Return to Answer

edited body

Source Link

edited Feb 16, 2010 at 2:54

28.9k
4
92
161

I would like to make one naive suggestion, related to the work of Voisin, who constructed Hilbert scheme of every almost complex manifolds $X$ of real dimension $4$. And to a more recent work of Julien Grivaux on this topic

http://arxiv.org/PS_cache/arxiv/pdf/1001/1001.0119v1.pdf

As far as I understand Julien can express the cohomology ring of this scheme in the case when $X$ is a symplectic manifold (Theorem 1.1). Symplectic structure is used in order to construct symplectic surfaces on $X$. These surfaces are quite plentiful on $X$ by a theorem of Donaldson. For any such symplectic surface $C$ one can find an almost complex sturcture on $X$, integrable in a neighborhood of $C$. This is discussed Section 3.2 (in particular Theorem 3.5 and Corollary 3.3).

Now if we have a smooth almost complex curve $C$ inside of $X$, such that the complex structure is integrable in its neighborhood then $C^{[n]}$ seem to be well defined as a submanifold of $X^{[n]}$ (the Voisin Hilbert scheme). Moreover, Lemma 3.8 seem to suggest that the homology class of $[C^{[n]}]$ is given by a formula analgous to the one in the case when $X$ is an algebraic surface (the formula of Nakajima).

I have an extremely superficial understanding of Julien's artice and of the whole topic, so I may make some here some silly mistake. Moreover it is not clear for me if after all the class $[C^{[n]}]$ in homologies of $X^{[n]}$ depends only on the homolgy class $[C]$ in $H_2(X)$, but will not depend on the choice of a sympectic curve $C$ that realises it. And, of course, there are some obvious (and not obvious) restictions on $C$, such as $\int_S \omega>0$$\int_C \omega>0$.

I would like to make one naive suggestion, related to the work of Voisin, who constructed Hilbert scheme of every almost complex manifolds $X$ of real dimension $4$. And to a more recent work of Julien Grivaux on this topic

http://arxiv.org/PS_cache/arxiv/pdf/1001/1001.0119v1.pdf

As far as I understand Julien can express the cohomology ring of this scheme in the case when $X$ is a symplectic manifold (Theorem 1.1). Symplectic structure is used in order to construct symplectic surfaces on $X$. These surfaces are quite plentiful on $X$ by a theorem of Donaldson. For any such symplectic surface $C$ one can find an almost complex sturcture on $X$, integrable in a neighborhood of $C$. This is discussed Section 3.2 (in particular Theorem 3.5 and Corollary 3.3).

Now if we have a smooth almost complex curve $C$ inside of $X$, such that the complex structure is integrable in its neighborhood then $C^{[n]}$ seem to be well defined as a submanifold of $X^{[n]}$ (the Voisin Hilbert scheme). Moreover, Lemma 3.8 seem to suggest that the homology class of $[C^{[n]}]$ is given by a formula analgous to the one in the case when $X$ is an algebraic surface (the formula of Nakajima).

I have an extremely superficial understanding of Julien's artice and of the whole topic, so I may make some here some silly mistake. Moreover it is not clear for me if after all the class $[C^{[n]}]$ in homologies of $X^{[n]}$ depends only on the homolgy class $[C]$ in $H_2(X)$, but will not depend on the choice of a sympectic curve $C$ that realises it. And, of course, there are some obvious (and not obvious) restictions on $C$, such as $\int_S \omega>0$.

I would like to make one naive suggestion, related to the work of Voisin, who constructed Hilbert scheme of every almost complex manifolds $X$ of real dimension $4$. And to a more recent work of Julien Grivaux on this topic

http://arxiv.org/PS_cache/arxiv/pdf/1001/1001.0119v1.pdf

As far as I understand Julien can express the cohomology ring of this scheme in the case when $X$ is a symplectic manifold (Theorem 1.1). Symplectic structure is used in order to construct symplectic surfaces on $X$. These surfaces are quite plentiful on $X$ by a theorem of Donaldson. For any such symplectic surface $C$ one can find an almost complex sturcture on $X$, integrable in a neighborhood of $C$. This is discussed Section 3.2 (in particular Theorem 3.5 and Corollary 3.3).

Now if we have a smooth almost complex curve $C$ inside of $X$, such that the complex structure is integrable in its neighborhood then $C^{[n]}$ seem to be well defined as a submanifold of $X^{[n]}$ (the Voisin Hilbert scheme). Moreover, Lemma 3.8 seem to suggest that the homology class of $[C^{[n]}]$ is given by a formula analgous to the one in the case when $X$ is an algebraic surface (the formula of Nakajima).

I have an extremely superficial understanding of Julien's artice and of the whole topic, so I may make some here some silly mistake. Moreover it is not clear for me if after all the class $[C^{[n]}]$ in homologies of $X^{[n]}$ depends only on the homolgy class $[C]$ in $H_2(X)$, but will not depend on the choice of a sympectic curve $C$ that realises it. And, of course, there are some obvious (and not obvious) restictions on $C$, such as $\int_C \omega>0$.

Source Link

created Feb 16, 2010 at 2:10

28.9k
4
92
161

I would like to make one naive suggestion, related to the work of Voisin, who constructed Hilbert scheme of every almost complex manifolds $X$ of real dimension $4$. And to a more recent work of Julien Grivaux on this topic

http://arxiv.org/PS_cache/arxiv/pdf/1001/1001.0119v1.pdf

As far as I understand Julien can express the cohomology ring of this scheme in the case when $X$ is a symplectic manifold (Theorem 1.1). Symplectic structure is used in order to construct symplectic surfaces on $X$. These surfaces are quite plentiful on $X$ by a theorem of Donaldson. For any such symplectic surface $C$ one can find an almost complex sturcture on $X$, integrable in a neighborhood of $C$. This is discussed Section 3.2 (in particular Theorem 3.5 and Corollary 3.3).

Now if we have a smooth almost complex curve $C$ inside of $X$, such that the complex structure is integrable in its neighborhood then $C^{[n]}$ seem to be well defined as a submanifold of $X^{[n]}$ (the Voisin Hilbert scheme). Moreover, Lemma 3.8 seem to suggest that the homology class of $[C^{[n]}]$ is given by a formula analgous to the one in the case when $X$ is an algebraic surface (the formula of Nakajima).

I have an extremely superficial understanding of Julien's artice and of the whole topic, so I may make some here some silly mistake. Moreover it is not clear for me if after all the class $[C^{[n]}]$ in homologies of $X^{[n]}$ depends only on the homolgy class $[C]$ in $H_2(X)$, but will not depend on the choice of a sympectic curve $C$ that realises it. And, of course, there are some obvious (and not obvious) restictions on $C$, such as $\int_S \omega>0$.