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On the connectedness principle and dual complexes for generalized pairs
Authors
Stefano Filipazzi
Roberto Svaldi
Publication date
1 December 2020
Publisher
View
on
arXiv
Abstract
Let
(
X
,
B
)
(X,B)
(
X
,
B
)
be a pair, and let
f
ββ£
:
X
β
S
f \colon X \rightarrow S
f
:
X
β
S
be a contraction with
β
(
K
X
+
B
)
-(K_X + B)
β
(
K
X
β
+
B
)
nef over
S
S
S
. A conjecture, known as the Shokurov-Koll\'{a}r connectedness principle, predicts that
f
β
1
(
s
)
β©
N
k
l
t
(
X
,
B
)
f^{-1} (s) \cap \mathrm{Nklt}(X,B)
f
β
1
(
s
)
β©
Nklt
(
X
,
B
)
has at most two connected components, where
s
β
S
s \in S
s
β
S
is an arbitrary schematic point and
N
k
l
t
(
X
,
B
)
\mathrm{Nklt}(X,B)
Nklt
(
X
,
B
)
denotes the non-klt locus of
(
X
,
B
)
(X,B)
(
X
,
B
)
. In this work, we prove this conjecture, characterizing those cases in which
N
k
l
t
(
X
,
B
)
\mathrm{Nklt}(X,B)
Nklt
(
X
,
B
)
fails to be connected, and we extend these same results also to the category of generalized pairs. Finally, we apply these results and the techniques to the study of the dual complex for generalized log Calabi-Yau pairs, generalizing results of Koll\'{a}r-Xu and Nakamura.Comment: Minor correction
Similar works
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oai:arXiv.org:2010.08018
Last time updated on 30/10/2020