The monodromy theorem for compact K ahler manifolds and smooth - PowerPoint PPT Presentation

The monodromy theorem for compact K¨ ahler manifolds and smooth quasi-projective varieties Yongqiang (Ted) Liu (Joint work with Nero Budur and Botong Wang) KU Leuven Geometry, Algebra and Combinatorics of Moduli Spaces and Configurations Dobbiaco, February 22 2017 Yongqiang Liu (KU Leuven) The Monodromy Theorem Dobbiaco, February 22 2017 1 / 17

Overview Motivation 1 Milnor fibration The classical Monodromy Theorem Yongqiang Liu (KU Leuven) The Monodromy Theorem Dobbiaco, February 22 2017 2 / 17

Overview Motivation 1 Milnor fibration The classical Monodromy Theorem Main Results 2 Yongqiang Liu (KU Leuven) The Monodromy Theorem Dobbiaco, February 22 2017 2 / 17

Motivation Milnor fibration Singularities Let f : ( C n , 0) → ( C , 0) be a germ of analytic function. Yongqiang Liu (KU Leuven) The Monodromy Theorem Dobbiaco, February 22 2017 3 / 17

Motivation Milnor fibration Singularities Let f : ( C n , 0) → ( C , 0) be a germ of analytic function. f is called singular at 0, if ∂ f (0) = 0. Yongqiang Liu (KU Leuven) The Monodromy Theorem Dobbiaco, February 22 2017 3 / 17

Motivation Milnor fibration Singularities Let f : ( C n , 0) → ( C , 0) be a germ of analytic function. f is called singular at 0, if ∂ f (0) = 0. Set V f = ( { f = 0 } , 0) ⊂ ( C n , 0). Yongqiang Liu (KU Leuven) The Monodromy Theorem Dobbiaco, February 22 2017 3 / 17

Motivation Milnor fibration Singularities Let f : ( C n , 0) → ( C , 0) be a germ of analytic function. f is called singular at 0, if ∂ f (0) = 0. Set V f = ( { f = 0 } , 0) ⊂ ( C n , 0). Definition We say that two analytic germ f and g have the same topological type if there is a homeomorphism ϕ : ( C n , 0) → ( C n , 0) such that ϕ ( V f ) = V g . Yongqiang Liu (KU Leuven) The Monodromy Theorem Dobbiaco, February 22 2017 3 / 17

Motivation Milnor fibration Milnor fibration Theorem (J. Milnor 1968) Let f : ( C n , 0) → ( C , 0) be a germ of analytic function. Let B ǫ be a small open ball at the origin in C n . Let D δ ⊂ C be a disc around the origin with 0 < δ ≪ ǫ . Set D ∗ δ = D δ \ { 0 } . Then there exists a fibration f : B ǫ ∩ f − 1 ( D ∗ δ ) → D ∗ δ . Yongqiang Liu (KU Leuven) The Monodromy Theorem Dobbiaco, February 22 2017 4 / 17

Motivation Milnor fibration Milnor fibration Theorem (J. Milnor 1968) Let f : ( C n , 0) → ( C , 0) be a germ of analytic function. Let B ǫ be a small open ball at the origin in C n . Let D δ ⊂ C be a disc around the origin with 0 < δ ≪ ǫ . Set D ∗ δ = D δ \ { 0 } . Then there exists a fibration f : B ǫ ∩ f − 1 ( D ∗ δ ) → D ∗ δ . The fibre F = B ǫ ∩ f − 1 ( δ ) is called the Milnor fibre. Yongqiang Liu (KU Leuven) The Monodromy Theorem Dobbiaco, February 22 2017 4 / 17

Motivation Milnor fibration Here is the picture of Milnor fibration from D. Massey: fibration.jpg fibration.jpg Yongqiang Liu (KU Leuven) The Monodromy Theorem Dobbiaco, February 22 2017 5 / 17

Motivation Milnor fibration Example Set f = � n i =1 x 2 i : ( C n , 0) → ( C , 0). Yongqiang Liu (KU Leuven) The Monodromy Theorem Dobbiaco, February 22 2017 6 / 17

Motivation Milnor fibration Example Set f = � n i =1 x 2 i : ( C n , 0) → ( C , 0). Then the Milnor fibre F is diffeomorphic to the total space of the tangent bundle of the sphere S n − 1 , hence F ≃ S n − 1 . Yongqiang Liu (KU Leuven) The Monodromy Theorem Dobbiaco, February 22 2017 6 / 17

Motivation Milnor fibration Monodromy Parallel translation along the path γ : [0 , 1] �→ D δ , t �→ δ e 2 π it gives a homeomorphism h : F → F called the geometric monodromy of the singularity. Yongqiang Liu (KU Leuven) The Monodromy Theorem Dobbiaco, February 22 2017 7 / 17

Motivation Milnor fibration Monodromy Parallel translation along the path γ : [0 , 1] �→ D δ , t �→ δ e 2 π it gives a homeomorphism h : F → F called the geometric monodromy of the singularity. The total space can be identified with F × [0 , 1] / ( x , 1) ∼ ( h ( x ) , 0) for any x ∈ F . Yongqiang Liu (KU Leuven) The Monodromy Theorem Dobbiaco, February 22 2017 7 / 17

Motivation The classical Monodromy Theorem The Monodromy Theorem The geometric monodromy h : F → F induces an linear automorphism h i : H i ( F , C ) → H i ( F , C ) . Yongqiang Liu (KU Leuven) The Monodromy Theorem Dobbiaco, February 22 2017 8 / 17

Motivation The classical Monodromy Theorem The Monodromy Theorem The geometric monodromy h : F → F induces an linear automorphism h i : H i ( F , C ) → H i ( F , C ) . Theorem With the above assumptions and notations, we have that (1) the eigenvalues of h i are all roots of unity for all i. (2) the sizes of the blocks in the Jordan normal form of h i are at most i + 1 . Yongqiang Liu (KU Leuven) The Monodromy Theorem Dobbiaco, February 22 2017 8 / 17

Motivation The classical Monodromy Theorem The Monodromy Theorem The geometric monodromy h : F → F induces an linear automorphism h i : H i ( F , C ) → H i ( F , C ) . Theorem With the above assumptions and notations, we have that (1) the eigenvalues of h i are all roots of unity for all i. (2) the sizes of the blocks in the Jordan normal form of h i are at most i + 1 . There are many different proofs of this theorem by A. Borel, E. Brieskorn, G. M. Greuel, C. H. Clements, P. Deligne, P. A. Griffiths, A. Grothendieck, N. M. Katz, A. Landman, Lˆ e D˜ ung Tr´ ang, E. Looijenga, B. Malgrange, W. Schmid ....... from 70s to 80s. Yongqiang Liu (KU Leuven) The Monodromy Theorem Dobbiaco, February 22 2017 8 / 17

Motivation The classical Monodromy Theorem F is homotopy equivalent to a finite ( n − 1)-dimensional CW complex. The possible maximal size of Jordan block is n . Examples of B. Malgrange (1973) show that the bounds on the sizes of the Jordan blocks are sharp. Yongqiang Liu (KU Leuven) The Monodromy Theorem Dobbiaco, February 22 2017 9 / 17

Motivation The classical Monodromy Theorem Now we give a different point view of the monodromy map. Yongqiang Liu (KU Leuven) The Monodromy Theorem Dobbiaco, February 22 2017 10 / 17

Motivation The classical Monodromy Theorem Now we give a different point view of the monodromy map. Due to the existence of Milnor fibration, one has a short exact sequence: δ ) ∼ 0 → π 1 ( F ) → π 1 ( X ) → π 1 ( S 1 = Z → 0 . Here X = B ǫ ∩ f − 1 ( S 1 δ ) and S 1 δ = ∂ D δ . Yongqiang Liu (KU Leuven) The Monodromy Theorem Dobbiaco, February 22 2017 10 / 17

Motivation The classical Monodromy Theorem Now we give a different point view of the monodromy map. Due to the existence of Milnor fibration, one has a short exact sequence: δ ) ∼ 0 → π 1 ( F ) → π 1 ( X ) → π 1 ( S 1 = Z → 0 . Here X = B ǫ ∩ f − 1 ( S 1 δ ) and S 1 δ = ∂ D δ . The covering space of X can be taken as F × R . Yongqiang Liu (KU Leuven) The Monodromy Theorem Dobbiaco, February 22 2017 10 / 17

Motivation The classical Monodromy Theorem Now we give a different point view of the monodromy map. Due to the existence of Milnor fibration, one has a short exact sequence: δ ) ∼ 0 → π 1 ( F ) → π 1 ( X ) → π 1 ( S 1 = Z → 0 . Here X = B ǫ ∩ f − 1 ( S 1 δ ) and S 1 δ = ∂ D δ . The covering space of X can be taken as F × R . The generator of the deck transformation group acts on F × R as ( h , +1): ( h , +1) : F × R �→ F × R ( x , s ) �→ ( h ( x ) , s + 1) . Yongqiang Liu (KU Leuven) The Monodromy Theorem Dobbiaco, February 22 2017 10 / 17

Motivation The classical Monodromy Theorem Now we give a different point view of the monodromy map. Due to the existence of Milnor fibration, one has a short exact sequence: δ ) ∼ 0 → π 1 ( F ) → π 1 ( X ) → π 1 ( S 1 = Z → 0 . Here X = B ǫ ∩ f − 1 ( S 1 δ ) and S 1 δ = ∂ D δ . The covering space of X can be taken as F × R . The generator of the deck transformation group acts on F × R as ( h , +1): ( h , +1) : F × R �→ F × R ( x , s ) �→ ( h ( x ) , s + 1) . h i H i ( F , C ) → H i ( F C ) h i H i ( F × R , C ) → H i ( F × R , C ) Yongqiang Liu (KU Leuven) The Monodromy Theorem Dobbiaco, February 22 2017 10 / 17

Main Results Let X be either a smooth complex quasi-projective variety or a compact K¨ ahler manifold. Yongqiang Liu (KU Leuven) The Monodromy Theorem Dobbiaco, February 22 2017 11 / 17

Main Results Let X be either a smooth complex quasi-projective variety or a compact K¨ ahler manifold. Fix an epimorphism ρ : π 1 ( X ) → Z → 0. Such kind of map exists if and only if b 1 ( X ) � = 0. Yongqiang Liu (KU Leuven) The Monodromy Theorem Dobbiaco, February 22 2017 11 / 17

Main Results Let X be either a smooth complex quasi-projective variety or a compact K¨ ahler manifold. Fix an epimorphism ρ : π 1 ( X ) → Z → 0. Such kind of map exists if and only if b 1 ( X ) � = 0. Let X ρ denote the corresponding covering space of X , where ρ 0 → π 1 ( X ρ ) → π 1 ( X ) → Z → 0 . Yongqiang Liu (KU Leuven) The Monodromy Theorem Dobbiaco, February 22 2017 11 / 17

Main Results Under the deck group Z action, H i ( X ρ , C ) becomes a finitely generated C [ Z ] = C [ t , t − 1 ]-module. Yongqiang Liu (KU Leuven) The Monodromy Theorem Dobbiaco, February 22 2017 12 / 17