Integral formulas in Riemannian geometry

Sasakian Geometry in Dimensions 3 and 5

The Seiberg-Witten equation with multiple spinors generalises the classical Seiberg-Witten equation in dimension three. In contrast to the classical case, the moduli space of solutions $\mathcal{M}$ can be non-compact due to the appearance of so-called Fueter sections. In the absence of Fueter sections we define a signed count of points in $\mathcal{M}$ and show its invariance under small perturbations. We then study the equation on the product of a Riemann surface and a circle, describing $\mathcal{M}$ in terms of holomorphic data over the surface. We define analytic and algebro-geometric compactifications of $\mathcal{M}$, and construct a homeomorphism between them. For a generic choice of circle-invariant parameters of the equation, Fueter sections do not appear and $\mathcal{M}$ is a compact Kahler manifold. After a perturbation it splits into isolated points which can be counted with signs, yielding a number independent of the initial choice of the parameters. We compute this number for surfaces of low genus.

Seiberg-Witten monopoles with multiple spinors on a surface times a circle

Seiberg–Witten monopoles with multiple spinors on a surface times a circle

We establish the equality of classical and tropical curve counts for elliptic curves on toric surfaces with fixed j-invariant, refining results of Mikhalkin and Nishinou–Siebert. As an application, we determine a formula for such counts on P2 and all Hirzebruch surfaces. This formula relates the count of elliptic curves with the number of rational curves on the surface satisfying a small number of tangency conditions with the toric boundary. Furthermore, the combinatorial tropical multiplicities of Kerber and Markwig for counts in P2 are derived and explained algebro-geometrically, using Berkovich geometry and logarithmic Gromov–Witten theory. As a consequence, a new proof of Pandharipande’s formula for counts of elliptic curves in P2 with fixed j-invariant is obtained.

Enumerative geometry of elliptic curves on toric surfaces

In this paper we consider twice-dimensionally reduced, generalized Seiberg-Witten (S-W) equations, defined on a compact Riemann surface. A novel feature of the reduction technique is that the resulting equations produce an extra “Higgs field”. Under suitable regularity assumptions, we show that the moduli space of gauge-equivalent classes of solutions to the reduced equations, is a smooth Kahler manifold and construct a pre-quantum line bundle over the moduli space of solutions.

Generalized Seiberg-Witten Equations on a Riemann Surface

Generalised Seiberg–Witten equations and almost-Hermitian geometry

In this article, we establish a Hitchin-Kobayashi type correspondence for generalised Seiberg-Witten monopole equations on Kahler surfaces. We show that the "stability" criterion we obtain, for the existence of solutions, coincides with that of the usual Seiberg-Witten monopole equations. This enables us to construct a map from the moduli space of solutions to the generalised equations to effective divisors.

Generalised monopole equations on Kahler surfaces

Genus one enumerative invariants in del-Pezzo surfaces with a fixed complex structure

In this paper we consider twice-dimensionally reduced, generalized Seiberg-Witten equations, defined on a compact Riemann surface. A novel feature of the reduction technique is that the resulting equations produce an extra "Higgs field". Under suitable regularity assumptions, we show that the moduli space of gauge-equivalent classes of solutions to the reduced equations, is a smooth Kahler manifold and construct a pre-quantum line bundle over the moduli space of solutions.

Varun Thakre

Papers

Generalized Seiberg-Witten Equations on a Riemann Surface

Generalised Seiberg–Witten equations and almost-Hermitian geometry

Generalised monopole equations on Kahler surfaces

Genus one enumerative invariants in del-Pezzo surfaces with a fixed complex structure

Generalized Seiberg-Witten equations on Riemann surface