Local invariant cycle theorem explained

p

from a Kähler manifold

X

to the unit disk that has maximal rank everywhere except over 0, each cohomology class on

p-1(t),t\ne0

is the restriction of some cohomology class on the entire

X

if the cohomology class is invariant under a circle action (monodromy action); in short,

\operatorname{H}*(X)\to\operatorname{H}*(p-1

S1
(t))
is surjective. The conjecture was first proved by Clemens. The theorem is also a consequence of the BBD decomposition.

X\toS

over the spectrum

S

of the henselization of

k[T]

,

k

an algebraically closed field, if

X

is essentially smooth over

k

and

X\overline{η

} smooth over

\overline{η}

, then the homomorphism on

Q

-cohomology:
*(X
\operatorname{H}
s)

\to

*(X
\operatorname{H}
\overline{η
})^is surjective, where

s,η

are the special and generic points and the homomorphism is the composition
*(X
\operatorname{H}
s)

\simeq\operatorname{H}*(X)\to

*(X
\operatorname{H}
η

)\to

*(X
\operatorname{H}
\overline{η
}).

See also

References

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