Mixing
Trivial canonical divisor of the Calabi--Yau fibres of a holomorphic submersion between compact Kähler...
$begingroup$
Let $X^{n+m}$ and $B^m$ be two compact Kähler manifolds of respective (complex) dimension $n+m$ and $m$, for $m > 0$. Let $pi : X to B$ be a holomorphic submersion with connected Calabi$-$Yau fibers and suppose $c_1(B) < 0$.
The following is well-known: The relative pluricanonical bundle of $pi$ is $$K_{X/B}^{ell} = K_X^{ell} otimes (pi^{ast} K_B^{ell})^{ast},$$ where $ell$ is any positive integer. By the projection formula, we know that $$pi_{ast}(K_X^{ell}) = (pi_{ast} (K_{X/B}^{ell})) otimes K_B^{ell},$$ and when restricted to any fibre $X_y = pi^{-1}(y)$, $$K_{X/B}^{ell} vert_{X_y} cong K_{X_y}^{ell}.$$
Note that since $X_y$ is Calabi$-$Yau, there is a positive integer $ell$ such that $K_{X_y}^{ell}$ is trivial.
Question: Can $ell$ be taken to be $ell =1$? By Calabi$-$Yau, at least in the context of Kähler geometry, we always mean $c_1(X_y) = -c_1(K_{X_y}) =0$, but perhaps when considering Calabi$-$Yau fibres, only a sufficiently large power of the canonical divisor is trivial?
If we do indeed know only that a sufficiently large power of the canonical divisor is trivial, can someone provide an example of this situation where $K_{X_y}$ need not be trivial?
References:
Fong, F. T.-H., Zhang, Z., The collapsing rate of the Kähler--Ricci flow with regular infinite time singularity, J. reine angew. Math. 703 (2015), 95$-$113.
Tosatti, V., Weinkove, B., Yang, X., The Kähler--Ricci flow, Ricci-flat metrics and collapsing limits, arXiv: 1408.0161v2 (2017)
algebraic-geometry riemannian-geometry kahler-manifolds
asked Jan 26 at 22:39
Kyle BroderKyle Broder
437
$endgroup$
add a comment |
$begingroup$
Let $X^{n+m}$ and $B^m$ be two compact Kähler manifolds of respective (complex) dimension $n+m$ and $m$, for $m > 0$. Let $pi : X to B$ be a holomorphic submersion with connected Calabi$-$Yau fibers and suppose $c_1(B) < 0$.
The following is well-known: The relative pluricanonical bundle of $pi$ is $$K_{X/B}^{ell} = K_X^{ell} otimes (pi^{ast} K_B^{ell})^{ast},$$ where $ell$ is any positive integer. By the projection formula, we know that $$pi_{ast}(K_X^{ell}) = (pi_{ast} (K_{X/B}^{ell})) otimes K_B^{ell},$$ and when restricted to any fibre $X_y = pi^{-1}(y)$, $$K_{X/B}^{ell} vert_{X_y} cong K_{X_y}^{ell}.$$
Note that since $X_y$ is Calabi$-$Yau, there is a positive integer $ell$ such that $K_{X_y}^{ell}$ is trivial.
Question: Can $ell$ be taken to be $ell =1$? By Calabi$-$Yau, at least in the context of Kähler geometry, we always mean $c_1(X_y) = -c_1(K_{X_y}) =0$, but perhaps when considering Calabi$-$Yau fibres, only a sufficiently large power of the canonical divisor is trivial?
If we do indeed know only that a sufficiently large power of the canonical divisor is trivial, can someone provide an example of this situation where $K_{X_y}$ need not be trivial?
References:
Fong, F. T.-H., Zhang, Z., The collapsing rate of the Kähler--Ricci flow with regular infinite time singularity, J. reine angew. Math. 703 (2015), 95$-$113.
Tosatti, V., Weinkove, B., Yang, X., The Kähler--Ricci flow, Ricci-flat metrics and collapsing limits, arXiv: 1408.0161v2 (2017)
algebraic-geometry riemannian-geometry kahler-manifolds
asked Jan 26 at 22:39
Kyle BroderKyle Broder
437
$endgroup$
2
$begingroup$
Enriques surfaces have $K^2=0neq K$.
$endgroup$
– Oven
Jan 26 at 22:56
$begingroup$
@Oven Thanks, I missed such a simple example!
$endgroup$
– Kyle Broder
Feb 2 at 2:59
add a comment |
$begingroup$
Let $X^{n+m}$ and $B^m$ be two compact Kähler manifolds of respective (complex) dimension $n+m$ and $m$, for $m > 0$. Let $pi : X to B$ be a holomorphic submersion with connected Calabi$-$Yau fibers and suppose $c_1(B) < 0$.
The following is well-known: The relative pluricanonical bundle of $pi$ is $$K_{X/B}^{ell} = K_X^{ell} otimes (pi^{ast} K_B^{ell})^{ast},$$ where $ell$ is any positive integer. By the projection formula, we know that $$pi_{ast}(K_X^{ell}) = (pi_{ast} (K_{X/B}^{ell})) otimes K_B^{ell},$$ and when restricted to any fibre $X_y = pi^{-1}(y)$, $$K_{X/B}^{ell} vert_{X_y} cong K_{X_y}^{ell}.$$
Note that since $X_y$ is Calabi$-$Yau, there is a positive integer $ell$ such that $K_{X_y}^{ell}$ is trivial.
Question: Can $ell$ be taken to be $ell =1$? By Calabi$-$Yau, at least in the context of Kähler geometry, we always mean $c_1(X_y) = -c_1(K_{X_y}) =0$, but perhaps when considering Calabi$-$Yau fibres, only a sufficiently large power of the canonical divisor is trivial?
If we do indeed know only that a sufficiently large power of the canonical divisor is trivial, can someone provide an example of this situation where $K_{X_y}$ need not be trivial?
References:
Fong, F. T.-H., Zhang, Z., The collapsing rate of the Kähler--Ricci flow with regular infinite time singularity, J. reine angew. Math. 703 (2015), 95$-$113.
Tosatti, V., Weinkove, B., Yang, X., The Kähler--Ricci flow, Ricci-flat metrics and collapsing limits, arXiv: 1408.0161v2 (2017)
algebraic-geometry riemannian-geometry kahler-manifolds
asked Jan 26 at 22:39
Kyle BroderKyle Broder
437
$endgroup$
Let $X^{n+m}$ and $B^m$ be two compact Kähler manifolds of respective (complex) dimension $n+m$ and $m$, for $m > 0$. Let $pi : X to B$ be a holomorphic submersion with connected Calabi$-$Yau fibers and suppose $c_1(B) < 0$.
The following is well-known: The relative pluricanonical bundle of $pi$ is $$K_{X/B}^{ell} = K_X^{ell} otimes (pi^{ast} K_B^{ell})^{ast},$$ where $ell$ is any positive integer. By the projection formula, we know that $$pi_{ast}(K_X^{ell}) = (pi_{ast} (K_{X/B}^{ell})) otimes K_B^{ell},$$ and when restricted to any fibre $X_y = pi^{-1}(y)$, $$K_{X/B}^{ell} vert_{X_y} cong K_{X_y}^{ell}.$$
Note that since $X_y$ is Calabi$-$Yau, there is a positive integer $ell$ such that $K_{X_y}^{ell}$ is trivial.
Question: Can $ell$ be taken to be $ell =1$? By Calabi$-$Yau, at least in the context of Kähler geometry, we always mean $c_1(X_y) = -c_1(K_{X_y}) =0$, but perhaps when considering Calabi$-$Yau fibres, only a sufficiently large power of the canonical divisor is trivial?
If we do indeed know only that a sufficiently large power of the canonical divisor is trivial, can someone provide an example of this situation where $K_{X_y}$ need not be trivial?
References:
Fong, F. T.-H., Zhang, Z., The collapsing rate of the Kähler--Ricci flow with regular infinite time singularity, J. reine angew. Math. 703 (2015), 95$-$113.
Tosatti, V., Weinkove, B., Yang, X., The Kähler--Ricci flow, Ricci-flat metrics and collapsing limits, arXiv: 1408.0161v2 (2017)
algebraic-geometry riemannian-geometry kahler-manifolds
algebraic-geometry riemannian-geometry kahler-manifolds
asked Jan 26 at 22:39
Kyle BroderKyle Broder
437
asked Jan 26 at 22:39
Kyle BroderKyle Broder
437
asked Jan 26 at 22:39
Kyle BroderKyle Broder
437
asked Jan 26 at 22:39
Kyle BroderKyle Broder
437
asked Jan 26 at 22:39
Kyle BroderKyle Broder
437
437
2
$begingroup$
Enriques surfaces have $K^2=0neq K$.
$endgroup$
– Oven
Jan 26 at 22:56
$begingroup$
@Oven Thanks, I missed such a simple example!
$endgroup$
– Kyle Broder
Feb 2 at 2:59
add a comment |
2
$begingroup$
Enriques surfaces have $K^2=0neq K$.
$endgroup$
– Oven
Jan 26 at 22:56
$begingroup$
@Oven Thanks, I missed such a simple example!
$endgroup$
– Kyle Broder
Feb 2 at 2:59
2
2
$begingroup$
Enriques surfaces have $K^2=0neq K$.
$endgroup$
– Oven
Jan 26 at 22:56
$begingroup$
Enriques surfaces have $K^2=0neq K$.
$endgroup$
– Oven
Jan 26 at 22:56
$begingroup$
@Oven Thanks, I missed such a simple example!
$endgroup$
– Kyle Broder
Feb 2 at 2:59
$begingroup$
@Oven Thanks, I missed such a simple example!
$endgroup$
– Kyle Broder
Feb 2 at 2:59
add a comment |
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$begingroup$
Enriques surfaces have $K^2=0neq K$.
$endgroup$
– Oven
Jan 26 at 22:56
$begingroup$
@Oven Thanks, I missed such a simple example!
$endgroup$
– Kyle Broder
Feb 2 at 2:59