Examples of smooth manifolds admitting inbetween one and a continuum of complex structures Announcing the arrival of Valued Associate #679: Cesar Manara Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern)Nonalgebraic complex manifoldsSmooth and analytic structures on low dimensional euclidian spacesExistence of closed manifolds with more than 3 linearly independent complex structures?Non-Integrable Almost-Complex Structures for Homogeneous SpacesDoes every smoothly embedded surface $mathbbR^3$ inherit a natural complex structure, and if so, which one?Obstructions to deformations of complex manifoldsSpin^c structures on manifolds with almost complex structureOpen subsets of Euclidean space in dimension 5 and higher admitting exotic smooth structuresHave complex manifolds with dual number structure on the holomorphic tangent bundle been studied?Do smooth manifolds admit unique cubical structures?
Examples of smooth manifolds admitting inbetween one and a continuum of complex structures
Announcing the arrival of Valued Associate #679: Cesar Manara
Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern)Nonalgebraic complex manifoldsSmooth and analytic structures on low dimensional euclidian spacesExistence of closed manifolds with more than 3 linearly independent complex structures?Non-Integrable Almost-Complex Structures for Homogeneous SpacesDoes every smoothly embedded surface $mathbbR^3$ inherit a natural complex structure, and if so, which one?Obstructions to deformations of complex manifoldsSpin^c structures on manifolds with almost complex structureOpen subsets of Euclidean space in dimension 5 and higher admitting exotic smooth structuresHave complex manifolds with dual number structure on the holomorphic tangent bundle been studied?Do smooth manifolds admit unique cubical structures?
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For smooth manifolds it is known that they can admit a unique, finitely many, or a continuum of distinct smooth structures (I don't know whether there are any examples admitting precisely a countably infinite number).
For complex manifolds there are examples of smooth manifolds admitting a unique complex structure ($mathbbCP^1$) or a continuum (compact Riemann surfaces, K3 surfaces, etc.)
Q. Are there examples admitting only finitely many or a countably infinite number?
By deformation theory the tangent space to the moduli space of complex structures on $X$ should be given by $H^1(X, TX)$ (at least morally) so it must be necessary for this to vanish for every possible complex structure on $X$ to have any hope.
dg.differential-geometry complex-geometry differential-topology
edited Apr 5 at 9:42
Francesco Polizzi
48.8k3132214
asked Apr 3 at 10:52
John McCarthyJohn McCarthy
785
$endgroup$
add a comment |
$begingroup$
For smooth manifolds it is known that they can admit a unique, finitely many, or a continuum of distinct smooth structures (I don't know whether there are any examples admitting precisely a countably infinite number).
For complex manifolds there are examples of smooth manifolds admitting a unique complex structure ($mathbbCP^1$) or a continuum (compact Riemann surfaces, K3 surfaces, etc.)
Q. Are there examples admitting only finitely many or a countably infinite number?
By deformation theory the tangent space to the moduli space of complex structures on $X$ should be given by $H^1(X, TX)$ (at least morally) so it must be necessary for this to vanish for every possible complex structure on $X$ to have any hope.
dg.differential-geometry complex-geometry differential-topology
edited Apr 5 at 9:42
Francesco Polizzi
48.8k3132214
asked Apr 3 at 10:52
John McCarthyJohn McCarthy
785
$endgroup$
4
$begingroup$
if you allow non-closed manifolds, smooth unit disc admits exactly two complex structures I believe (one its own, the other coming from the diffeomorphism with $mathbbR^2$)
$endgroup$
– Aknazar Kazhymurat
Apr 3 at 16:02
add a comment |
$begingroup$
For smooth manifolds it is known that they can admit a unique, finitely many, or a continuum of distinct smooth structures (I don't know whether there are any examples admitting precisely a countably infinite number).
For complex manifolds there are examples of smooth manifolds admitting a unique complex structure ($mathbbCP^1$) or a continuum (compact Riemann surfaces, K3 surfaces, etc.)
Q. Are there examples admitting only finitely many or a countably infinite number?
By deformation theory the tangent space to the moduli space of complex structures on $X$ should be given by $H^1(X, TX)$ (at least morally) so it must be necessary for this to vanish for every possible complex structure on $X$ to have any hope.
dg.differential-geometry complex-geometry differential-topology
edited Apr 5 at 9:42
Francesco Polizzi
48.8k3132214
asked Apr 3 at 10:52
John McCarthyJohn McCarthy
785
$endgroup$
For smooth manifolds it is known that they can admit a unique, finitely many, or a continuum of distinct smooth structures (I don't know whether there are any examples admitting precisely a countably infinite number).
For complex manifolds there are examples of smooth manifolds admitting a unique complex structure ($mathbbCP^1$) or a continuum (compact Riemann surfaces, K3 surfaces, etc.)
Q. Are there examples admitting only finitely many or a countably infinite number?
By deformation theory the tangent space to the moduli space of complex structures on $X$ should be given by $H^1(X, TX)$ (at least morally) so it must be necessary for this to vanish for every possible complex structure on $X$ to have any hope.
dg.differential-geometry complex-geometry differential-topology
dg.differential-geometry complex-geometry differential-topology
edited Apr 5 at 9:42
Francesco Polizzi
48.8k3132214
asked Apr 3 at 10:52
John McCarthyJohn McCarthy
785
edited Apr 5 at 9:42
Francesco Polizzi
48.8k3132214
asked Apr 3 at 10:52
John McCarthyJohn McCarthy
785
edited Apr 5 at 9:42
Francesco Polizzi
48.8k3132214
edited Apr 5 at 9:42
Francesco Polizzi
48.8k3132214
edited Apr 5 at 9:42
Francesco Polizzi
48.8k3132214
48.8k3132214
asked Apr 3 at 10:52
John McCarthyJohn McCarthy
785
asked Apr 3 at 10:52
John McCarthyJohn McCarthy
785
asked Apr 3 at 10:52
John McCarthyJohn McCarthy
785
785
4
$begingroup$
if you allow non-closed manifolds, smooth unit disc admits exactly two complex structures I believe (one its own, the other coming from the diffeomorphism with $mathbbR^2$)
$endgroup$
– Aknazar Kazhymurat
Apr 3 at 16:02
add a comment |
4
$begingroup$
if you allow non-closed manifolds, smooth unit disc admits exactly two complex structures I believe (one its own, the other coming from the diffeomorphism with $mathbbR^2$)
$endgroup$
– Aknazar Kazhymurat
Apr 3 at 16:02
4
4
$begingroup$
if you allow non-closed manifolds, smooth unit disc admits exactly two complex structures I believe (one its own, the other coming from the diffeomorphism with $mathbbR^2$)
$endgroup$
– Aknazar Kazhymurat
Apr 3 at 16:02
$begingroup$
if you allow non-closed manifolds, smooth unit disc admits exactly two complex structures I believe (one its own, the other coming from the diffeomorphism with $mathbbR^2$)
$endgroup$
– Aknazar Kazhymurat
Apr 3 at 16:02
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
Examples of smooth manifolds admitting finitely many complex structures are provided by some holomorphically rigid complex varieties.
For instance, considering fake projective planes (i.e., smooth compact complex surfaces which are not the complex projective plane but have the same Betti numbers as the
complex projective plane), there are $100$ examples determined up to biholomorphism, but only $50$ examples determined up to isometry.
In fact, the real $4$-manifold underlying any fake projective plane admits precisely two inequivalent complex structures, that are exchanged by complex conjugation.
Moreover, by standard results on rigidity, any minimal Kähler surface with the same fundamental group as a fake projective plane is actually biholomorphic or conjugate biholomorphic to it.
answered Apr 3 at 11:00
Francesco PolizziFrancesco Polizzi
48.8k3132214
$endgroup$
add a comment |
$begingroup$
There are countably many complex structures on $S^2 times S^2$ up to isomorphism. Specifically, the even Hirzebruch surfaces $F_2k$ are the only options. This is the main result of
Qin, Zhenbo, Complex structures on certain differentiable 4-manifolds, Topology 32, No. 3, 551-566 (1993). ZBL0796.57010.
The author also proves that the odd Hirzebruch surfaces are the only complex structures on the $4$-fold $mathbbCP^2 # overlinemathbbCP^2$.
answered Apr 3 at 23:57
David E SpeyerDavid E Speyer
108k9282540
$endgroup$
add a comment |
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2 Answers
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2 Answers
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$begingroup$
Examples of smooth manifolds admitting finitely many complex structures are provided by some holomorphically rigid complex varieties.
For instance, considering fake projective planes (i.e., smooth compact complex surfaces which are not the complex projective plane but have the same Betti numbers as the
complex projective plane), there are $100$ examples determined up to biholomorphism, but only $50$ examples determined up to isometry.
In fact, the real $4$-manifold underlying any fake projective plane admits precisely two inequivalent complex structures, that are exchanged by complex conjugation.
Moreover, by standard results on rigidity, any minimal Kähler surface with the same fundamental group as a fake projective plane is actually biholomorphic or conjugate biholomorphic to it.
answered Apr 3 at 11:00
Francesco PolizziFrancesco Polizzi
48.8k3132214
$endgroup$
add a comment |
$begingroup$
Examples of smooth manifolds admitting finitely many complex structures are provided by some holomorphically rigid complex varieties.
For instance, considering fake projective planes (i.e., smooth compact complex surfaces which are not the complex projective plane but have the same Betti numbers as the
complex projective plane), there are $100$ examples determined up to biholomorphism, but only $50$ examples determined up to isometry.
In fact, the real $4$-manifold underlying any fake projective plane admits precisely two inequivalent complex structures, that are exchanged by complex conjugation.
Moreover, by standard results on rigidity, any minimal Kähler surface with the same fundamental group as a fake projective plane is actually biholomorphic or conjugate biholomorphic to it.
answered Apr 3 at 11:00
Francesco PolizziFrancesco Polizzi
48.8k3132214
$endgroup$
add a comment |
$begingroup$
Examples of smooth manifolds admitting finitely many complex structures are provided by some holomorphically rigid complex varieties.
For instance, considering fake projective planes (i.e., smooth compact complex surfaces which are not the complex projective plane but have the same Betti numbers as the
complex projective plane), there are $100$ examples determined up to biholomorphism, but only $50$ examples determined up to isometry.
In fact, the real $4$-manifold underlying any fake projective plane admits precisely two inequivalent complex structures, that are exchanged by complex conjugation.
Moreover, by standard results on rigidity, any minimal Kähler surface with the same fundamental group as a fake projective plane is actually biholomorphic or conjugate biholomorphic to it.
answered Apr 3 at 11:00
Francesco PolizziFrancesco Polizzi
48.8k3132214
$endgroup$
Examples of smooth manifolds admitting finitely many complex structures are provided by some holomorphically rigid complex varieties.
For instance, considering fake projective planes (i.e., smooth compact complex surfaces which are not the complex projective plane but have the same Betti numbers as the
complex projective plane), there are $100$ examples determined up to biholomorphism, but only $50$ examples determined up to isometry.
In fact, the real $4$-manifold underlying any fake projective plane admits precisely two inequivalent complex structures, that are exchanged by complex conjugation.
Moreover, by standard results on rigidity, any minimal Kähler surface with the same fundamental group as a fake projective plane is actually biholomorphic or conjugate biholomorphic to it.
answered Apr 3 at 11:00
Francesco PolizziFrancesco Polizzi
48.8k3132214
edited Apr 5 at 5:15
answered Apr 3 at 11:00
Francesco PolizziFrancesco Polizzi
48.8k3132214
answered Apr 3 at 11:00
Francesco PolizziFrancesco Polizzi
48.8k3132214
answered Apr 3 at 11:00
Francesco PolizziFrancesco Polizzi
48.8k3132214
48.8k3132214
add a comment |
add a comment |
$begingroup$
There are countably many complex structures on $S^2 times S^2$ up to isomorphism. Specifically, the even Hirzebruch surfaces $F_2k$ are the only options. This is the main result of
Qin, Zhenbo, Complex structures on certain differentiable 4-manifolds, Topology 32, No. 3, 551-566 (1993). ZBL0796.57010.
The author also proves that the odd Hirzebruch surfaces are the only complex structures on the $4$-fold $mathbbCP^2 # overlinemathbbCP^2$.
answered Apr 3 at 23:57
David E SpeyerDavid E Speyer
108k9282540
$endgroup$
add a comment |
$begingroup$
There are countably many complex structures on $S^2 times S^2$ up to isomorphism. Specifically, the even Hirzebruch surfaces $F_2k$ are the only options. This is the main result of
Qin, Zhenbo, Complex structures on certain differentiable 4-manifolds, Topology 32, No. 3, 551-566 (1993). ZBL0796.57010.
The author also proves that the odd Hirzebruch surfaces are the only complex structures on the $4$-fold $mathbbCP^2 # overlinemathbbCP^2$.
answered Apr 3 at 23:57
David E SpeyerDavid E Speyer
108k9282540
$endgroup$
add a comment |
$begingroup$
There are countably many complex structures on $S^2 times S^2$ up to isomorphism. Specifically, the even Hirzebruch surfaces $F_2k$ are the only options. This is the main result of
Qin, Zhenbo, Complex structures on certain differentiable 4-manifolds, Topology 32, No. 3, 551-566 (1993). ZBL0796.57010.
The author also proves that the odd Hirzebruch surfaces are the only complex structures on the $4$-fold $mathbbCP^2 # overlinemathbbCP^2$.
answered Apr 3 at 23:57
David E SpeyerDavid E Speyer
108k9282540
$endgroup$
There are countably many complex structures on $S^2 times S^2$ up to isomorphism. Specifically, the even Hirzebruch surfaces $F_2k$ are the only options. This is the main result of
Qin, Zhenbo, Complex structures on certain differentiable 4-manifolds, Topology 32, No. 3, 551-566 (1993). ZBL0796.57010.
The author also proves that the odd Hirzebruch surfaces are the only complex structures on the $4$-fold $mathbbCP^2 # overlinemathbbCP^2$.
answered Apr 3 at 23:57
David E SpeyerDavid E Speyer
108k9282540
answered Apr 3 at 23:57
David E SpeyerDavid E Speyer
108k9282540
answered Apr 3 at 23:57
David E SpeyerDavid E Speyer
108k9282540
answered Apr 3 at 23:57
David E SpeyerDavid E Speyer
108k9282540
108k9282540
add a comment |
add a comment |
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$begingroup$
if you allow non-closed manifolds, smooth unit disc admits exactly two complex structures I believe (one its own, the other coming from the diffeomorphism with $mathbbR^2$)
$endgroup$
– Aknazar Kazhymurat
Apr 3 at 16:02