Pragacz, Sch¨urmann;Ohmoto, Maxim, . . . (’90s →) Prototype: Vector bundles have Chern classes;

Characteristic classes in Intersection theory Characteristic classes of singular/noncompact algebraic varieties

Characteristic classes

One topic among many: Characteristic classes. Some names: Deligne-Grothendieck, Marie-H´el`eneSchwartz (’60s); Mather, MacPherson, Fulton, Johnson, Brasselet, Yokura, . . . (’70s-’80s);

9 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory Prototype: Vector bundles have Chern classes;

Characteristic classes in Intersection theory Characteristic classes of singular/noncompact algebraic varieties

Characteristic classes

One topic among many: Characteristic classes. Some names: Deligne-Grothendieck, Marie-HélèneSchwartz (’60s); Mather, MacPherson, Fulton, Johnson, Brasselet, Yokura, . . . (’70s-’80s); (—), Parusinski, Pragacz, Schürmann;Ohmoto, Maxim, . . . (’90s →)

9 / 26 Paolo Aluffi Characteristic classes in Intersection theory ci (E) = obstruction to finding rk E + 1 − i linearly independent global sections. V smooth compact ‘(total) Chern class of V ’, c(TV ) ∩ [V ]. Components of c(TV ) ∩ [V ]: obstructions to defining global frames of vector fields on V . Dually (up to sign): obstruction to defining linearly independent global differential forms.

Characteristic classes in Intersection theory Characteristic classes of singular/noncompact algebraic varieties

Characteristic classes

9 / 26 Paolo Aluffi Characteristic classes in Intersection theory V smooth compact ‘(total) Chern class of V ’, c(TV ) ∩ [V ]. Components of c(TV ) ∩ [V ]: obstructions to defining global frames of vector fields on V . Dually (up to sign): obstruction to defining linearly independent global differential forms.

Characteristic classes in Intersection theory Characteristic classes of singular/noncompact algebraic varieties

Characteristic classes

ci (E) = obstruction to ﬁnding rk E + 1 − i linearly independent global sections.

9 / 26 Paolo Aluffi Characteristic classes in Intersection theory Components of c(TV ) ∩ [V ]: obstructions to defining global frames of vector fields on V . Dually (up to sign): obstruction to defining linearly independent global differential forms.

Characteristic classes in Intersection theory Characteristic classes of singular/noncompact algebraic varieties

Characteristic classes

ci (E) = obstruction to ﬁnding rk E + 1 − i linearly independent global sections. V smooth compact ‘(total) Chern class of V ’, c(TV ) ∩ [V ].

9 / 26 Paolo Aluffi Characteristic classes in Intersection theory Dually (up to sign): obstruction to defining linearly independent global differential forms.

Characteristic classes in Intersection theory Characteristic classes of singular/noncompact algebraic varieties

Characteristic classes

9 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory Characteristic classes in Intersection theory Characteristic classes of singular/noncompact algebraic varieties

Characteristic classes

9 / 26 Paolo Aluffi Characteristic classes in Intersection theory Poincaré-Hopf: R c(TV ) ∩ [V ] = χ(V ), topological Euler characteristic. Question: What is the ‘right’ analogue for possibly singular or noncompact varieties? E.g., Does M0,n have a natural notion of “total Chern class”? Naive answer: Replace TV with a coherent sheaf; the sheaf ∨ 1 T V = ΩV is defined for all V . Short summary: This does not work too well. Coherent sheaves on singular varieties don’t have Chern classes. (There are alternatives Chern-Mather classes.) Chow group / Homology of noncompact varieties: losing too much information; e.g., can’t recover χ.

Characteristic classes in Intersection theory Characteristic classes of singular/noncompact algebraic varieties

c1(TV ) = −KV , canonical bundle. E.g., V Calabi-Yau =⇒ c1(TV ) = 0;

10 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory Question: What is the ‘right’ analogue for possibly singular or noncompact varieties? E.g., Does M0,n have a natural notion of “total Chern class”? Naive answer: Replace TV with a coherent sheaf; the sheaf ∨ 1 T V = ΩV is deﬁned for all V . Short summary: This does not work too well. Coherent sheaves on singular varieties don’t have Chern classes. (There are alternatives Chern-Mather classes.) Chow group / Homology of noncompact varieties: losing too much information; e.g., can’t recover χ.

Characteristic classes in Intersection theory Characteristic classes of singular/noncompact algebraic varieties

c1(TV ) = −KV , canonical bundle. E.g., V Calabi-Yau =⇒ c1(TV ) = 0; Poincar´e-Hopf: R c(TV ) ∩ [V ] = χ(V ), topological Euler characteristic.

10 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory E.g., Does M0,n have a natural notion of “total Chern class”? Naive answer: Replace TV with a coherent sheaf; the sheaf ∨ 1 T V = ΩV is deﬁned for all V . Short summary: This does not work too well. Coherent sheaves on singular varieties don’t have Chern classes. (There are alternatives Chern-Mather classes.) Chow group / Homology of noncompact varieties: losing too much information; e.g., can’t recover χ.

Characteristic classes in Intersection theory Characteristic classes of singular/noncompact algebraic varieties

10 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory Naive answer: Replace TV with a coherent sheaf; the sheaf ∨ 1 T V = ΩV is deﬁned for all V . Short summary: This does not work too well. Coherent sheaves on singular varieties don’t have Chern classes. (There are alternatives Chern-Mather classes.) Chow group / Homology of noncompact varieties: losing too much information; e.g., can’t recover χ.

Characteristic classes in Intersection theory Characteristic classes of singular/noncompact algebraic varieties

10 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory Short summary: This does not work too well. Coherent sheaves on singular varieties don’t have Chern classes. (There are alternatives Chern-Mather classes.) Chow group / Homology of noncompact varieties: losing too much information; e.g., can’t recover χ.

Characteristic classes in Intersection theory Characteristic classes of singular/noncompact algebraic varieties

10 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory Coherent sheaves on singular varieties don’t have Chern classes. (There are alternatives Chern-Mather classes.) Chow group / Homology of noncompact varieties: losing too much information; e.g., can’t recover χ.

Characteristic classes in Intersection theory Characteristic classes of singular/noncompact algebraic varieties

10 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory Chow group / Homology of noncompact varieties: losing too much information; e.g., can’t recover χ.

Characteristic classes in Intersection theory Characteristic classes of singular/noncompact algebraic varieties

c1(TV ) = −KV , canonical bundle. E.g., V Calabi-Yau =⇒ c1(TV ) = 0; Poincar´e-Hopf: R c(TV ) ∩ [V ] = χ(V ), topological Euler characteristic. Question: What is the ‘right’ analogue for possibly singular or noncompact varieties? E.g., Does M0,n have a natural notion of “total Chern class”? Naive answer: Replace TV with a coherent sheaf; the sheaf ∨ 1 T V = ΩV is deﬁned for all V . Short summary: This does not work too well. Coherent sheaves on singular varieties don’t have Chern classes. (There are alternatives Chern-Mather classes.)

10 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory Characteristic classes in Intersection theory Characteristic classes of singular/noncompact algebraic varieties

10 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory Aside: Quotients may acquire mild ‘orbifold’ singularities. Dixon-Harvey-Vafa-Witten: orbifold Euler number (’85). (Batyrev ’99): Generalization to stringy Euler number. (—, deFernex-Lupercio-Nevins-Uribe ’06): stringy Chern class, whose degree is the stringy Euler number.

Interesting story, but still not the ‘right’ generalization of the ordinary Chern class.

(E.g., V not compact ?)

Characteristic classes in Intersection theory Characteristic classes of singular/noncompact algebraic varieties

Moral: We need something less naive.

11 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory (Batyrev ’99): Generalization to stringy Euler number. (—, deFernex-Lupercio-Nevins-Uribe ’06): stringy Chern class, whose degree is the stringy Euler number.

Interesting story, but still not the ‘right’ generalization of the ordinary Chern class.

(E.g., V not compact ?)

Characteristic classes in Intersection theory Characteristic classes of singular/noncompact algebraic varieties

Moral: We need something less naive. Aside: Quotients may acquire mild ‘orbifold’ singularities. Dixon-Harvey-Vafa-Witten: orbifold Euler number (’85).

11 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory (—, deFernex-Lupercio-Nevins-Uribe ’06): stringy Chern class, whose degree is the stringy Euler number.

Interesting story, but still not the ‘right’ generalization of the ordinary Chern class.

(E.g., V not compact ?)

Characteristic classes in Intersection theory Characteristic classes of singular/noncompact algebraic varieties

11 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory Interesting story, but still not the ‘right’ generalization of the ordinary Chern class.

(E.g., V not compact ?)

Characteristic classes in Intersection theory Characteristic classes of singular/noncompact algebraic varieties

Moral: We need something less naive. Aside: Quotients may acquire mild ‘orbifold’ singularities. Dixon-Harvey-Vafa-Witten: orbifold Euler number (’85). (Batyrev ’99): Generalization to stringy Euler number. (—, deFernex-Lupercio-Nevins-Uribe ’06): stringy Chern class, whose degree is the stringy Euler number.

11 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory (E.g., V not compact ?)

Characteristic classes in Intersection theory Characteristic classes of singular/noncompact algebraic varieties

Interesting story, but still not the ‘right’ generalization of the ordinary Chern class.

11 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory Characteristic classes in Intersection theory Characteristic classes of singular/noncompact algebraic varieties

Interesting story, but still not the ‘right’ generalization of the ordinary Chern class.

(E.g., V not compact ?)

11 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory X : complex projective variety. F (X ) =group of integer-valued constructible functions. P ϕ ∈ F (X ) ⇐⇒ ϕ = mW 11W , W ⊆ X closed subvarieties, mW ∈ Z, mW 6= 0 for ﬁn. many W . F is a covariant functor from {varieties} to {abelian groups}! f : X → Y proper map f∗ : F (X ) → F (Y ) −1 W ⊆ X ; p ∈ Y : f∗(11W )(p) = χ(W ∩ f (p)).

Fact (not diﬃcult): (f ◦ g)∗ = f∗ ◦ g∗.

Remark: homology, H∗ (or Chow, A∗) is also a covariant functor {varieties} to {abelian groups}

Characteristic classes in Intersection theory Characteristic classes of singular/noncompact algebraic varieties

Right generalization: Grothendieck-Deligne (’65)

12 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory P ϕ ∈ F (X ) ⇐⇒ ϕ = mW 11W , W ⊆ X closed subvarieties, mW ∈ Z, mW 6= 0 for ﬁn. many W . F is a covariant functor from {varieties} to {abelian groups}! f : X → Y proper map f∗ : F (X ) → F (Y ) −1 W ⊆ X ; p ∈ Y : f∗(11W )(p) = χ(W ∩ f (p)).

Fact (not diﬃcult): (f ◦ g)∗ = f∗ ◦ g∗.

Remark: homology, H∗ (or Chow, A∗) is also a covariant functor {varieties} to {abelian groups}

Characteristic classes in Intersection theory Characteristic classes of singular/noncompact algebraic varieties

Right generalization: Grothendieck-Deligne (’65)

X : complex projective variety. F (X ) =group of integer-valued constructible functions.

12 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory F is a covariant functor from {varieties} to {abelian groups}! f : X → Y proper map f∗ : F (X ) → F (Y ) −1 W ⊆ X ; p ∈ Y : f∗(11W )(p) = χ(W ∩ f (p)).

Fact (not diﬃcult): (f ◦ g)∗ = f∗ ◦ g∗.

Remark: homology, H∗ (or Chow, A∗) is also a covariant functor {varieties} to {abelian groups}

Characteristic classes in Intersection theory Characteristic classes of singular/noncompact algebraic varieties

Right generalization: Grothendieck-Deligne (’65)

X : complex projective variety. F (X ) =group of integer-valued constructible functions. P ϕ ∈ F (X ) ⇐⇒ ϕ = mW 11W , W ⊆ X closed subvarieties, mW ∈ Z, mW 6= 0 for ﬁn. many W .

12 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory f : X → Y proper map f∗ : F (X ) → F (Y ) −1 W ⊆ X ; p ∈ Y : f∗(11W )(p) = χ(W ∩ f (p)).

Fact (not diﬃcult): (f ◦ g)∗ = f∗ ◦ g∗.

Remark: homology, H∗ (or Chow, A∗) is also a covariant functor {varieties} to {abelian groups}

Characteristic classes in Intersection theory Characteristic classes of singular/noncompact algebraic varieties

Right generalization: Grothendieck-Deligne (’65)

X : complex projective variety. F (X ) =group of integer-valued constructible functions. P ϕ ∈ F (X ) ⇐⇒ ϕ = mW 11W , W ⊆ X closed subvarieties, mW ∈ Z, mW 6= 0 for ﬁn. many W . F is a covariant functor from {varieties} to {abelian groups}!

12 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory −1 W ⊆ X ; p ∈ Y : f∗(11W )(p) = χ(W ∩ f (p)).

Fact (not diﬃcult): (f ◦ g)∗ = f∗ ◦ g∗.

Remark: homology, H∗ (or Chow, A∗) is also a covariant functor {varieties} to {abelian groups}

Characteristic classes in Intersection theory Characteristic classes of singular/noncompact algebraic varieties

Right generalization: Grothendieck-Deligne (’65)

12 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory −1 f∗(11W )(p) = χ(W ∩ f (p)).

Fact (not diﬃcult): (f ◦ g)∗ = f∗ ◦ g∗.

Remark: homology, H∗ (or Chow, A∗) is also a covariant functor {varieties} to {abelian groups}

Characteristic classes in Intersection theory Characteristic classes of singular/noncompact algebraic varieties

Right generalization: Grothendieck-Deligne (’65)

12 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory χ(W ∩ f −1(p)).

Fact (not diﬃcult): (f ◦ g)∗ = f∗ ◦ g∗.

Remark: homology, H∗ (or Chow, A∗) is also a covariant functor {varieties} to {abelian groups}

Characteristic classes in Intersection theory Characteristic classes of singular/noncompact algebraic varieties

Right generalization: Grothendieck-Deligne (’65)

12 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory Fact (not diﬃcult): (f ◦ g)∗ = f∗ ◦ g∗.

Remark: homology, H∗ (or Chow, A∗) is also a covariant functor {varieties} to {abelian groups}

Characteristic classes in Intersection theory Characteristic classes of singular/noncompact algebraic varieties

Right generalization: Grothendieck-Deligne (’65)

12 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory Remark: homology, H∗ (or Chow, A∗) is also a covariant functor {varieties} to {abelian groups}

Characteristic classes in Intersection theory Characteristic classes of singular/noncompact algebraic varieties

Right generalization: Grothendieck-Deligne (’65)

Fact (not diﬃcult): (f ◦ g)∗ = f∗ ◦ g∗.

12 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory Characteristic classes in Intersection theory Characteristic classes of singular/noncompact algebraic varieties

Right generalization: Grothendieck-Deligne (’65)

Fact (not diﬃcult): (f ◦ g)∗ = f∗ ◦ g∗.

Remark: homology, H∗ (or Chow, A∗) is also a covariant functor {varieties} to {abelian groups}

12 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory Question [Deligne-Grothendieck]

Is there a natural transformation c∗ : F H∗, such that if V is a nonsingular variety, then c∗(11V ) = c(TV ) ∩ [V ]?

Theorem (MacPherson ’74) Yes Please appreciate this. Content: There is a functorial theory of ‘Chern classes’ for arbitrarily singular projective varieties, satisfying the same combinatorial properties of the topological Euler characteristic.

Characteristic classes in Intersection theory Characteristic classes of singular/noncompact algebraic varieties

Two covariant functors {varieties} → {abelian groups}: F , H∗.

13 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory Theorem (MacPherson ’74) Yes Please appreciate this. Content: There is a functorial theory of ‘Chern classes’ for arbitrarily singular projective varieties, satisfying the same combinatorial properties of the topological Euler characteristic.

Characteristic classes in Intersection theory Characteristic classes of singular/noncompact algebraic varieties

Two covariant functors {varieties} → {abelian groups}: F , H∗. Question [Deligne-Grothendieck]

Is there a natural transformation c∗ : F H∗, such that if V is a nonsingular variety, then c∗(11V ) = c(TV ) ∩ [V ]?

13 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory Yes Please appreciate this. Content: There is a functorial theory of ‘Chern classes’ for arbitrarily singular projective varieties, satisfying the same combinatorial properties of the topological Euler characteristic.

Characteristic classes in Intersection theory Characteristic classes of singular/noncompact algebraic varieties

Two covariant functors {varieties} → {abelian groups}: F , H∗. Question [Deligne-Grothendieck]

Is there a natural transformation c∗ : F H∗, such that if V is a nonsingular variety, then c∗(11V ) = c(TV ) ∩ [V ]?

Theorem (MacPherson ’74)

13 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory Please appreciate this. Content: There is a functorial theory of ‘Chern classes’ for arbitrarily singular projective varieties, satisfying the same combinatorial properties of the topological Euler characteristic.

Characteristic classes in Intersection theory Characteristic classes of singular/noncompact algebraic varieties

Two covariant functors {varieties} → {abelian groups}: F , H∗. Question [Deligne-Grothendieck]

Is there a natural transformation c∗ : F H∗, such that if V is a nonsingular variety, then c∗(11V ) = c(TV ) ∩ [V ]?

Theorem (MacPherson ’74) Yes

13 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory Content: There is a functorial theory of ‘Chern classes’ for arbitrarily singular projective varieties, satisfying the same combinatorial properties of the topological Euler characteristic.

Characteristic classes in Intersection theory Characteristic classes of singular/noncompact algebraic varieties

Two covariant functors {varieties} → {abelian groups}: F , H∗. Question [Deligne-Grothendieck]

Is there a natural transformation c∗ : F H∗, such that if V is a nonsingular variety, then c∗(11V ) = c(TV ) ∩ [V ]?

Theorem (MacPherson ’74) Yes Please appreciate this.

13 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory Characteristic classes in Intersection theory Characteristic classes of singular/noncompact algebraic varieties

Two covariant functors {varieties} → {abelian groups}: F , H∗. Question [Deligne-Grothendieck]

Is there a natural transformation c∗ : F H∗, such that if V is a nonsingular variety, then c∗(11V ) = c(TV ) ∩ [V ]?

13 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory Deﬁnition The Chern-Schwartz-MacPherson class of X ⊆ V is

cSM(X ) := c∗(11X ) ∈ H∗V

(Finer theory: use A∗ rather than H∗)

Example: Well-deﬁned class cSM(M0,n) in homology of M0,n. (Not computed explicitly as far as I know.)

Characteristic classes in Intersection theory Characteristic classes of singular/noncompact algebraic varieties

n Fix a projective V (e.g., V = P ); X ⊆ V : locally closed subset.

14 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory (Finer theory: use A∗ rather than H∗)

Example: Well-deﬁned class cSM(M0,n) in homology of M0,n. (Not computed explicitly as far as I know.)

Characteristic classes in Intersection theory Characteristic classes of singular/noncompact algebraic varieties

n Fix a projective V (e.g., V = P ); X ⊆ V : locally closed subset. Deﬁnition The Chern-Schwartz-MacPherson class of X ⊆ V is

cSM(X ) := c∗(11X ) ∈ H∗V

14 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory Example: Well-deﬁned class cSM(M0,n) in homology of M0,n. (Not computed explicitly as far as I know.)

Characteristic classes in Intersection theory Characteristic classes of singular/noncompact algebraic varieties

n Fix a projective V (e.g., V = P ); X ⊆ V : locally closed subset. Deﬁnition The Chern-Schwartz-MacPherson class of X ⊆ V is

cSM(X ) := c∗(11X ) ∈ H∗V

(Finer theory: use A∗ rather than H∗)

14 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory (Not computed explicitly as far as I know.)

Characteristic classes in Intersection theory Characteristic classes of singular/noncompact algebraic varieties

n Fix a projective V (e.g., V = P ); X ⊆ V : locally closed subset. Deﬁnition The Chern-Schwartz-MacPherson class of X ⊆ V is

cSM(X ) := c∗(11X ) ∈ H∗V

(Finer theory: use A∗ rather than H∗)

Example: Well-deﬁned class cSM(M0,n) in homology of M0,n.

14 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory Characteristic classes in Intersection theory Characteristic classes of singular/noncompact algebraic varieties

n Fix a projective V (e.g., V = P ); X ⊆ V : locally closed subset. Deﬁnition The Chern-Schwartz-MacPherson class of X ⊆ V is

cSM(X ) := c∗(11X ) ∈ H∗V

(Finer theory: use A∗ rather than H∗)

Example: Well-deﬁned class cSM(M0,n) in homology of M0,n. (Not computed explicitly as far as I know.)

14 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory ‘Inclusion-exclusion’. X , Y ⊆ V .

cSM(X ∪ Y ) = cSM(X ) + cSM(Y ) − cSM(X ∩ Y )

Z ⊆ V , V nonsingular.

cSM(Z) = formula in terms of Chern classes/Segre classes (This + inclusion-exclusion → Macaulay2 procedures to compute cSM(X ) for X projective/toric, from ideal deﬁning X ) Z = V r D, D = ∪i Di simple normal crossings. 1 ∨ cSM(Z) = c(Ω (log D) ) ∩ [V ] (+generalization to ‘free’ divisors, Liao) n X ⊆ P : degrees of components of cSM(X ) ↔ Euler characteristics of linear sections of X .

Characteristic classes in Intersection theory Characteristic classes of singular/noncompact algebraic varieties

How to compute cSM(X )? What kind of info does it carry?

15 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory Z ⊆ V , V nonsingular.

Characteristic classes in Intersection theory Characteristic classes of singular/noncompact algebraic varieties

How to compute cSM(X )? What kind of info does it carry?

‘Inclusion-exclusion’. X , Y ⊆ V .

cSM(X ∪ Y ) = cSM(X ) + cSM(Y ) − cSM(X ∩ Y )

15 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory (This + inclusion-exclusion → Macaulay2 procedures to compute cSM(X ) for X projective/toric, from ideal deﬁning X ) Z = V r D, D = ∪i Di simple normal crossings. 1 ∨ cSM(Z) = c(Ω (log D) ) ∩ [V ] (+generalization to ‘free’ divisors, Liao) n X ⊆ P : degrees of components of cSM(X ) ↔ Euler characteristics of linear sections of X .

Characteristic classes in Intersection theory Characteristic classes of singular/noncompact algebraic varieties

How to compute cSM(X )? What kind of info does it carry?

‘Inclusion-exclusion’. X , Y ⊆ V .

cSM(X ∪ Y ) = cSM(X ) + cSM(Y ) − cSM(X ∩ Y )

Z ⊆ V , V nonsingular.

cSM(Z) = formula in terms of Chern classes/Segre classes

15 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory Z = V r D, D = ∪i Di simple normal crossings. 1 ∨ cSM(Z) = c(Ω (log D) ) ∩ [V ] (+generalization to ‘free’ divisors, Liao) n X ⊆ P : degrees of components of cSM(X ) ↔ Euler characteristics of linear sections of X .

Characteristic classes in Intersection theory Characteristic classes of singular/noncompact algebraic varieties

How to compute cSM(X )? What kind of info does it carry?

‘Inclusion-exclusion’. X , Y ⊆ V .

cSM(X ∪ Y ) = cSM(X ) + cSM(Y ) − cSM(X ∩ Y )

Z ⊆ V , V nonsingular.

cSM(Z) = formula in terms of Chern classes/Segre classes (This + inclusion-exclusion → Macaulay2 procedures to compute cSM(X ) for X projective/toric, from ideal deﬁning X )

15 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory n X ⊆ P : degrees of components of cSM(X ) ↔ Euler characteristics of linear sections of X .

Characteristic classes in Intersection theory Characteristic classes of singular/noncompact algebraic varieties

How to compute cSM(X )? What kind of info does it carry?

‘Inclusion-exclusion’. X , Y ⊆ V .

cSM(X ∪ Y ) = cSM(X ) + cSM(Y ) − cSM(X ∩ Y )

Z ⊆ V , V nonsingular.

15 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory Characteristic classes in Intersection theory Characteristic classes of singular/noncompact algebraic varieties

How to compute cSM(X )? What kind of info does it carry?

‘Inclusion-exclusion’. X , Y ⊆ V .

cSM(X ∪ Y ) = cSM(X ) + cSM(Y ) − cSM(X ∩ Y )

Z ⊆ V , V nonsingular.

15 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory V compact, so κ : V → {pt} proper. Functoriality:

F (V ) / H∗(V )

κ∗ κ∗ F (pt) Z H∗(pt)

commutes. For X ⊆ V locally close (note: not nec. compact!)

11X / cSM(X ) _ _

R χ(X ) cSM(X )

‘Singular/noncompact Poincar´e-Hopf’

Characteristic classes in Intersection theory Characteristic classes of singular/noncompact algebraic varieties

Functoriality: vast generalization of Poincar´e-Hopf.

16 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory For X ⊆ V locally close (note: not nec. compact!)

11X / cSM(X ) _ _

R χ(X ) cSM(X )

‘Singular/noncompact Poincar´e-Hopf’

Characteristic classes in Intersection theory Characteristic classes of singular/noncompact algebraic varieties

Functoriality: vast generalization of Poincar´e-Hopf. V compact, so κ : V → {pt} proper. Functoriality:

F (V ) / H∗(V )

κ∗ κ∗ F (pt) Z H∗(pt)

commutes.

16 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory ‘Singular/noncompact Poincar´e-Hopf’

Characteristic classes in Intersection theory Characteristic classes of singular/noncompact algebraic varieties

Functoriality: vast generalization of Poincar´e-Hopf. V compact, so κ : V → {pt} proper. Functoriality:

F (V ) / H∗(V )

κ∗ κ∗ F (pt) Z H∗(pt)

commutes. For X ⊆ V locally close (note: not nec. compact!)

11X / cSM(X ) _ _

R χ(X ) cSM(X )

16 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory Characteristic classes in Intersection theory Characteristic classes of singular/noncompact algebraic varieties

Functoriality: vast generalization of Poincar´e-Hopf. V compact, so κ : V → {pt} proper. Functoriality:

F (V ) / H∗(V )

κ∗ κ∗ F (pt) Z H∗(pt)

commutes. For X ⊆ V locally close (note: not nec. compact!)

11X / cSM(X ) _ _

R χ(X ) cSM(X )

‘Singular/noncompact Poincar´e-Hopf’

16 / 26 Paolo Aluffi Characteristic classes in Intersection theory Barthel-Brasselet-Fieseler-Kaup, —: toric varieties — -Mihalcea, Schürmann, Su, Feher, Rimanyi, Weber, . . . : Schubert varieties in Grassmannians/flag manifolds ... Generalizations: Motivic Chern classes (Brasselet-Schürmann-Yokura ’04)

Equivariant cSM classes (Ohmoto ’06) ··· Many stories! Rest of the talk: ‘Graph hypersurfaces’ (— -Marcolli, ∼’10).

Characteristic classes in Intersection theory Characteristic classes of singular/noncompact algebraic varieties

cSM known for many interesting varieties: Parusinski-Pragacz (’95): degeneracy loci

17 / 26 Paolo Aluffi Characteristic classes in Intersection theory — -Mihalcea, Schürmann, Su, Feher, Rimanyi, Weber, . . . : Schubert varieties in Grassmannians/flag manifolds ... Generalizations: Motivic Chern classes (Brasselet-Schürmann-Yokura ’04)

Equivariant cSM classes (Ohmoto ’06) ··· Many stories! Rest of the talk: ‘Graph hypersurfaces’ (— -Marcolli, ∼’10).

Characteristic classes in Intersection theory Characteristic classes of singular/noncompact algebraic varieties

cSM known for many interesting varieties: Parusinski-Pragacz (’95): degeneracy loci Barthel-Brasselet-Fieseler-Kaup, —: toric varieties

17 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory ... Generalizations: Motivic Chern classes (Brasselet-Sch¨urmann-Yokura ’04)

Equivariant cSM classes (Ohmoto ’06) ··· Many stories! Rest of the talk: ‘Graph hypersurfaces’ (— -Marcolli, ∼’10).

Characteristic classes in Intersection theory Characteristic classes of singular/noncompact algebraic varieties

cSM known for many interesting varieties: Parusinski-Pragacz (’95): degeneracy loci Barthel-Brasselet-Fieseler-Kaup, —: toric varieties — -Mihalcea, Sch¨urmann, Su, Feher, Rimanyi, Weber, . . . : Schubert varieties in Grassmannians/ﬂag manifolds

17 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory Equivariant cSM classes (Ohmoto ’06) ··· Many stories! Rest of the talk: ‘Graph hypersurfaces’ (— -Marcolli, ∼’10).

Characteristic classes in Intersection theory Characteristic classes of singular/noncompact algebraic varieties

17 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory ··· Many stories! Rest of the talk: ‘Graph hypersurfaces’ (— -Marcolli, ∼’10).

Characteristic classes in Intersection theory Characteristic classes of singular/noncompact algebraic varieties

Equivariant cSM classes (Ohmoto ’06)

17 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory Many stories! Rest of the talk: ‘Graph hypersurfaces’ (— -Marcolli, ∼’10).

Characteristic classes in Intersection theory Characteristic classes of singular/noncompact algebraic varieties

Equivariant cSM classes (Ohmoto ’06) ···

17 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory Rest of the talk: ‘Graph hypersurfaces’ (— -Marcolli, ∼’10).

Characteristic classes in Intersection theory Characteristic classes of singular/noncompact algebraic varieties

Equivariant cSM classes (Ohmoto ’06) ··· Many stories!

17 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory Characteristic classes in Intersection theory Characteristic classes of singular/noncompact algebraic varieties

Equivariant cSM classes (Ohmoto ’06) ··· Many stories! Rest of the talk: ‘Graph hypersurfaces’ (— -Marcolli, ∼’10).

17 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory P Q → polynomialΨ G (t1,..., tn) := T e6∈T te where T ranges over maximal spanning forests of G. “Kirchhoﬀ-Tutte-Symanzik” polynomial. Example (‘banana graph’):

t1 t2 t3

ΨG (t) = t2t3 + t1t3 + t1t2.

Characteristic classes in Intersection theory

cSM classes of graph hypersurfaces cSM classes of graph hypersurfaces

G: graph (think: Feynman); edges e1,..., en ↔ variables te : t1,..., tn.

18 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory “Kirchhoﬀ-Tutte-Symanzik” polynomial. Example (‘banana graph’):

t1 t2 t3

ΨG (t) = t2t3 + t1t3 + t1t2.

Characteristic classes in Intersection theory

cSM classes of graph hypersurfaces cSM classes of graph hypersurfaces

G: graph (think: Feynman); edges e1,..., en ↔ variables te : t1,..., tn. P Q → polynomialΨ G (t1,..., tn) := T e6∈T te where T ranges over maximal spanning forests of G.

18 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory Example (‘banana graph’):

t1 t2 t3

ΨG (t) = t2t3 + t1t3 + t1t2.

Characteristic classes in Intersection theory

cSM classes of graph hypersurfaces cSM classes of graph hypersurfaces

G: graph (think: Feynman); edges e1,..., en ↔ variables te : t1,..., tn. P Q → polynomialΨ G (t1,..., tn) := T e6∈T te where T ranges over maximal spanning forests of G. “Kirchhoﬀ-Tutte-Symanzik” polynomial.

18 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory ΨG (t) = t2t3 + t1t3 + t1t2.

Characteristic classes in Intersection theory

cSM classes of graph hypersurfaces cSM classes of graph hypersurfaces

t1 t2 t3

18 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory Characteristic classes in Intersection theory

cSM classes of graph hypersurfaces cSM classes of graph hypersurfaces

t1 t2 t3

ΨG (t) = t2t3 + t1t3 + t1t2.

18 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory XG has degree b1(G), and is irreducible if and only if G cannot be separated as a disjoint union of two graphs, possibly joined at a vertex.

Task: Study invariants of XG . Motivation: Extensive computations suggested that contributions of G to Feynman amplitudes may be multiple zeta values. This was veriﬁed for ‘small’ graphs (∼ up to 13 edges)! Fact (Belkale-Brosnan; Brown; . . . ): not true!

Characteristic classes in Intersection theory

cSM classes of graph hypersurfaces

n−1 ΨG (t) is homogeneous, so it defines a hypersurface in P . Definition n−1 The graph hypersurface of G is the hypersurface XG ⊆ P defined by ΨG (t) = 0.

19 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory Task: Study invariants of XG . Motivation: Extensive computations suggested that contributions of G to Feynman amplitudes may be multiple zeta values. This was veriﬁed for ‘small’ graphs (∼ up to 13 edges)! Fact (Belkale-Brosnan; Brown; . . . ): not true!

Characteristic classes in Intersection theory

cSM classes of graph hypersurfaces

n−1 ΨG (t) is homogeneous, so it defines a hypersurface in P . Definition n−1 The graph hypersurface of G is the hypersurface XG ⊆ P defined by ΨG (t) = 0.

XG has degree b1(G), and is irreducible if and only if G cannot be separated as a disjoint union of two graphs, possibly joined at a vertex.

19 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory This was veriﬁed for ‘small’ graphs (∼ up to 13 edges)! Fact (Belkale-Brosnan; Brown; . . . ): not true!

Characteristic classes in Intersection theory

cSM classes of graph hypersurfaces

n−1 ΨG (t) is homogeneous, so it defines a hypersurface in P . Definition n−1 The graph hypersurface of G is the hypersurface XG ⊆ P defined by ΨG (t) = 0.

XG has degree b1(G), and is irreducible if and only if G cannot be separated as a disjoint union of two graphs, possibly joined at a vertex.

Task: Study invariants of XG . Motivation: Extensive computations suggested that contributions of G to Feynman amplitudes may be multiple zeta values.

19 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory Fact (Belkale-Brosnan; Brown; . . . ): not true!

Characteristic classes in Intersection theory

cSM classes of graph hypersurfaces

n−1 ΨG (t) is homogeneous, so it defines a hypersurface in P . Definition n−1 The graph hypersurface of G is the hypersurface XG ⊆ P defined by ΨG (t) = 0.

XG has degree b1(G), and is irreducible if and only if G cannot be separated as a disjoint union of two graphs, possibly joined at a vertex.

19 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory Characteristic classes in Intersection theory

cSM classes of graph hypersurfaces

n−1 ΨG (t) is homogeneous, so it defines a hypersurface in P . Definition n−1 The graph hypersurface of G is the hypersurface XG ⊆ P defined by ΨG (t) = 0.

XG has degree b1(G), and is irreducible if and only if G cannot be separated as a disjoint union of two graphs, possibly joined at a vertex.

19 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory Periods are controlled by the motive of XG . ‘Motive’ a universal Euler characteristic, dominating all invariants satisfying (e.g.) inclusion-exclusion. (Such as periods!).

E.g.: ΨG deﬁned over Z, hence over Fq.

n Nq(G) := #{(t1,..., tn) ∈ Fq | ΨG (t1,..., tn) = 0}

Original conjecture =⇒ Nq(G) is a polynomial in q.

(Kontsevich?) Is the motive of XG a mixed-Tate motive? Disproved!

Characteristic classes in Intersection theory

cSM classes of graph hypersurfaces

Feynman contribution of G: a period of the complement of XG .

20 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory ‘Motive’ a universal Euler characteristic, dominating all invariants satisfying (e.g.) inclusion-exclusion. (Such as periods!).

E.g.: ΨG deﬁned over Z, hence over Fq.

n Nq(G) := #{(t1,..., tn) ∈ Fq | ΨG (t1,..., tn) = 0}

Original conjecture =⇒ Nq(G) is a polynomial in q.

(Kontsevich?) Is the motive of XG a mixed-Tate motive? Disproved!

Characteristic classes in Intersection theory

cSM classes of graph hypersurfaces

Feynman contribution of G: a period of the complement of XG . Periods are controlled by the motive of XG .

20 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory E.g.: ΨG deﬁned over Z, hence over Fq.

n Nq(G) := #{(t1,..., tn) ∈ Fq | ΨG (t1,..., tn) = 0}

Original conjecture =⇒ Nq(G) is a polynomial in q.

(Kontsevich?) Is the motive of XG a mixed-Tate motive? Disproved!

Characteristic classes in Intersection theory

cSM classes of graph hypersurfaces

Feynman contribution of G: a period of the complement of XG . Periods are controlled by the motive of XG . ‘Motive’ a universal Euler characteristic, dominating all invariants satisfying (e.g.) inclusion-exclusion. (Such as periods!).

20 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory Original conjecture =⇒ Nq(G) is a polynomial in q.

(Kontsevich?) Is the motive of XG a mixed-Tate motive? Disproved!

Characteristic classes in Intersection theory

cSM classes of graph hypersurfaces

E.g.: ΨG deﬁned over Z, hence over Fq.

n Nq(G) := #{(t1,..., tn) ∈ Fq | ΨG (t1,..., tn) = 0}

20 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory (Kontsevich?) Is the motive of XG a mixed-Tate motive? Disproved!

Characteristic classes in Intersection theory

cSM classes of graph hypersurfaces

E.g.: ΨG deﬁned over Z, hence over Fq.

n Nq(G) := #{(t1,..., tn) ∈ Fq | ΨG (t1,..., tn) = 0}

Original conjecture =⇒ Nq(G) is a polynomial in q.

20 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory Disproved!

Characteristic classes in Intersection theory

cSM classes of graph hypersurfaces

E.g.: ΨG deﬁned over Z, hence over Fq.

n Nq(G) := #{(t1,..., tn) ∈ Fq | ΨG (t1,..., tn) = 0}

Original conjecture =⇒ Nq(G) is a polynomial in q.

(Kontsevich?) Is the motive of XG a mixed-Tate motive?

20 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory Characteristic classes in Intersection theory

cSM classes of graph hypersurfaces

E.g.: ΨG deﬁned over Z, hence over Fq.

n Nq(G) := #{(t1,..., tn) ∈ Fq | ΨG (t1,..., tn) = 0}

Original conjecture =⇒ Nq(G) is a polynomial in q.

(Kontsevich?) Is the motive of XG a mixed-Tate motive? Disproved!

20 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory Operations on G preserving mixed-Tate property?

Broad theme: Study properties of G which only depend on XG : Algebro-geometric Feynman rules.

The cSM class is also a generalized Euler characteristic. . . . but not controlled by the motive. (Reason: ‘moving target’) → motivation to study cSM(XG ).

Can cSM be used to construct an algebro-geometric Feynman rule?

Characteristic classes in Intersection theory

cSM classes of graph hypersurfaces

Next obvious questions: For what G is it true (↔ for which QFTs would Feynman contributions be multiple zeta values. . . )

21 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory Broad theme: Study properties of G which only depend on XG : Algebro-geometric Feynman rules.

The cSM class is also a generalized Euler characteristic. . . . but not controlled by the motive. (Reason: ‘moving target’) → motivation to study cSM(XG ).

Can cSM be used to construct an algebro-geometric Feynman rule?

Characteristic classes in Intersection theory

cSM classes of graph hypersurfaces

Next obvious questions: For what G is it true (↔ for which QFTs would Feynman contributions be multiple zeta values. . . ) Operations on G preserving mixed-Tate property?

21 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory Algebro-geometric Feynman rules.

The cSM class is also a generalized Euler characteristic. . . . but not controlled by the motive. (Reason: ‘moving target’) → motivation to study cSM(XG ).

Can cSM be used to construct an algebro-geometric Feynman rule?

Characteristic classes in Intersection theory

cSM classes of graph hypersurfaces

Next obvious questions: For what G is it true (↔ for which QFTs would Feynman contributions be multiple zeta values. . . ) Operations on G preserving mixed-Tate property?

Broad theme: Study properties of G which only depend on XG :

21 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory The cSM class is also a generalized Euler characteristic. . . . but not controlled by the motive. (Reason: ‘moving target’) → motivation to study cSM(XG ).

Can cSM be used to construct an algebro-geometric Feynman rule?

Characteristic classes in Intersection theory

cSM classes of graph hypersurfaces

Next obvious questions: For what G is it true (↔ for which QFTs would Feynman contributions be multiple zeta values. . . ) Operations on G preserving mixed-Tate property?

Broad theme: Study properties of G which only depend on XG : Algebro-geometric Feynman rules.

21 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory . . . but not controlled by the motive. (Reason: ‘moving target’) → motivation to study cSM(XG ).

Can cSM be used to construct an algebro-geometric Feynman rule?

Characteristic classes in Intersection theory

cSM classes of graph hypersurfaces

Next obvious questions: For what G is it true (↔ for which QFTs would Feynman contributions be multiple zeta values. . . ) Operations on G preserving mixed-Tate property?

Broad theme: Study properties of G which only depend on XG : Algebro-geometric Feynman rules.

The cSM class is also a generalized Euler characteristic.

21 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory → motivation to study cSM(XG ).

Can cSM be used to construct an algebro-geometric Feynman rule?

Characteristic classes in Intersection theory

cSM classes of graph hypersurfaces

Next obvious questions: For what G is it true (↔ for which QFTs would Feynman contributions be multiple zeta values. . . ) Operations on G preserving mixed-Tate property?

Broad theme: Study properties of G which only depend on XG : Algebro-geometric Feynman rules.

The cSM class is also a generalized Euler characteristic. . . . but not controlled by the motive. (Reason: ‘moving target’)

21 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory Can cSM be used to construct an algebro-geometric Feynman rule?

Characteristic classes in Intersection theory

cSM classes of graph hypersurfaces

Next obvious questions: For what G is it true (↔ for which QFTs would Feynman contributions be multiple zeta values. . . ) Operations on G preserving mixed-Tate property?

Broad theme: Study properties of G which only depend on XG : Algebro-geometric Feynman rules.

The cSM class is also a generalized Euler characteristic. . . . but not controlled by the motive. (Reason: ‘moving target’) → motivation to study cSM(XG ).

21 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory Characteristic classes in Intersection theory

cSM classes of graph hypersurfaces

Next obvious questions: For what G is it true (↔ for which QFTs would Feynman contributions be multiple zeta values. . . ) Operations on G preserving mixed-Tate property?

Broad theme: Study properties of G which only depend on XG : Algebro-geometric Feynman rules.

The cSM class is also a generalized Euler characteristic. . . . but not controlled by the motive. (Reason: ‘moving target’) → motivation to study cSM(XG ).

Can cSM be used to construct an algebro-geometric Feynman rule?

21 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory Here H is the hyperplane class, so this means

n n c ( n−1 Γ ) = [ n] + [ n−2] − [ n−3] + · · · ± [ 0] . SM P r n P 2 P 3 P P

‘Proof’: Never mind. ΨΓn ↔ Cremona transfromation; use functoriality. . . One moral is that the work needed to compute the motive of Γn can often be retooled to say something about cSM.

Characteristic classes in Intersection theory

cSM classes of graph hypersurfaces

(joint with Matilde Marcolli)

Example: G = Γn, ‘banana graph’, n edges. Then

n−1 n n cSM(P r Γn) = ((1 − H) + nH) ∩ [P ] .

22 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory ‘Proof’: Never mind. ΨΓn ↔ Cremona transfromation; use functoriality. . . One moral is that the work needed to compute the motive of Γn can often be retooled to say something about cSM.

Characteristic classes in Intersection theory

cSM classes of graph hypersurfaces

(joint with Matilde Marcolli)

Example: G = Γn, ‘banana graph’, n edges. Then

n−1 n n cSM(P r Γn) = ((1 − H) + nH) ∩ [P ] . Here H is the hyperplane class, so this means

n n c ( n−1 Γ ) = [ n] + [ n−2] − [ n−3] + · · · ± [ 0] . SM P r n P 2 P 3 P P

22 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory One moral is that the work needed to compute the motive of Γn can often be retooled to say something about cSM.

Characteristic classes in Intersection theory

cSM classes of graph hypersurfaces

(joint with Matilde Marcolli)

Example: G = Γn, ‘banana graph’, n edges. Then

n−1 n n cSM(P r Γn) = ((1 − H) + nH) ∩ [P ] . Here H is the hyperplane class, so this means

n n c ( n−1 Γ ) = [ n] + [ n−2] − [ n−3] + · · · ± [ 0] . SM P r n P 2 P 3 P P

‘Proof’: Never mind. ΨΓn ↔ Cremona transfromation; use functoriality. . .

22 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory Characteristic classes in Intersection theory

cSM classes of graph hypersurfaces

(joint with Matilde Marcolli)

Example: G = Γn, ‘banana graph’, n edges. Then

n−1 n n cSM(P r Γn) = ((1 − H) + nH) ∩ [P ] . Here H is the hyperplane class, so this means

n n c ( n−1 Γ ) = [ n] + [ n−2] − [ n−3] + · · · ± [ 0] . SM P r n P 2 P 3 P P

‘Proof’: Never mind. ΨΓn ↔ Cremona transfromation; use functoriality. . . One moral is that the work needed to compute the motive of Γn can often be retooled to say something about cSM. 22 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory n−1 (I have never seen a graph G for which χ(P r XG ) 6= 0 or ±1.) Companion algebro-geometric Feynman rule?

Deﬁnition

CG (t) determined by

n−1 n n−1 cSM(P r XG ) = (H CG (1/H)) ∩ [P ]

This satisﬁes CG1qG2 (t) = CG1 (t)CG2 (t). The corresponding ‘propagator’ is 1 + t.

Characteristic classes in Intersection theory

cSM classes of graph hypersurfaces

n−1 Remark: In particular, χ(Γn) = (−1) for n > 1.

23 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory Companion algebro-geometric Feynman rule?

Deﬁnition

CG (t) determined by

n−1 n n−1 cSM(P r XG ) = (H CG (1/H)) ∩ [P ]

This satisﬁes CG1qG2 (t) = CG1 (t)CG2 (t). The corresponding ‘propagator’ is 1 + t.

Characteristic classes in Intersection theory

cSM classes of graph hypersurfaces

n−1 Remark: In particular, χ(Γn) = (−1) for n > 1. n−1 (I have never seen a graph G for which χ(P r XG ) 6= 0 or ±1.)

23 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory Deﬁnition

CG (t) determined by

n−1 n n−1 cSM(P r XG ) = (H CG (1/H)) ∩ [P ]

This satisﬁes CG1qG2 (t) = CG1 (t)CG2 (t). The corresponding ‘propagator’ is 1 + t.

Characteristic classes in Intersection theory

cSM classes of graph hypersurfaces

n−1 Remark: In particular, χ(Γn) = (−1) for n > 1. n−1 (I have never seen a graph G for which χ(P r XG ) 6= 0 or ±1.) Companion algebro-geometric Feynman rule?

23 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory The corresponding ‘propagator’ is 1 + t.

Characteristic classes in Intersection theory

cSM classes of graph hypersurfaces

n−1 Remark: In particular, χ(Γn) = (−1) for n > 1. n−1 (I have never seen a graph G for which χ(P r XG ) 6= 0 or ±1.) Companion algebro-geometric Feynman rule?

Deﬁnition

CG (t) determined by

n−1 n n−1 cSM(P r XG ) = (H CG (1/H)) ∩ [P ]

This satisﬁes CG1qG2 (t) = CG1 (t)CG2 (t).

23 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory Characteristic classes in Intersection theory

cSM classes of graph hypersurfaces

n−1 Remark: In particular, χ(Γn) = (−1) for n > 1. n−1 (I have never seen a graph G for which χ(P r XG ) 6= 0 or ±1.) Companion algebro-geometric Feynman rule?

Deﬁnition

CG (t) determined by

n−1 n n−1 cSM(P r XG ) = (H CG (1/H)) ∩ [P ]

This satisﬁes CG1qG2 (t) = CG1 (t)CG2 (t). The corresponding ‘propagator’ is 1 + t.

23 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory CG (t) monic, degree = n:=#edges n−1 Coeﬀ. of t in CG (t) = n − b1(G) n G forest, n edges =⇒ CG (t) = (1 + t)

G not a forest =⇒ CG (0) = 0 0 n−1 CG (0) = χ(P r XG ) G 0 obtained from G by splitting/attaching edges: CG 0 (t) = t CG (t)

G not 1-particle irreducible =⇒ CG (−1) = 0

Characteristic classes in Intersection theory

cSM classes of graph hypersurfaces

Properties (— - Marcolli):

24 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory n−1 Coeﬀ. of t in CG (t) = n − b1(G) n G forest, n edges =⇒ CG (t) = (1 + t)

G not a forest =⇒ CG (0) = 0 0 n−1 CG (0) = χ(P r XG ) G 0 obtained from G by splitting/attaching edges: CG 0 (t) = t CG (t)

G not 1-particle irreducible =⇒ CG (−1) = 0

Characteristic classes in Intersection theory

cSM classes of graph hypersurfaces

Properties (— - Marcolli):

CG (t) monic, degree = n:=#edges

24 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory n G forest, n edges =⇒ CG (t) = (1 + t)

G not a forest =⇒ CG (0) = 0 0 n−1 CG (0) = χ(P r XG ) G 0 obtained from G by splitting/attaching edges: CG 0 (t) = t CG (t)

G not 1-particle irreducible =⇒ CG (−1) = 0

Characteristic classes in Intersection theory

cSM classes of graph hypersurfaces

Properties (— - Marcolli):

CG (t) monic, degree = n:=#edges n−1 Coeﬀ. of t in CG (t) = n − b1(G)

24 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory G not a forest =⇒ CG (0) = 0 0 n−1 CG (0) = χ(P r XG ) G 0 obtained from G by splitting/attaching edges: CG 0 (t) = t CG (t)

G not 1-particle irreducible =⇒ CG (−1) = 0

Characteristic classes in Intersection theory

cSM classes of graph hypersurfaces

Properties (— - Marcolli):

CG (t) monic, degree = n:=#edges n−1 Coeﬀ. of t in CG (t) = n − b1(G) n G forest, n edges =⇒ CG (t) = (1 + t)

24 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory 0 n−1 CG (0) = χ(P r XG ) G 0 obtained from G by splitting/attaching edges: CG 0 (t) = t CG (t)

G not 1-particle irreducible =⇒ CG (−1) = 0

Characteristic classes in Intersection theory

cSM classes of graph hypersurfaces

Properties (— - Marcolli):

CG (t) monic, degree = n:=#edges n−1 Coeﬀ. of t in CG (t) = n − b1(G) n G forest, n edges =⇒ CG (t) = (1 + t)

G not a forest =⇒ CG (0) = 0

24 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory G 0 obtained from G by splitting/attaching edges: CG 0 (t) = t CG (t)

G not 1-particle irreducible =⇒ CG (−1) = 0

Characteristic classes in Intersection theory

cSM classes of graph hypersurfaces

Properties (— - Marcolli):

CG (t) monic, degree = n:=#edges n−1 Coeﬀ. of t in CG (t) = n − b1(G) n G forest, n edges =⇒ CG (t) = (1 + t)

G not a forest =⇒ CG (0) = 0 0 n−1 CG (0) = χ(P r XG )

24 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory G not 1-particle irreducible =⇒ CG (−1) = 0

Characteristic classes in Intersection theory

cSM classes of graph hypersurfaces

Properties (— - Marcolli):

CG (t) monic, degree = n:=#edges n−1 Coeﬀ. of t in CG (t) = n − b1(G) n G forest, n edges =⇒ CG (t) = (1 + t)

G not a forest =⇒ CG (0) = 0 0 n−1 CG (0) = χ(P r XG ) G 0 obtained from G by splitting/attaching edges: CG 0 (t) = t CG (t)

24 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory Characteristic classes in Intersection theory

cSM classes of graph hypersurfaces

Properties (— - Marcolli):

CG (t) monic, degree = n:=#edges n−1 Coeﬀ. of t in CG (t) = n − b1(G) n G forest, n edges =⇒ CG (t) = (1 + t)

G not a forest =⇒ CG (0) = 0 0 n−1 CG (0) = χ(P r XG ) G 0 obtained from G by splitting/attaching edges: CG 0 (t) = t CG (t)

G not 1-particle irreducible =⇒ CG (−1) = 0

24 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory Question: are the ‘unfortunate technical conditions’ necessary?

Characteristic classes in Intersection theory

cSM classes of graph hypersurfaces

Theorem (—) e edges of G, satisfying unfortunate technical conditions. Then

CG2e (t) = (2t − 1)CG (t) − t(t − 1)CGre (t) + CG/e (t)

G2e : double edge e G r e: delete e G/e: contract e

25 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory Characteristic classes in Intersection theory

cSM classes of graph hypersurfaces

Theorem (—) e edges of G, satisfying unfortunate technical conditions. Then

CG2e (t) = (2t − 1)CG (t) − t(t − 1)CGre (t) + CG/e (t)

G2e : double edge e G r e: delete e G/e: contract e

Question: are the ‘unfortunate technical conditions’ necessary?

25 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory ‘Twisted’ algebro-geometric Feynman rules?

Do relations among cSM classes of relevant loci imply relations of corresponding Feynman amplitudes? ···

Characteristic classes in Intersection theory

cSM classes of graph hypersurfaces

Several natural questions!

Can construct a “cSM class” in twisted cohomology?

26 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory Do relations among cSM classes of relevant loci imply relations of corresponding Feynman amplitudes? ···

Characteristic classes in Intersection theory

cSM classes of graph hypersurfaces

Several natural questions!

Can construct a “cSM class” in twisted cohomology? ‘Twisted’ algebro-geometric Feynman rules?

26 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory Characteristic classes in Intersection theory

cSM classes of graph hypersurfaces

Several natural questions!

Can construct a “cSM class” in twisted cohomology? ‘Twisted’ algebro-geometric Feynman rules?

Do relations among cSM classes of relevant loci imply relations of corresponding Feynman amplitudes? ···

26 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory Characteristic classes in Intersection theory

cSM classes of graph hypersurfaces

Thank you for your attention!

27 / 26 Paolo Aluﬃ Characteristic classes in Intersection theory