Based on the orbifold string theory model in physics, we construct a new cohomology ring for any almost complex orbifold. The key theorem is the associativity of this new ring. Some examples are computed.

/pdf/a-new-cohomology-theory-of-orbifold-3s1i92lsly.pdf

A New Cohomology Theory of Orbifold

Motivated by orbifold string theory, we introduce orbifold cohomology group for any almost complex orbifold and orbifold Dolbeault cohomology for any complex orbifold. Then, we show that our new cohomology group satisfies Poincare duality and has a natural ring structure. Some examples of orbifold cohomology ring are computed.

A New Cohomology Theory for Orbifold

3 Invariants of Hamiltonian group actions 17 3.1 An action functional . . . . . . . . . . . . . . . . . . . . . . . 17 3.2 Symplectic reduction . . . . . . . . . . . . . . . . . . . . . . . 19 3.3 Hamiltonian perturbations . . . . . . . . . . . . . . . . . . . . 24 3.4 Moduli spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.5 Fredholm theory . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.6 Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.7 Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.8 Adiabatic limits . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.9 Wall crossing and localization . . . . . . . . . . . . . . . . . . 35

/pdf/j-holomorphic-curves-moment-maps-and-invariants-of-x3lrkcixcb.pdf

J-holomorphic curves, moment maps, and invariants of Hamiltonian group actions

In [CH3], Caporaso and Harris derive recursive formulas counting nodal plane curves of degree d and geometric genus g in the plane (through the appropriate number of fixed general points). We rephrase their arguments in the language of maps, and extend them to other rational surfaces, and other specified intersections with a divisor. As applications, (i) we count irreducible curves on Hirzebruch surfaces in a fixed divisor class and of fixed geometric genus, (ii) we compute the higher-genus Gromov–Witten invariants of (or equivalently, counting curves of any genus and divisor class on) del Pezzo surfaces of degree at least 3. In the case of the cubic surface in (ii), we first use a result of Graber to enumeratively interpret higher-genus Gromov–Witten invariants of certain K-nef surfaces, and then apply this to a degeneration of a cubic surface.

http://math.stanford.edu/~vakil/files/countingfinal.pdf

Counting curves on rational surfaces

For any smooth complex projective variety $X$ and any smooth very ample hypersurface $Y\subset X$, we develop the technique of genus zero relative Gromov-Witten invariants of $Y$ in $X$ in algebro-geometric terms. We prove an equality of cycles in the Chow groups of the moduli spaces of relative stable maps which relates these relative invariants to the Gromov-Witten invariants of $X$ and $Y$. Given the Gromov-Witten invariants of $X$, we show that these relations are sufficient to compute all relative invariants, as well as all genus zero Gromov-Witten invariants of $Y$ whose homology and cohomology classes are induced by $X$.

Absolute and relative Gromov-Witten invariants of very ample hypersurfaces

This is an expository article about the recent results on the relation between the naturality problem of quantum cohomology and birational geometry. We give a short survey on Li-Ruan's results. Some conjecture-proposals are discussed.

Yongbin Ruan

Papers

Surgery, quantum cohomology and birational geometry