on a Deformation long exact sequence of moduli space of stable maps - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T01:29:56Z http://mathoverflow.net/feeds/question/108722 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/108722/on-a-deformation-long-exact-sequence-of-moduli-space-of-stable-maps on a Deformation long exact sequence of moduli space of stable maps ZhuangXiaobo 2012-10-03T16:50:24Z 2012-10-04T14:17:15Z <p>I am reading the book "mirror symmetry" by Hori,Katz,Klemm,etc. And I want to understand the following Deformation long exact sequence</p> <p>\begin{align} 0 &amp; \to Aut(Σ, p_1, . . . , p_n, f)\to Aut(Σ, p_1, . . . , p_n) &amp;\newline \to Def(f) &amp;→ Def(Σ, p_1, . . . , p_n, f) → Def(Σ, p_1, . . . , p_n) &amp;\newline \to Ob(f) &amp;\to Ob(Σ, p_1, . . . , p_n, f) \to 0 \end{align}</p> <p>it connects three deformation theory:<br> 1. deformation of stable curves<br> 2. deformation of maps(with fixed source)<br> 3. deformation of stable maps(with possible changing source curves)</p> <p>And <strong>my understanding</strong> goes as follows:<br> Let $\mathscr{X}=M_{g,n}$ be the moduli stack of algebraic curves(genus $g$, n-marked point), and let $\mathscr{Y}=M_{g,n}(X,\beta)$ be the moduli stack of stable maps. Then there is a natural "forgetful" morphism:<br> $\pi : \mathscr{Y} \to \mathscr{X}$<br> by forgeting the "map". </p> <p>We have a distinguished triangle of cotangent complexes in the derived category $D^{-} (\mathscr O_{\mathcal{Y}})$: </p> <p>$$\pi^* L_{\mathscr{X}}\to L_{\mathscr Y}\to L_{\mathscr{Y}/\mathscr{X}}\to \cdot$$</p> <p>Now apply $R\mathscr{Hom}$, we have a long exact sequence:</p> <p>\begin{align} \mathscr Ext ^0 (L_{\mathscr Y/\mathscr X },\mathcal O_{\mathscr Y }) &amp; \to \mathscr Ext^0 (L_{\mathscr Y}, \mathcal O_{\mathscr Y} ) \to \mathscr Ext^0 (\pi^* L_{\mathscr X},\mathcal O_{\mathscr Y} )&amp; \newline \to \mathscr Ext ^1 (L_{\mathscr Y/\mathscr X },\mathcal O_{\mathscr Y }) &amp; \to \mathscr Ext^1 (L_{\mathscr Y}, \mathcal O_{\mathscr Y} ) \to \mathscr Ext^1 (\pi^* L_{\mathscr X},\mathcal O_{\mathscr Y} )&amp; \newline \to \mathscr Ext ^2 (L_{\mathscr Y/\mathscr X },\mathcal O_{\mathscr Y }) &amp; \to \mathscr Ext^2 (L_{\mathscr Y}, \mathcal O_{\mathscr Y} ) \to \mathscr Ext^2 (\pi^* L_{\mathscr X},\mathcal O_{\mathscr Y} ) \end{align}</p> <p><strong>My questions:</strong><br> (1). is it an exact sequence of sheaves on $\mathscr Y$ with the first long exact sequence as its stalks?<br> (2). If (1) is true, then how to see $\mathscr Ext^i (\pi^* L_{\mathscr X},\mathcal O_{\mathscr Y} )$ (i=0,1,2) corresponds to Aut,Def,Ob of curves? And why the two ends of the exact seqence vanishes?</p> http://mathoverflow.net/questions/108722/on-a-deformation-long-exact-sequence-of-moduli-space-of-stable-maps/108738#108738 Answer by Simon Rose for on a Deformation long exact sequence of moduli space of stable maps Simon Rose 2012-10-03T19:15:52Z 2012-10-04T14:17:15Z <p>I don't believe that this is correct. The easiest way to see this is to look at your second question: The automorphisms/deformations/obstructions of a curve come from $H^i(C, T_C)$, i.e. they are the sheaves</p> <p><code>$R^i p_*\omega_{U/\overline{\mathcal{M}_{g,n}}}^\vee$</code></p> <p>where <code>$p : U \to \overline{\mathcal{M}_{g,n}}$</code> is the universal family, and <code>$\omega_{U/\overline{\mathcal{M}_{g,n}}}$</code> the relative dualizing sheaf. But these do not depend on <code>$\overline{\mathcal{M}_{g,n}}(X, \beta)$</code> !</p> <p>In the end, I think the issue is that you have the wrong exact sequence. What you want (to produce the relative obstruction theory) is the complex</p> <p><code>$R^i p_*f^*T_X$</code></p> <p>where the maps $p, f$ arise in the universal diagram </p> <p><code>$\overline{\mathcal{M}_{g,n}}(X, \beta) \longleftarrow_p U \longrightarrow_f X$</code></p> <p>It is not obvious to me that your sheaves should be the same as these ones.</p>