EQUITOP IN ALGGEO · Equivariant topology in algebraic geometry

Equivariant topology in algebraic geometry

(1) With L. Feher and A. Nemethi we studied the geometry of matroids and their Gromov-Witten (GW) invariants. We define the GW invariant as the number of ways a given point configuration can be represented by points of C^n lying on given subspaces. The first main result of this project is the proof of the fact that the GW invariants are determined by a rather rough, topological invariant of the matroid representation variety. This is the cohomology class represented by the matroid representation variety in the space of all configurations. The second main result is that the equivariant cohomology class satisfies certain interpolation properties. The third main result describes a stabilisation property of the cohomology classes. Computer evidence shows that there is only one class satisfying the interpolation and stabilisation properties. Putting all these results together now we have an effective way of calculating GW invariants for matroids. Future study may involve the connection to quantum cohomology. (2) In another joint work with L. Feher we studied global properties of singularities. It is classically know that the global behaviour of singularities is governed by their Thom polynomias. Namely, the number of given singularities of a map equals the value of the corresponding Thom polynomial, if we substitute the so-called characteristic classes of the source and target manifolds. Following pioneering works of Szenes and Berczi we studied natural infinite sequences of Thom polynomials. Our main result is that all these infinitely many Thom polynomials follow from a finite set of data. This set of data is certain cohomology classes represented by natural varieties in a Hilbert scheme. We made a great number of calculations in the cohomology of Hilbert schemes, and thus found several so-far unknown infinite sequences of Thom polynomials. Some of these infinite sequences reach beyond the realm of 'nice' singularities - meaning that not even individual ones of these series can be obtained by the classical methods. (3) With V. Schechtman, A. Varchenko we considered conformal blocks on the Riemann sphere in the Wess-Zimono-Novikov-Witten conformal field theory. The associated space of conformal blocks can be realised as a vector subspace of a well understood tensor product space, and the tensor product can be realised as a suitable vector space of polynomials. The subspace of conformal blocks is defined as the set of solutions to a system of differential equations. We solve that system for level 1. In that case the dimension of the conformal blocks is 1 and we give a formula for one remarkable polynomial generating the one-dimensional space of conformal blocks. A striking property of the formula is its similarity to equivariant localisation formulas. More conceptual discussion of this connection is a topic for future research. According to a general principle by Mukhin-Varchenko, if the space of conformal blocks is one-dimensional, then the hypergeometric integral representing the conformal block can be calculated explicitly giving a Selberg-type integral. Therefore, our formula for the conformal block at level 1 produces a new Selberg-type integral. Broader impacts of the project are: better understanding sudden changes caused by smooth alteration of parameters (physics), and bringing research close to teaching.

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