Hodge Theory and Moduli ∗ Phillip Griffiths ∗ Talk given in Miami at the inaugural conference for the Institute of the Mathematical Sciences of the Americas (IMSA) on September 7, 2019. Based in part on joint work in progress with Mark Green, Radu Laza, Colleen Robles and on discussions with Marco Franciosi, Rita Pardini and S¨ onke Rollenske (FPR). 1/40 1 / 40
Outline I. Introduction II. Hodge theory III. Moduli IV. I -surfaces and M I Both Hodge theory and birational geometry/moduli are highly developed subjects in their own right. The theme of this talk will be on the uses of Hodge theory to study an interesting geometric question and to illustrate how this works in one particular non-classical example of an algebraic surface. 2/40 2 / 40
I.A. Introduction ◮ classification of algebraic varieties is a central problem in the algebraic geometry (minimal model program) discrete invariants Kodaira dimension ✟ ✟✟✟ Chern numbers ◮ It falls into two parts � ❍❍❍ continuous invariants ❍ moduli space M . ◮ Under the second part a basic issue is What singular varieties does one add to construct a canonical completion M of M ? 3/40 3 / 40
◮ Basically, given a family { X t } t ∈ ∆ ∗ of smooth varieties, how can one determine a unique limit X 0 ? ◮ A fundamental invariant of a smooth variety X is the Hodge structure Φ( X ) given by linear algebra data on its cohomology H ∗ ( X ). how Φ( X ) varies in families ✑ ◮ one knows ✑✑✑ how to define Φ( X ) when X is singular ◗◗◗ ◗ how to uniquely define lim Φ( X t ) for { X t } t ∈ ∆ ∗ t → 0 4/40 4 / 40
Goal: Use Hodge theory in combination with standard algebraic geometry to help understand M (A) general theory ✟ ✟✟✟ ◮ two aspects ❍❍❍ ❍ (B) interesting examples 5/40 5 / 40
◮ under (B) there are ◮ the classical case (curves, abelian varieties, K3’s, hyperK¨ ahlers, cubic 4-fold) — space of Hodge structures is a Hermitian symmetric domain ◮ some results for Calabi-Yau varieties (especially those motivated by physics) ◮ existence of M for X ’s of general type — not yet any examples of ∂ M (the global structure the singular X ’s nor the stratification of M \ M ). ◮ First non-classical general type surface is the I-surface ( p g ( X ) = 2, q ( X ) = 0, K 2 X = 1 , dim M I = 28) — informally stated we have the 6/40 6 / 40
Main result: The extended period mapping Φ e : M I → P has degree 1 and faithfully captures the boundary structure of Gor M . I Gor ◮ Analysis of M was initiated by FPR — first case I beyond M g ( g ≧ 2) where the boundary structure of the Koll´ ar-Shepherd-Barron-Alexeev (KSBA) canonical Gor ⊂ M is understood. completion M ◮ Hodge theory (using Lie theory, differential geometry, complex analysis) gives us P ⊃ P = Φ( M I ) — the result says that Φ extends to Φ e and the stratification of P Gor determines that of M — the non-Gorenstein case is I only partially understood. 7/40 7 / 40
II. Hodge theory A. Selected uses of Hodge theory These include ◮ topology of algebraic varieties: � � smooth case (PHS’s) — (Hard Lefschetz) ✑ ✑✑✑ � � singular case (MHS’s) — also general case, relative case ◗◗◗ ◗ families of algebraic varieties (LMHS’s) — monodromy (local and global) 8/40 8 / 40
◮ geometry of algebraic varieties: � � Torelli questions; rationality and stable rationality; character varieties ✏ ✏✏✏ � � algebraic cycles — conjectures ◗◗◗ of Hodge and Beilinson-Bloch ◗ � direct study of the geometry � of algebraic varieties/Riemann Θ-divisor, IVHS classical case ✟ ✟✟✟ ◮ moduli of algebraic varieties ❍❍❍ ❍ non-classical case 9/40 9 / 40
We will see that geometry, analysis and topology enter here. Not discussed in this talk are other interesting uses of Hodge theory including: mirror symmetry ✏ ✏✏✏ ◮ physics � � PPP homological mirror symmetry — P Landau-Ginsberg models etc. ◮ Hodge theory and combinatorics 10/40 10 / 40
B. Objects of Hodge theory These include ◮ polarized Hodge structures ( V , Q , F ) (PHS’s); ◮ period domains D and period mappings Φ : B → Γ \ D ; ◮ first order variation ( V , Q , F , T , δ ) of PHS’s (IVHS); ◮ mixed Hodge structures ( V , W , F ) ◮ limiting mixed Hodge structures ( V , W ( N ) , F lim ) (LMHS’s); ◮ IVLMHS. All of these enter in the result mentioned above. 11/40 11 / 40
PHS ( V , Q , F ) of weight n ◮ F n ⊂ F n − 1 ⊂ · · · ⊂ F 0 = V C Hodge filtration satisfying F p ⊕ F n − p +1 ∼ − → V C 0 ≦ p ≦ n ◮ setting V p , q = F p ∩ F q , this is equivalent to a Hodge decomposition V p , q = V q , p . p + q = n V p , q , V C = ⊕ Given such a decomposition F p = ⊕ p ′ ≧ p V p ′ , q gives a Hodge filtration. 12/40 12 / 40
� (HRI) Q ( F p , F n − p +1 ) = 0 ✟ ✟ ◮ Hodge-Riemann bilinear relations � ❍ ❍ (HRII) p ) > 0 i p − q ( Q )( F p , F Notes: One usually defines Hodge structures ( V , F ) without reference to a Q and HRI, II — only HS’s I have seen used in algebraic geometry are polarizable — PHS’s form a semi-simple category — in practice there is also usually a lattice V Z ⊂ V . 13/40 13 / 40
Example: The cohomology H n ( X , Q ) of a smooth, projective variety is a polarizable Hodge structure of weight n . The class L ⊂ H 2 ( X , Q ) of an ample line bundle satisfies L k : H n − k ( X , Q ) → H n + k ( X , Q ) ∼ − (Hard Lefschetz) This then completes to the action of an sl 2 { L , H , Λ } on H ∗ ( X , Q ). This is the “tip of the iceberg” for the uses of the Lie theory in Hodge theory. Note: The reason for using the Hodge filtration rather than the Hodge decomposition is that F varies holomorphically with X . 14/40 14 / 40
Period mapping Φ : B → Γ \ D : For given ( V , Q ) and h p , q ’s ◮ period domain D = { ( V , Q , F ) = PHS, dim V p , q = h p , q } ◮ D = G R / H where G = Aut ( V , Q ), H = compact isotropy group of a fixed PHS. Example: D = H = { z ∈ C : Im z > 0 } = SL 2 ( R ) / SO (2) ◮ period mapping is given by a complex manifold B and a holomorphic mapping Φ · B → Γ \ D where Γ ⊂ G Z and ρ : π 1 ( B ) → Γ is the induced map on fundamental groups. 15/40 15 / 40
MHS: ( V , W , F ) ◮ F k ⊂ F k − 1 ⊂ · · · ⊂ F 0 = V C Hodge filtration ◮ W 0 ⊂ W 1 ⊂ · · · ⊂ W ℓ = V weight filtration ◮ F induces a HS of weight m on Gr W m V = W m / W m − 1 MHS’s form an abelian category. A most useful property is that morphisms ψ ( V , W , F ) − → ( V ′ , W ′ , F ′ ) are strict ; i.e., � ψ ( V ) ∩ W ′ n = ψ ( W n ) ′ p = ψ ( F p ) . ψ ( V C ) ∩ F Example: For X a complete algebraic variety H n ( X , Q ) has a functorial MHS (where k = ℓ = n ). 16/40 16 / 40
π Example: X − → B is a family of smooth projective varieties X b = π − 1 ( b ) and ρ : π 1 ( B , b 0 ) → Aut ( H n ( X b 0 , Q )) is the monodromy representation. Then Φ( b ) = PHS on H n ( X b , Q ). Special case: B = ∆ ∗ = { t · 0 < | t | < 1 } and we have ◮ ρ (generator) = T ∈ Aut H m ( X b 0 , Q ) T m ss = Id ✏ ✏✏ ◮ T = T ss T u where PP P T u = e N where N m +1 = 0 17/40 17 / 40
LMHS: ( V , W ( N ) , F lim ) is a MHS where ◮ N ∈ End Q ( V ) and N m +1 = 0 gives unique monodromy weight filtration W 0 ( N ) ⊂ W 1 ( N ) ⊂ · · · ⊂ W 2 m ( N ) satisfying � N : W k ( N ) → W k − 2 ( N ) N k : Gr W ( N ) → Gr W ( N ) ∼ m + k ( V ) − m − k ( V ); ◮ N : F p lim → F p − 1 lim . Example: Above example where B = ∆ ∗ — here Γ = { T } . 18/40 18 / 40
Classic Example: X is a compact Riemann surface of genus 1 ◮ topological picture γ δ y 2 = x 3 + a ( t ) x 2 + b ( t ) x + c ( t ), algebraic picture ◮ ω = dx / y w ◮ C / Z · w + Z analytic picture 1 ◮ w = ´ ´ γ ω/ δ ω . 19/40 19 / 40
◮ The space of LMHS’s ( V , Q , W ( N ) , F lim ) has a symmetry group ◮ G acts on conjugacy classes of N ’s; ◮ G C acts transitively on ˇ D = { ( V , F ) : Q ( F p , F n − p +1 ) = 0 } ; ◮ F lim ∈ ˇ D . Thus one may imagine using Lie-theoretic methods to attach to the space Γ \ D of Γ- equivalence classes of PHS’s a set of equivalence classes of LMHS’s — then informally stated one has the result the images P ⊂ Γ \ D of global period mappings have natural completions P . The proof that P has the structure of a projective variety is a work in progress. 20/40 20 / 40
Example: The moduli space of elliptic curves j algebro-geometric object M 1 − → C � Hodge theoretic object SL 2 ( Z ) \ H completes by adding ∞ corresponding to the LMHS associated to ✲ ◮ ✲ ◮ y 2 = x ( x − t )( x − 1) − → y 2 = x 2 ( x − 1) 21/40 21 / 40
w w → i ∞ C / Z w + Z → C / Z ∼ = C ∗ IVHS: ( V , F , T , δ ) where ( V , F ) is a HS and � � � p Gr p F V C , Gr p − 1 δ : T → ⊕ Hom V C F [ δ, δ ] = 0 . 22/40 22 / 40
Example: Φ ∗ for a period mapping. Example: For M g , g ≧ 3, the IVHS is equivalent to the quadrics through the canonical curve C → P g − 1 = P H 0 (Ω 1 C ) ∗ . Example (work in progress): The equation of a smooth I -surface can be reconstructed from Φ ∗ . 23/40 23 / 40
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