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3
votes
0
answers
78
views
Cup-product ($\cup$) vs. cross-product ($\times$) on the space of graph homology
I'm currently reading Nieper-Wißkirchen's Chern numbers and Rozansky-Witten invariants of compact Hyper-Kähler manifolds. He introduces the space of graph homology $\mathcal B$ as the free $K[\circ]$- …
1
vote
1
answer
134
views
Are horizontal divisors on abelian fibered hyperkähler manifolds proportional in $NS(X)$ up ...
Oguiso writes[1]
Theorem 1.1 Let $f: X \to \mathbf P^n$ be an abelian fibered HK [hyperkähler] manifold. Let $K = \mathbf C(\mathbf P^n)$ and let $A_k$ be the generic fiber of $f$. Then, $\rho(A_K)= …
2
votes
0
answers
145
views
Describing singular fibers of the lagrangian fibration $\mathcal M^s(0, [C], 1) \to |C|$
Let $S \to \mathbb P^2$ be a two-to-one cover branched over a sextic, i.e. $S$ is a K3-surface. Let $C \subset S$ be the preimage of a (smooth) quadric, so that by Hurwitz' formula, $g(C) = 5$. Accord …
4
votes
0
answers
214
views
Can Lagrangian fibrations have multiple fibres in codimension $1$?
I know that if $\pi: S \to \mathbb P^1$ is an elliptic fibration of a K3-surface $S$, then $\pi$ does not have multiple fibers. A proof of this can be found in Huybrechts' Lectures on K3 surfaces, Cha …
2
votes
Accepted
Does miracle flatness always fail for a non-regular base?
The answer lies in Theorem 23.7 from Matsumura's Commutative Ring Theory:
Theorem 23.7. Let $(A, \mathfrak m, k)$ and $(B, \mathfrak n, k')$ be local Noetherian local rings, and $A \to B$ a local hom …
3
votes
1
answer
306
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Does miracle flatness always fail for a non-regular base?
In [1], Huybrechts and Mauri argue that a holomorphic Lagrangian fibration $f: X \to B$ with smooth base $B$ is flat. This is an application of so called miracle flatness [2, Thm 23.1], because Lagran …