Skeleton and Dual Complex Chenyang Xu Beijing International Center - PowerPoint PPT Presentation

Skeleton and Dual Complex Computation of dual complex Skeleton and Dual Complex Chenyang Xu Beijing International Center of Mathematics Research Papetee, 2015 August Chenyang Xu Skeleton and Dual Complexx

Berkovich skeleton Skeleton and Dual Complex Essential skeleton after KS and MN Computation of dual complex Dual complex of a minimal model Let K = k (( t )) , where char ( k ) = 0 and R = k [[ t ]] . We fix a t -adic absolute value on K by setting | t | K = 1 / e . Let X be a smooth proper variety over K and X a proper model over R . Denote by X k special fiber and X red the k reduced special fiber. We assume X is obtained from a base change of an algebraic model over O p , C , where C is a curve over k and p ∈ C is a k point. Let X an be the analytification defined by Berkovich. Chenyang Xu Skeleton and Dual Complexx

Berkovich skeleton Skeleton and Dual Complex Essential skeleton after KS and MN Computation of dual complex Dual complex of a minimal model Assume ( X , X red k ) is simple normal crossing. Let = � E i . X red k Then we can define the dual complex D ( X red k ) in the following way: For each component E i of E , we put a vertex v E i ; for each irreducible component of v E i ∩ v E j , we associate an edge connecting v E i and v E j ; and for each irreducible component of v E i ∩ v E j ∩ v E k , we associate a 2-dimensional face, etc.. Eventually, we obtain a cell complex. Properties needed on � E i : each component of the intersections has the expected dimension; the irreducible components of � E i coincide with the connected components. Chenyang Xu Skeleton and Dual Complexx

Berkovich skeleton Skeleton and Dual Complex Essential skeleton after KS and MN Computation of dual complex Dual complex of a minimal model Theorem (Berkovich, Thuillier) k ) is simple normal crossing, X an has a strong If ( X , X red deformation ρ X : X an → Sk ( X ) ≃ D ( X red k ) . Corollary (Arapura-Bakhtary-Wlodarczyk, Payne, Stepanov, Thuillier) D ( X red 1 , k ) and D ( X red 2 , k ) are (simple)-homotopical equivalent to each other if X i are two snc models. Chenyang Xu Skeleton and Dual Complexx

Berkovich skeleton Skeleton and Dual Complex Essential skeleton after KS and MN Computation of dual complex Dual complex of a minimal model The corollary can be proved by using Theorem (Abramovich-Karu-Matsuki-Wlodarczyk) If f i : ( X i , X i , k ) are two snc birational models of X. Then there exists a sequence of admissible blow ups X 1 = Y 1 ��� Y 2 ��� · · · ��� Y n = X 2 , such that Y i ��� Y i + 1 is an admissible blow up or its inverse. Chenyang Xu Skeleton and Dual Complexx

Berkovich skeleton Skeleton and Dual Complex Essential skeleton after KS and MN Computation of dual complex Dual complex of a minimal model Fix ( X , ω ) where ω is a m -th pluri-canonical form, Kontsevich-Soibelman (2006) and Musta¸ tˇ a-Nicaise (2012) define some interesting (sub)-skeleta via a weight function. Let x ∈ X an be a monomial point, the weight function wt ω ( x ) = v x ( div X ( ω ) + m ( X red k )) . If E i is a divisor on the special fiber with multiplicity N i in X k , and let µ i = m + order ω ( E i ) , then wt ω ( x E i ) = µ i / N i . Chenyang Xu Skeleton and Dual Complexx

Berkovich skeleton Skeleton and Dual Complex Essential skeleton after KS and MN Computation of dual complex Dual complex of a minimal model wt ω can be naturally extended as a lower semi-continuous function on X an , by wt ω ( x ) = Sup X { wt ω ( ρ X ( x )) } ∈ R ∪ { + ∞} . We define the Kontsevich-Soibelman skeleton Sk ( X , ω ) = { p ∈ X an | wt ω ( X )( p ) takes the minimum } and the essential skeleton Sk ( X ) = � ω Sk ( X , ω ) where ω runs over all pluri-canonical forms. Sk ( X , ω ) ⊂ Sk ( X ) for any snc model X . Chenyang Xu Skeleton and Dual Complexx

Berkovich skeleton Skeleton and Dual Complex Essential skeleton after KS and MN Computation of dual complex Dual complex of a minimal model Assume that | ω ⊗ m | is base point free for some m ∈ Z > 0 , X i.e., X is a good minimal model. For example X is a CY manifold, i.e., ω X ∼ O X . Applying the minimal model program, we can construct a minimal model X min over R such that m ( K X min + X min , red ) is k base point free over R for some m ∈ Z > 0 . Caveat: ( X min , X min , red ) has divisorial log terminal (dlt) k singularities. Nevertheless, we can define D ( X min , red ) . k Chenyang Xu Skeleton and Dual Complexx

Berkovich skeleton Skeleton and Dual Complex Essential skeleton after KS and MN Computation of dual complex Dual complex of a minimal model Theorem (Nicaise-X. 2013) D ( X min , red ) ≃ Sk ( X ) . k D ( X min , red ) ⊂ Sk ( X ) is easy. k For other side, assume x ∈ Sk ( X , ω ) . If red ( x ) ∈ X min , snc , then x ∈ D ( X min , red ) . k If red ( x ) is not in X min , snc . Let θ be a generator of ω ⊗ m ( mX k ) and write ω = g · θ . Then wt ω ( x ) > In | g ( x ) | ≥ − In | g ( x ′ ) | = wt ω ( x ′ ) for some divisorial point x ′ corresponding to a component of X min , red . k Chenyang Xu Skeleton and Dual Complexx

Berkovich skeleton Skeleton and Dual Complex Essential skeleton after KS and MN Computation of dual complex Dual complex of a minimal model Corollary D ( X min , red ) and D ( X min , red ) are homeomorphism to each other if 1 , k 2 , k X red are two minimal models. i This can be proved by weak factorization, dlt properties and the + X min , red fact that the pull back of K X min on a common i , k i resolution are the same. Chenyang Xu Skeleton and Dual Complexx

Berkovich skeleton Skeleton and Dual Complex Essential skeleton after KS and MN Computation of dual complex Dual complex of a minimal model Theorem (de Fernex-Kollár-X. 2012) k ) collapses to D ( X min , red D ( X red ) . k The Theorem is proved by tracking how dual complexes vary during the minimal model program. Corollary X an has a strong deformation retract to Sk ( X ) . Chenyang Xu Skeleton and Dual Complexx

Rationally connected varieties Skeleton and Dual Complex Calabi-Yau manifolds Computation of dual complex Character variety Theorem (de Fernex-Kollár-X. 2012) If X is a rationally connected varieties, then Sk ( X ) is contractible to a point for any snc model X . Brown-Foster generalizes this result to a relative setting. Chenyang Xu Skeleton and Dual Complexx

Rationally connected varieties Skeleton and Dual Complex Calabi-Yau manifolds Computation of dual complex Character variety Theorem (Kollár-Kovács 2009) The essential skeleton of a Calabi-Yau is a pseudo-manifold with boundary. Question Is the essential skeleton of a Calabi-Yau always a finite quotient of a sphere? Question (Kontsevich-Soibelman) If X is a simply connected CY admitted a minimal semi-stable degeneration. Assume dim D ( X k ) = dim ( X ) − 1. Is D ( X k ) a PL-sphere? Chenyang Xu Skeleton and Dual Complexx

Rationally connected varieties Skeleton and Dual Complex Calabi-Yau manifolds Computation of dual complex Character variety Consider a projective dlt pair ( D , E ) such that K D + E ∼ Q 0, we call it a log Calabi-Yau (CY) . Example: Let ( X , X red 0 ) be a minimal degeneration of Calabi-Yau, i.e. ( X , red ( X 0 )) is dlt and K X + X red 0 ) ∼ Q 0. If D is a component of X 0 , then ( D , E ) is a log CY, where � K X + X red � | D = K D + E . 0 D ( E ) is the link of D ( X red 0 ) at v E . Question Is D ( E ) a finite quotient of a sphere? Chenyang Xu Skeleton and Dual Complexx

Rationally connected varieties Skeleton and Dual Complex Calabi-Yau manifolds Computation of dual complex Character variety Theorem (Kollár-X. 2015) Let ( D , E ) be a log CY. Assume dim ( D ( E )) ≥ 2 . Then H i ( D ( E ) , Q ) = 0 for 1 ≤ i ≤ dim D ( E ) − 1 . 1 There is a surjection π 1 ( D sm ) → π 1 ( D ( E )) . 2 The profinite completion is ˆ π 1 ( D ( E )) finite. 3 the finite cover of D ( E ) given by ˆ π 1 ( D ( E )) is the dual 4 complex of a log CY. We will only discuss the case that ( D , E ) is snc and dim ( D ( E )) = dim ( D ) − 1. Then (2)-(4) just says that π 1 ( D ) = π 1 ( D ( E )) = { e } . Chenyang Xu Skeleton and Dual Complexx

Rationally connected varieties Skeleton and Dual Complex Calabi-Yau manifolds Computation of dual complex Character variety Vanishing of rational cohomology There is an injection H i ( D ( E ) , C ) → H i ( E , O E ) . We have the exact sequence H i ( O X ( − E )) → H i ( O X ) → H i ( O E ) , and H i ( O X ( − E )) ∼ = H dim X − i ( O X ) , when ( X , E ) is a log CY, the first two terms vanish. Chenyang Xu Skeleton and Dual Complexx

Rationally connected varieties Skeleton and Dual Complex Calabi-Yau manifolds Computation of dual complex Character variety Change the models Theorem (Maximal Boundary Theorem, Kollár-X) There is a birational model ( G , ∆) of ( D , E ) such that ( G , ∆) is log canonical, ( D , E ) and ( G , ∆) are crepant 1 birationally equivalent. ∆ supports a divisor H which is ample over a variety Z 2 such that dim Z ≤ dim ( D ) − dim ( D ( E )) − 1 . Furthermore, D ��� G is isomorphic over G \ ∆ . 3 Chenyang Xu Skeleton and Dual Complexx