Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Virtual Segre and Verlinde numbers of projective surfaces

L. Göttsche, M. Kool

TL;DR

This work develops a universal, algebraic framework for virtual Segre and Verlinde numbers on surfaces, expressing their generating functions through rank- and topology-dependent universal series tied to Mochizuki’s formula and Seiberg–Witten data. It proves that canonical virtual Segre and Verlinde numbers depend only on topological data for broad classes of surfaces, and establishes algebraicity and Galois-structure properties for the associated universal functions. The authors formulate a Segre–Verlinde correspondence, verify it in multiple ranks (including up to $ ho=4$ in several examples), and connect these invariants to the higher-rank Mariño–Moore program, with explicit conjectures for K3 and general type surfaces. They also develop a detailed blow-up and disconnection theory, along with a robust K3-based reduction that aligns with known Witten-type formulas for Donaldson invariants. Overall, the paper provides both concrete computational evidence and a deep structural conjecture set linking Donaldson theory, Hilbert schemes, and Seiberg–Witten theory in higher rank.

Abstract

Recently, Marian-Oprea-Pandharipande established (a generalization of) Lehn's conjecture for Segre numbers associated to Hilbert schemes of points on surfaces. Extending work of Johnson, they provided a conjectural correspondence between Segre and Verlinde numbers. For surfaces with holomorphic 2-form, we propose conjectural generalizations of their results to moduli spaces of stable sheaves of any rank. Using Mochizuki's formula, we derive a universal function which expresses virtual Segre and Verlinde numbers of surfaces with holomorphic 2-form in terms of Seiberg-Witten invariants and intersection numbers on products of Hilbert schemes of points. We prove that certain canonical virtual Segre and Verlinde numbers of general type surfaces are topological invariants and we verify our conjectures in examples. The power series in our conjectures are algebraic functions, for which we find expressions in several cases and which are permuted under certain Galois actions. Our conjectures imply an algebraic analog of the Mariño-Moore conjecture for higher rank Donaldson invariants. For ranks $3$ and $4$, we obtain explicit expressions for Donaldson invariants in terms of Seiberg-Witten invariants.

Virtual Segre and Verlinde numbers of projective surfaces

TL;DR

This work develops a universal, algebraic framework for virtual Segre and Verlinde numbers on surfaces, expressing their generating functions through rank- and topology-dependent universal series tied to Mochizuki’s formula and Seiberg–Witten data. It proves that canonical virtual Segre and Verlinde numbers depend only on topological data for broad classes of surfaces, and establishes algebraicity and Galois-structure properties for the associated universal functions. The authors formulate a Segre–Verlinde correspondence, verify it in multiple ranks (including up to in several examples), and connect these invariants to the higher-rank Mariño–Moore program, with explicit conjectures for K3 and general type surfaces. They also develop a detailed blow-up and disconnection theory, along with a robust K3-based reduction that aligns with known Witten-type formulas for Donaldson invariants. Overall, the paper provides both concrete computational evidence and a deep structural conjecture set linking Donaldson theory, Hilbert schemes, and Seiberg–Witten theory in higher rank.

Abstract

Recently, Marian-Oprea-Pandharipande established (a generalization of) Lehn's conjecture for Segre numbers associated to Hilbert schemes of points on surfaces. Extending work of Johnson, they provided a conjectural correspondence between Segre and Verlinde numbers. For surfaces with holomorphic 2-form, we propose conjectural generalizations of their results to moduli spaces of stable sheaves of any rank. Using Mochizuki's formula, we derive a universal function which expresses virtual Segre and Verlinde numbers of surfaces with holomorphic 2-form in terms of Seiberg-Witten invariants and intersection numbers on products of Hilbert schemes of points. We prove that certain canonical virtual Segre and Verlinde numbers of general type surfaces are topological invariants and we verify our conjectures in examples. The power series in our conjectures are algebraic functions, for which we find expressions in several cases and which are permuted under certain Galois actions. Our conjectures imply an algebraic analog of the Mariño-Moore conjecture for higher rank Donaldson invariants. For ranks and , we obtain explicit expressions for Donaldson invariants in terms of Seiberg-Witten invariants.

Paper Structure

This paper contains 20 sections, 13 theorems, 214 equations.

Table of Contents

  1. Introduction
  2. Universal function
  3. Mochizuki's formula
  4. Main theorems
  5. Main conjectures
  6. Verifications
  7. Rank 1
  8. Higher rank: calculations
  9. Algebraicity
  10. Rank 1
  11. Rank 2
  12. Rank 3
  13. Rank 4
  14. Galois actions
  15. $K3$, Serre duality, Mariño-Moore conjecture
  16. ...and 5 more sections

Key Result

Theorem 1.1

For any $s \in \mathbb Z$, there exist $V_s$, $W_s$, $X_s$, $Y_s$, $Z_s \in \mathbb Q[[z]]$ with the following properties. For any $K$-theory class $\alpha$ of rank $s$ on a smooth projective surface $S$, we have Moreover, under the formal change of variables $z = t(1+(1-s)t)^{1-s}$, we have

Theorems & Definitions (48)

  • Theorem 1.1: Marian-Oprea-Pandharipande
  • Theorem 1.2: Ellingsrud-Göttsche-Lehn
  • Conjecture 1.3: Johnson, Marian-Oprea-Pandharipande
  • Conjecture 1.4
  • Remark 1.5
  • Conjecture 1.6
  • Conjecture 1.7
  • Theorem 1.8
  • Theorem 2.1: Mochizuki
  • Theorem 2.2
  • ...and 38 more