Order complexes of singular sets and topology ofspaces of nonsingular projective varieties
by
Victor A. Vassiliev
Steklov Institute of Mathematics, Independent University of Moscow

This is a short description of the work , in which we present a method of computing homology groups of spaces N(d, n) of all nonsingular hypersurfaces of degree d in CPⁿ or, equivalently, of the space \Pi(d, n) \\Sigma of all homogeneous (of degree d) polynomials Cⁿ⁺¹ --> C defining such nonsingular hypersurfaces. This problem is the natural "complexification" of the rigid isotopy classification of real algebraic hypersurfaces, and our methods can be applied also to this problem.

In the simplest case, when d=2, it is easy to see that the space \Pi(d, n) \\Sigma is homotopy equivalent to the Lagrangian Grassmannian U(n+1)/O(n+1). For some other pairs (d, n) our first calculations give the following answers, formulated in the terms of the Poincaré polynomials P_{d, n} of groups H^*(N(d, n), Q).

Theorem. P_{3, 2}=(1+t³)(1+t⁵). P_{3, 3}=(1+t³)(1+t⁵)(1+t⁷). P_{4, 2}=(1+t³)(1+t⁵)(1+t⁶).

Conjecture. For any n, H^*(N(3, n), Q) =~ H^*(PGL(n+1, C), Q).

On the other hand, I do not know any explicit realization of the 6-dimensional generator of H^*(N(4, 2)) or of any cycle on which it takes nonzero value.

The method of calculation is based on the stuidy of the discriminant variety \Sigma of singular polynomials (whose locally finite homology group is related to the cohomology group of P(d, n)\\Sigma by the Alexander duality, cf. . The main technical tools are the conical resolutions of discriminant varieties (which are a far generalization of the combinatorial inclusion-exclusion formula, and also of simplicial resolutions used in the algebraic geometry) and the continuous order complex of singular sets generalizing the notion of buildings. See also Chapter 7 in .

Our calculations lead to many natural and beautiful problems of algebraic geometry and topology, in particular on the classification of all possible singular sets of hypersurfaces of a given degree, and on homology groups of such classes.

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V.I.Arnold, On some topological invariants of algebraic functions, Trans. Moscow Math. Soc. 21 (1970), 27-46.

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V.A.Vassiliev, How to calculate homology groups of spaces of nonsingular algebraic projective varieties, Proc. Steklov Mat. Inst., 1999.

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-, Topology of Complements of Discriminants, Phasis Publ., Moscow, 1997 (in Russian).

Date received: June 1, 1999