I wonder if anyone knows anything about the cohomology with compact supports for determinantal varieties, such as the varieties of $m \times n$ matrices of full rank.
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5$\begingroup$ Let $X$ be the space of matrices of full rank. $X$ is a codimension zero submanifold of the vector space of all matrices. By a suitable version of Poincare duality, $H^k_c(X)\cong H_{mn-k}(X)$. Moreover, $X$ is homotopy equivalent to to the Stiefel manifold $V_{m,n}$, whose homology is well-known. Am I missing something? $\endgroup$– Gregory AroneCommented Oct 29, 2017 at 18:56
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6$\begingroup$ And welcome to mathoverflow! $\endgroup$– Gregory AroneCommented Oct 29, 2017 at 18:57
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7$\begingroup$ Welcome to MO, see p.20 worksing.icmc.usp.br/main_site/2016/JDamon.pdf . It is known theorem of Miller. arxiv.org/pdf/1512.03391.pdf . I learned it in his talk in Lille $\endgroup$– user21574Commented Oct 29, 2017 at 19:31
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1$\begingroup$ Thanks, of course you are right. Ultimately I am interested in somewhat more complicated problems, in including varieties of rank exactly k, where k < min(m,n). I guess there the situation is much more complicated, but the cohomology still carries the information. $\endgroup$– Gunnar CarlssonCommented Oct 29, 2017 at 20:20
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3$\begingroup$ Scorza variety, tangents variety, and secants variety like this ? , please edit your question. $\endgroup$– user21574Commented Oct 29, 2017 at 20:36
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