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Using the work of Guillen and Navarro Aznar we associate to each real algebraic variety a filtered chain complex, the weight complex, which is well-defined up to filtered quasi-isomorphism, and which induces on Borel-Moore homology with Z/2 coefficients an analog of the weight filtration for complex algebraic varieties.
We extend the Altmann-Hausen presentation of normal affine algebraic C-varieties endowed with effective torus actions to the real setting. In particular, we focus on actions of quasi-split real tori, in which case we obtain a simpler presentation.
We give some explicit bounds for the number of cobordism classes of real algebraic manifolds of real degree less than $d$, and for the size of the sum of $mod 2$ Betti numbers for the real form of complex manifolds of complex degree less than $d$.
Let $ Y subseteq Bbb P^N $ be a possibly singular projective variety, defined over the field of complex numbers. Let $X$ be the intersection of $Y$ with $h$ general hypersurfaces of sufficiently large degrees. Let $d>0$ be an integer, and assume that
We seek to determine a real algebraic variety from a fixed finite subset of points. Existing methods are studied and new methods are developed. Our focus lies on aspects of topology and algebraic geometry, such as dimension and defining polynomials.
Let $n$ be a positive integer. The main result of this manuscript is a construction of a filtration on the cohomology ring of a regular nilpotent Hessenberg variety in $GL(n,{mathbb{C}})/B$ such that its associated graded ring has graded pieces (i.e.