# Mathematical Research Letters 2, (1995) A VANISHING THEOREM FOR SEIBERG-WITTEN INVARIANTS. Shuguang Wang

Size: px
Start display at page:

Download "Mathematical Research Letters 2, (1995) A VANISHING THEOREM FOR SEIBERG-WITTEN INVARIANTS. Shuguang Wang"

Transcription

1 Mathematical Research Letters 2, (1995) A VANISHING THEOREM FOR SEIBERG-WITTEN INVARIANTS Shuguang Wang Abstract. It is shown that the quotients of Kähler surfaces under free anti-holomorphic involutions have vanishing Seiberg-Witten invariants. Various vanishing theorems have played important roles in gauge theory. The first among them, due to S. K. Donaldson [2], states that the Donaldson invariants vanish for any smooth closed oriented 4-manifold X which decomposes to a connected sum X 1 #X 2 with b + 2 (X 1) > 0, b + 2 (X 2) > 0. (Here b + 2 (X i) is the dimension of a maximal subspace of H 2 (X i,z) on which the intersection pairing is positive definite.) E. Witten [12] has shown more recently that such a connected-sum manifold has also vanishing Seiberg- Witten invariants. Even though its proof is simple, Witten s vanishing theorem is equally useful as Donaldson s vanishing theorem; for example, it implies that any symplectic 4-manifold cannot be decomposed into the above connected sum by combining with Taubes theorem in [9]. In this note we show that Seiberg-Witten invariants vanish for another class of 4-manifolds. These manifolds are obtained in connection with real algebraic geometry, and the construction was originally proposed in Donaldson [3]. See Remark 4below. To state our theorem, recall that a map σ between two almost complex manifolds is called anti-holomorphic if σ J 1 = J 2 σ on the tangent bundles, where J i are the almost complex structures of the manifolds. In the following, K denotes the canonical bundle of an almost complex manifold (underlying a Kähler manifold). Theorem 1. Let X beakähler surface with K 2 X > 0 and b+ 2 ( X) > 3. Suppose that σ : X X is an anti-holomorphic involution without fixed 1991 Mathematics Subject Classification. Primary 57R55, 57R57, 57N13; Secondary 14P25. Received April 14, Work supported by the Research Board grant of University of Missouri. 305

2 306 SHUGUANG WANG points. Then the quotient manifold X = X/σ has vanishing Seiberg-Witten invariants. In view of Witten s vanishing theorem, it is natural to examine whether the quotient X can be decomposed into a connected sum: Proposition 2. If X is a simply-connected Kähler surface or more generally symplectic 4-manifold with b + 2 ( X) > 1 and σ : X X isafree involution, then the quotient manifold X = X/σ is not diffeomorphic to any connected sum X 1 #X 2 with both b + 2 (X i) > 0. Proof. Suppose on the contrary, that there is such a decomposition X = X 1 #X 2. Since π 1 (X) =Z 2, we can assume π 1 (X 1 )=Z 2 and π 1 (X 2 )=1. A universal (double) cover X 1 of X 1 yields a universal cover X 1 #X 2 #X 2 of X, which should then be diffeomorphic to X. This is however impossible as X, being symplectic, cannot be decomposed into such a connected sum with b + 2 ( X 1 #X 2 ) > 0 and b + 2 (X 2) > 0 [9]. (This kind of covering trick was initially used in [7] in a different context.) Proposition 2 therefore indicates that at least in the case where X is simply-connected, the situation in Theorem 1 is not covered by Witten s vanishing theorem. To the author s knowledge, the quotient manifold X given here, with b 1 (X)+b + 2 (X) =b+ 2 (X) odd, is the first kind of example which is known to satisfy the conclusions in Theorem 1 and Proposition 2. Before proving Theorem 1, we recall briefly the definition of Seiberg- Witten invariants from [8] and [12], for example. Let Y be a smooth oriented Riemannian 4-manifold. In this case, a spin c structure λ on Y consists of a pair of U(2) bundles W ± over Y, whose sum W + W is the usual spinor bundle. Let L denote the determinant (complex line) bundle of W +. The perturbed Seiberg-Witten equations are the following pair of equations for a unitary connection A on L and a section φ of W + : ρ(f + A D A φ =0 ( ) + iδ) =iθ(φ, φ), Here D A :Γ(W + ) Γ(W ) is the Dirac operator, F + A is the self-dual part of the curvature of A, ρ is the Clifford multiplication, and θ is a pairing defined by matrix multiplication. (See [8] for details.) For a generic perturbation δ Ω + (X), the moduli space M λ,δ of irreducible solutions (A, φ) is either empty or a smooth manifold of dimension d λ = 1 4 [c 1(L) 2 (2e Y +3s Y )],

3 A VANISHING THEOREM FOR SEIBERG-WITTEN INVARIANTS 307 where e Y,s Y are respectively the Euler characteristic and signature of Y. (The pair (A, φ) is called irreducible if φ 0.) The moduli space M λ,δ will be compact if the metric on Y is chosen so that ( ) admit no reducible solutions, which can be achieved in a path-connected subset of metrics if b + 2 (Y ) > 1. In this situation and if d λ 0 also, the Seiberg-Witten invariant SW(Y,λ) ofy with respect to λ is then defined to be the integral of the maximal power of the Chern class of the circle bundle Mλ,δ 0 M λ,δ, where Mλ,δ 0 is the framed moduli space. (The integral makes sense only if b 1 (Y )+b + 2 (Y ) is odd; if b 1(Y )+b + 2 (Y ) is even, the Seiberg-Witten invariant is defined to be zero.) The full details of the definition of Seiberg-Witten invariants will not be needed in this paper. In fact, for the purpose of proving Theorem 1, it is enough to note that the perturbation δ will be generic and the Seiberg- Witten invariant will be zero if there are no reducible and irreducible solutions to ( ). The issue of reducible solutions is dealt with in the following simple observation. Lemma 3. Set δ =0in ( ). If the smooth 4-manifold Y satisfies 2e Y + 3s Y > 0, then there is no reducible solution to ( ) for any metric and any spin c structure λ on Y with d λ 0. Proof. Since d λ = 1 4 [c 1(L) 2 (2e Y +3s Y )], its nonnegativeness yields c 1 (L) 2 > 0. If there is a reducible Seiberg-Witten solution for some metric, that is, a connection A on L with i 2π F A being anti-self dual, then c 1 (L) 2 = Y ( i 2π F A) 2 0. This is a contradiction. It is interesting to observe that the situation in Lemma 3 is different from the Donaldson theory. There the inequality c 1 (L) 2 0 (from a reducible anti-self dual connection on an SU(2) bundle E) does not contradict the nonnegativity of the (virtual) dimension d E =8c 2 (E) 3(e Y + s Y )/2 of the moduli space of ASD connections, as c 2 (E) = c 1 (L) 2 0. Thus a condition such as e Y + s Y > 0 is not helpful to rule out the existence of reducible ASD connections. Proof of Theorem 1. Let h be the Kähler metric on X and ω the associated Kähler form. As σ is anti-holomorphic, one sees easily that g = h + σ h is an equivariant Kähler metric on X with Kähler form ω = ω σ ω. Through the double cover p : X X, g pushes down to a metric g on X. Both g and g will be fixed for the rest of the proof. The standard Euler characteristic and signature formulae applied to p also yield b + 2 (X) = 1 2 [b+ 2 ( X) 1] > 1. Consider an arbitrary spin c structure λ on X. It pulls back to a spin c structure λ on X through the double covering p. Without much difficulty one verifies that the associated bundles W ± and L of λ pull back to the

4 308 SHUGUANG WANG corresponding associated bundles W ± and L of λ. Assume the dimension d λ 0 from now on. Suppose there is a solution pair (A, φ) to the Seiberg-Witten equations: D A φ =0 ρ(f + ( ) A )=iθ(φ, φ), where A and φ are respectively a connection on L and a section on W +. Then the pull-back (Ã, φ) is a solution pair to the Seiberg-Witten equations on X with spin c structure λ and Kähler metric g. Since 2e X +3s X =(2e X + 3s X )/2 /2 > 0, there are no reducible Seiberg-Witten solutions for λ =K2 X from Lemma 3; thus (A, φ) and hence (Ã, φ) are both irreducible. It follows easily from the irreducibility of (Ã, φ) that ω L 0 (see [12] for example). This is however not possible; σ L = L and σ ω = ω have already forced ω L = 0. (Note that σ preserves the orientation of X.) Thus the argument above shows that there is no reducible or irreducible solution to ( ). As noted before Lemma 3, this means that in the perturbed Seiberg-Witten equations of ( ), the perturbation δ = 0 is generic and the Seiberg-Witten invariant of X with respect to λ is zero. Since λ is an arbitrary spin c structure, the Seiberg-Witten invariants of X all vanish. Note that all minimal complex surfaces of general type satisfy the condition K > 0 in Theorem 1 [1; page 208]. (Moreover a necessary condition 2 X for K 2 X > 0 is that X be projective by using Grauert s ampleness criterion [1; Page 127].) Therefore the conditions in both Theorem 1 and Proposition 2 are satisfied by all simply-connected minimal complex surfaces of general type with b + 2 > 3, and these include lots of examples. For one set of examples, one can take X to be algebraic surfaces in CP n defined by real polynomials and σ to be the complex conjugation. These include hypersurfaces 4 j=1 x2n j =0 in CP 3, where n>2. For another set of examples, consider a Kähler surface Y with anti-holomorphic involution τ. Suppose that there exits a complex curve C Y such that 2 [C] H 2 (Y,Z), and that C is invariant under τ, disjoint from Fix τ. Then τ lifts to two anti-holomorphic involutions on the double cover X of Y branched over C, and one of the lifting involutions has no fixed point. As a special case, one can take Y =CP 2, C = { 3 j=1 x2n j =0} (n >3) and τ to be the complex conjugation on CP 2.

5 A VANISHING THEOREM FOR SEIBERG-WITTEN INVARIANTS 309 Remark 4. (i) Since the Seiberg-Witten invariants vanish, it follows from [9] that the quotient X in Theorem 1 does not admit any symplectic structure. Moreover it is possible to give examples of X (e.g., degree 4m+2 hypersurfaces in CP 3 ), from which b + 2 (X),b 2 (X) are both even. Thus X does not even have an almost complex structure with either of its orientations in this case. (ii) Fix a complex surface X but vary free anti-holomorphic involutions σ. It is an open problem whether the quotients X are diffeomorphic to each other. (It is not difficult to see that the quotients are homeomorphic to each other for most cases using [6].) Perhaps the d-complex structures introduced in [11] are a useful tool. (iii) It is also interesting to investigate the quotient X when σ is not free. By using the generalized adjunction inequality [4], one can show easily that the Seiberg-Witten invariants of X vanish again for many cases. (Furthermore, combining with the rational blowdown formula [5] as well as a surgery formula [10], one can recover the vanishing result in Theorem 1 for hypersurfaces in CP 3.) It remains unknown [3] whether there exits such an X which cannot be decomposed into a sum X 1 #X 2 with both b + 2 (X i) > 0 (other than a couple of trivial cases with b + 2 (X) = 0). Acknowledgements The author wishes to thank Ron Fintushel for pointing out the incompleteness of a version of the paper, and Ron Stern for valuable encouragement. In addition, he thanks Jan Segert for sharing the enthusiasm about the new theory. References 1. W. Barth, C. Peters and A. Van de Ven, Compact complex surfaces, Springer-Verlag, Berlin-Heidelberg, S. Donaldson, Polynomial invariants for smooth 4-manifolds, Topology 29 (1990), , Yang-Mills invariants of 4-manifolds, in Geometry of low-dimensional manifolds (S. K. Donaldson and C. B. Thomas Eds, eds.), vol. 1, Cambridge University Press, 1990, pp R. Fintushel, P. Kronheimer, T. Mrowka, R. Stern and C. Taubes, in preparation. 5. R. Fintushel and R. Stern, in preparation. 6. I. Hambleton and M. Kreck, Smooth structures on algebraic surfaces with cyclic fundamental group, Invt. Math. 91 (1988),

6 310 SHUGUANG WANG 7. D. Kotschick, On connected sum decompositions of algebraic surfaces and their fundamental groups, Intern. Math. Research Notices (1993), P. Kronheimer and T. Mrowka, The genus of embedded surfaces in the projective plane, Math. Research Letters 1 (1994), C. Taubes, The Seiberg-Witten invariants and symplectic forms, Math. Research Letters 1 (1994), S. Wang, On quotients of real algebraic surfaces in CP 3, Topology and Its Applications (to appear). 11., A Narasimhan-Seshadri-Donaldson correspondence on non-orientable surfaces, Forum Mathematicae (to appear). 12. E. Witten, Monopoles and 4-manifolds, Math. Research Letters 1 (1994), M athematics Department, University of M issouri, Columbia, M O address:

### Scalar curvature and the Thurston norm

Scalar curvature and the Thurston norm P. B. Kronheimer 1 andt.s.mrowka 2 Harvard University, CAMBRIDGE MA 02138 Massachusetts Institute of Technology, CAMBRIDGE MA 02139 1. Introduction Let Y be a closed,

### MONOPOLE CLASSES AND PERELMAN S INVARIANT OF FOUR-MANIFOLDS

MONOPOLE CLASSES AND PERELMAN S INVARIANT OF FOUR-MANIFOLDS D. KOTSCHICK ABSTRACT. We calculate Perelman s invariant for compact complex surfaces and a few other smooth four-manifolds. We also prove some

### Pin (2)-monopole theory I

Intersection forms with local coefficients Osaka Medical College Dec 12, 2016 Pin (2) = U(1) j U(1) Sp(1) H Pin (2)-monopole equations are a twisted version of the Seiberg-Witten (U(1)-monopole) equations.

### DISTINGUISHING EMBEDDED CURVES IN RATIONAL COMPLEX SURFACES

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 16, Number 1, January 1998, Pages 305 310 S 000-9939(98)04001-5 DISTINGUISHING EMBEDDED CURVES IN RATIONAL COMPLEX SURFACES TERRY FULLER (Communicated

### Higgs Bundles and Character Varieties

Higgs Bundles and Character Varieties David Baraglia The University of Adelaide Adelaide, Australia 29 May 2014 GEAR Junior Retreat, University of Michigan David Baraglia (ADL) Higgs Bundles and Character

### The topology of symplectic four-manifolds

The topology of symplectic four-manifolds Michael Usher January 12, 2007 Definition A symplectic manifold is a pair (M, ω) where 1 M is a smooth manifold of some even dimension 2n. 2 ω Ω 2 (M) is a two-form

### Donaldson and Seiberg-Witten theory and their relation to N = 2 SYM

Donaldson and Seiberg-Witten theory and their relation to N = SYM Brian Williams April 3, 013 We ve began to see what it means to twist a supersymmetric field theory. I will review Donaldson theory and

### HYPERKÄHLER MANIFOLDS

HYPERKÄHLER MANIFOLDS PAVEL SAFRONOV, TALK AT 2011 TALBOT WORKSHOP 1.1. Basic definitions. 1. Hyperkähler manifolds Definition. A hyperkähler manifold is a C Riemannian manifold together with three covariantly

### arxiv: v2 [math.gt] 26 Sep 2013

HOMOLOGY CLASSES OF NEGATIVE SQUARE AND EMBEDDED SURFACES IN 4-MANIFOLDS arxiv:1301.3733v2 [math.gt] 26 Sep 2013 M. J. D. HAMILTON ABSTRACT. Let X be a simply-connected closed oriented 4-manifold and A

### An exotic smooth structure on CP 2 #6CP 2

ISSN 1364-0380 (on line) 1465-3060 (printed) 813 Geometry & Topology Volume 9 (2005) 813 832 Published: 18 May 2005 G G G G T T T G T T T G T G T GG TT G G G G GG T T T TT An exotic smooth structure on

### Donaldson Invariants and Moduli of Yang-Mills Instantons

Donaldson Invariants and Moduli of Yang-Mills Instantons Lincoln College Oxford University (slides posted at users.ox.ac.uk/ linc4221) The ASD Equation in Low Dimensions, 17 November 2017 Moduli and Invariants

### LAGRANGIAN HOMOLOGY CLASSES WITHOUT REGULAR MINIMIZERS

LAGRANGIAN HOMOLOGY CLASSES WITHOUT REGULAR MINIMIZERS JON WOLFSON Abstract. We show that there is an integral homology class in a Kähler-Einstein surface that can be represented by a lagrangian twosphere

### LECTURE 6: J-HOLOMORPHIC CURVES AND APPLICATIONS

LECTURE 6: J-HOLOMORPHIC CURVES AND APPLICATIONS WEIMIN CHEN, UMASS, SPRING 07 1. Basic elements of J-holomorphic curve theory Let (M, ω) be a symplectic manifold of dimension 2n, and let J J (M, ω) be

### THE GENUS OF EMBEDDED SURFACES IN THE PROJECTIVE PLANE. P. B. Kronheimer and T. S. Mrowka

THE GENUS OF EMBEDDED SURFACES IN THE PROJECTIVE PLANE P. B. Kronheimer and T. S. Mrowka Abstract. We show how the new invariants of 4-manifolds resulting from the Seiberg-Witten monopole equation lead

### COUNT OF GENUS ZERO J-HOLOMORPHIC CURVES IN DIMENSIONS FOUR AND SIX arxiv: v5 [math.sg] 17 May 2013

COUNT OF GENUS ZERO J-HOLOMORPHIC CURVES IN DIMENSIONS FOUR AND SIX arxiv:0906.5472v5 [math.sg] 17 May 2013 AHMET BEYAZ Abstract. In this note, genus zero Gromov-Witten invariants are reviewed and then

### Classifying complex surfaces and symplectic 4-manifolds

Classifying complex surfaces and symplectic 4-manifolds UT Austin, September 18, 2012 First Cut Seminar Basics Symplectic 4-manifolds Definition A symplectic 4-manifold (X, ω) is an oriented, smooth, 4-dimensional

### Holomorphic line bundles

Chapter 2 Holomorphic line bundles In the absence of non-constant holomorphic functions X! C on a compact complex manifold, we turn to the next best thing, holomorphic sections of line bundles (i.e., rank

### Lefschetz pencils and the symplectic topology of complex surfaces

Lefschetz pencils and the symplectic topology of complex surfaces Denis AUROUX Massachusetts Institute of Technology Symplectic 4-manifolds A (compact) symplectic 4-manifold (M 4, ω) is a smooth 4-manifold

### INSTANTON MODULI AND COMPACTIFICATION MATTHEW MAHOWALD

INSTANTON MODULI AND COMPACTIFICATION MATTHEW MAHOWALD () Instanton (definition) (2) ADHM construction (3) Compactification. Instantons.. Notation. Throughout this talk, we will use the following notation:

### Instantons and Donaldson invariants

Instantons and Donaldson invariants George Korpas Trinity College Dublin IFT, November 20, 2015 A problem in mathematics A problem in mathematics Important probem: classify d-manifolds up to diffeomorphisms.

### Broken pencils and four-manifold invariants. Tim Perutz (Cambridge)

Broken pencils and four-manifold invariants Tim Perutz (Cambridge) Aim This talk is about a project to construct and study a symplectic substitute for gauge theory in 2, 3 and 4 dimensions. The 3- and

### Symplectic 4-manifolds, singular plane curves, and isotopy problems

Symplectic 4-manifolds, singular plane curves, and isotopy problems Denis AUROUX Massachusetts Inst. of Technology and Ecole Polytechnique Symplectic manifolds A symplectic structure on a smooth manifold

### MAPPING CLASS ACTIONS ON MODULI SPACES. Int. J. Pure Appl. Math 9 (2003),

MAPPING CLASS ACTIONS ON MODULI SPACES RICHARD J. BROWN Abstract. It is known that the mapping class group of a compact surface S, MCG(S), acts ergodically with respect to symplectic measure on each symplectic

### THE FUNDAMENTAL GROUP OF MANIFOLDS OF POSITIVE ISOTROPIC CURVATURE AND SURFACE GROUPS

THE FUNDAMENTAL GROUP OF MANIFOLDS OF POSITIVE ISOTROPIC CURVATURE AND SURFACE GROUPS AILANA FRASER AND JON WOLFSON Abstract. In this paper we study the topology of compact manifolds of positive isotropic

### k=0 /D : S + S /D = K 1 2 (3.5) consistently with the relation (1.75) and the Riemann-Roch-Hirzebruch-Atiyah-Singer index formula

20 VASILY PESTUN 3. Lecture: Grothendieck-Riemann-Roch-Hirzebruch-Atiyah-Singer Index theorems 3.. Index for a holomorphic vector bundle. For a holomorphic vector bundle E over a complex variety of dim

### Four-Manifold Geography and Superconformal Symmetry

YCTP-P30-98 math.dg/9812042 ariv:math/9812042v2 [math.dg] 9 Dec 1998 Four-Manifold Geography and Superconformal Symmetry Marcos Mariño, Gregory Moore, and Grigor Peradze Department of Physics, Yale University

### Periodic-end Dirac Operators and Positive Scalar Curvature. Daniel Ruberman Nikolai Saveliev

Periodic-end Dirac Operators and Positive Scalar Curvature Daniel Ruberman Nikolai Saveliev 1 Recall from basic differential geometry: (X, g) Riemannian manifold Riemannian curvature tensor tr scalar curvature

### DISSOLVING FOUR-MANIFOLDS AND POSITIVE SCALAR CURVATURE

DISSOLVING FOUR-MANIFOLDS AND POSITIVE SCALAR CURVATURE B. HANKE, D. KOTSCHICK, AND J. WEHRHEIM ABSTRACT. We prove that many simply connected symplectic fourmanifolds dissolve after connected sum with

### The Yang-Mills equations over Klein surfaces

The Yang-Mills equations over Klein surfaces Chiu-Chu Melissa Liu & Florent Schaffhauser Columbia University (New York City) & Universidad de Los Andes (Bogotá) Seoul ICM 2014 Outline 1 Moduli of real

### ORIENTABILITY AND REAL SEIBERG-WITTEN INVARIANTS

(To appear in Intern. J. Math.) ORIENTABILITY AND REAL SEIBERG-WITTEN INVARIANTS GANG TIAN AND SHUGUANG WANG Abstract. We investigate Seiberg-Witten theory in the presence of real structures. Certain conditions

### SMALL EXOTIC 4-MANIFOLDS. 0. Introduction

SMALL EXOTIC 4-MANIFOLDS ANAR AKHMEDOV Dedicated to Professor Ronald J. Stern on the occasion of his sixtieth birthday Abstract. In this article, we construct the first example of a simply-connected minimal

### A NOTE ON STEIN FILLINGS OF CONTACT MANIFOLDS

A NOTE ON STEIN FILLINGS OF CONTACT MANIFOLDS ANAR AKHMEDOV, JOHN B. ETNYRE, THOMAS E. MARK, AND IVAN SMITH Abstract. In this note we construct infinitely many distinct simply connected Stein fillings

### Linear connections on Lie groups

Linear connections on Lie groups The affine space of linear connections on a compact Lie group G contains a distinguished line segment with endpoints the connections L and R which make left (resp. right)

### KODAIRA DIMENSION OF LEFSCHETZ FIBRATIONS OVER TORI

KODAIRA DIMENSION OF LEFSCHETZ FIBRATIONS OVER TORI JOSEF G. DORFMEISTER Abstract. The Kodaira dimension for Lefschetz fibrations was defined in [1]. In this note we show that there exists no Lefschetz

### arxiv: v1 [math.ag] 13 Mar 2019

THE CONSTRUCTION PROBLEM FOR HODGE NUMBERS MODULO AN INTEGER MATTHIAS PAULSEN AND STEFAN SCHREIEDER arxiv:1903.05430v1 [math.ag] 13 Mar 2019 Abstract. For any integer m 2 and any dimension n 1, we show

### ALF spaces and collapsing Ricci-flat metrics on the K3 surface

ALF spaces and collapsing Ricci-flat metrics on the K3 surface Lorenzo Foscolo Stony Brook University Recent Advances in Complex Differential Geometry, Toulouse, June 2016 The Kummer construction Gibbons

### DIMENSION 4: GETTING SOMETHING FROM NOTHING

DIMENSION 4: GETTING SOMETHING FROM NOTHING RON STERN UNIVERSITY OF CALIFORNIA, IRVINE MAY 6, 21 JOINT WORK WITH RON FINTUSHEL Topological n-manifold: locally homeomorphic to R n TOPOLOGICAL VS. SMOOTH

### ALF spaces and collapsing Ricci-flat metrics on the K3 surface

ALF spaces and collapsing Ricci-flat metrics on the K3 surface Lorenzo Foscolo Stony Brook University Simons Collaboration on Special Holonomy Workshop, SCGP, Stony Brook, September 8 2016 The Kummer construction

### L6: Almost complex structures

L6: Almost complex structures To study general symplectic manifolds, rather than Kähler manifolds, it is helpful to extract the homotopy-theoretic essence of having a complex structure. An almost complex

### Constructing compact 8-manifolds with holonomy Spin(7)

Constructing compact 8-manifolds with holonomy Spin(7) Dominic Joyce, Oxford University Simons Collaboration meeting, Imperial College, June 2017. Based on Invent. math. 123 (1996), 507 552; J. Diff. Geom.

### CONSIDERATION OF COMPACT MINIMAL SURFACES IN 4-DIMENSIONAL FLAT TORI IN TERMS OF DEGENERATE GAUSS MAP

CONSIDERATION OF COMPACT MINIMAL SURFACES IN 4-DIMENSIONAL FLAT TORI IN TERMS OF DEGENERATE GAUSS MAP TOSHIHIRO SHODA Abstract. In this paper, we study a compact minimal surface in a 4-dimensional flat

### Rohlin s Invariant and 4-dimensional Gauge Theory. Daniel Ruberman Nikolai Saveliev

Rohlin s Invariant and 4-dimensional Gauge Theory Daniel Ruberman Nikolai Saveliev 1 Overall theme: relation between Rohlin-type invariants and gauge theory, especially in dimension 4. Background: Recall

### Morse-Bott Framework for the 4-dim SW-Equations

Morse-Bott Framework for the 4-dim SW-Equations Celso M. Doria UFSC - Depto. de Matemática august/2010 Figure 1: Ilha de Santa Catarina Figure 2: Floripa research partially supported by FAPESC 2568/2010-2

### REAL INSTANTONS, DIRAC OPERATORS AND QUATERNIONIC CLASSIFYING SPACES PAUL NORBURY AND MARC SANDERS Abstract. Let M(k; SO(n)) be the moduli space of ba

REAL INSTANTONS, DIRAC OPERATORS AND QUATERNIONIC CLASSIFYING SPACES PAUL NORBURY AND MARC SANDERS Abstract. Let M(k; SO(n)) be the moduli space of based gauge equivalence classes of SO(n) instantons on

### The Spinor Representation

The Spinor Representation Math G4344, Spring 2012 As we have seen, the groups Spin(n) have a representation on R n given by identifying v R n as an element of the Clifford algebra C(n) and having g Spin(n)

### Citation Osaka Journal of Mathematics. 43(1)

TitleA note on compact solvmanifolds wit Author(s) Hasegawa, Keizo Citation Osaka Journal of Mathematics. 43(1) Issue 2006-03 Date Text Version publisher URL http://hdl.handle.net/11094/11990 DOI Rights

### Singular fibers in Lefschetz fibrations on manifolds with b + 2 = 1

Topology and its Applications 117 (2002) 9 21 Singular fibers in Lefschetz fibrations on manifolds with b + 2 = 1 András I. Stipsicz a,b a Department of Analysis, ELTE TTK, 1088 Múzeum krt. 6-8, Budapest,

### Lecture 8: More characteristic classes and the Thom isomorphism

Lecture 8: More characteristic classes and the Thom isomorphism We begin this lecture by carrying out a few of the exercises in Lecture 1. We take advantage of the fact that the Chern classes are stable

### Flat connections on 2-manifolds Introduction Outline:

Flat connections on 2-manifolds Introduction Outline: 1. (a) The Jacobian (the simplest prototype for the class of objects treated throughout the paper) corresponding to the group U(1)). (b) SU(n) character

### MONOPOLES AND LENS SPACE SURGERIES

MONOPOLES AND LENS SPACE SURGERIES PETER KRONHEIMER, TOMASZ MROWKA, PETER OZSVÁTH, AND ZOLTÁN SZABÓ Abstract. Monopole Floer homology is used to prove that real projective three-space cannot be obtained

### arxiv:dg-ga/ v2 7 Feb 1997

A FAKE SMOOTH CP 2 #RP 4 DANIEL RUBERMAN AND RONALD J. STERN arxiv:dg-ga/9702003v2 7 Feb 1997 Abstract. We show that the manifold RP 4 # CP 2, which is homotopy equivalent but not homeomorphic to RP 4

### ON SEIBERG WITTEN EQUATIONS ON SYMPLECTIC 4 MANIFOLDS

SYPLECTIC SINGULARITIES AND GEOETRY OF GAUGE FIELDS BANACH CENTER PUBLICATIONS, VOLUE 39 INSTITUTE OF ATHEATICS POLISH ACADEY OF SCIENCES WARSZAWA 1997 ON SEIBERG WITTEN EQUATIONS ON SYPLECTIC 4 ANIFOLDS

### Two simple ideas from calculus applied to Riemannian geometry

Calibrated Geometries and Special Holonomy p. 1/29 Two simple ideas from calculus applied to Riemannian geometry Spiro Karigiannis karigiannis@math.uwaterloo.ca Department of Pure Mathematics, University

### The kernel of the Dirac operator

The kernel of the Dirac operator B. Ammann 1 M. Dahl 2 E. Humbert 3 1 Universität Regensburg Germany 2 Institutionen för Matematik Kungliga Tekniska Högskolan, Stockholm Sweden 3 Laboratoire de Mathématiques

### Morse theory and stable pairs

Richard A. SCGAS 2010 Joint with Introduction Georgios Daskalopoulos (Brown University) Jonathan Weitsman (Northeastern University) Graeme Wilkin (University of Colorado) Outline Introduction 1 Introduction

### Lecture I: Overview and motivation

Lecture I: Overview and motivation Jonathan Evans 23rd September 2010 Jonathan Evans () Lecture I: Overview and motivation 23rd September 2010 1 / 31 Difficulty of exercises is denoted by card suits in

### Einstein Metrics, Minimizing Sequences, and the. Differential Topology of Four-Manifolds. Claude LeBrun SUNY Stony Brook

Einstein Metrics, Minimizing Sequences, and the Differential Topology of Four-Manifolds Claude LeBrun SUNY Stony Brook Definition A Riemannian metric is said to be Einstein if it has constant Ricci curvature

### Periodic monopoles and difference modules

Periodic monopoles and difference modules Takuro Mochizuki RIMS, Kyoto University 2018 February Introduction In complex geometry it is interesting to obtain a correspondence between objects in differential

### Some new torsional local models for heterotic strings

Some new torsional local models for heterotic strings Teng Fei Columbia University VT Workshop October 8, 2016 Teng Fei (Columbia University) Strominger system 10/08/2016 1 / 30 Overview 1 Background and

### Exotic Lefschetz Fibrations and Stein Fillings with Arbitrary Fundamental Group

Exotic Lefschetz Fibrations and Stein Fillings with Arbitrary Fundamental Group Anar Akhmedov University of Minnesota, Twin Cities February 19, 2015 Anar Akhmedov (University of Minnesota, Minneapolis)Exotic

### 0.1 Complex Analogues 1

0.1 Complex Analogues 1 Abstract In complex geometry Kodaira s theorem tells us that on a Kähler manifold sufficiently high powers of positive line bundles admit global holomorphic sections. Donaldson

### September 27, :51 WSPC/INSTRUCTION FILE biswas-loftin. Hermitian Einstein connections on principal bundles over flat affine manifolds

International Journal of Mathematics c World Scientific Publishing Company Hermitian Einstein connections on principal bundles over flat affine manifolds Indranil Biswas School of Mathematics Tata Institute

### Math. Res. Lett. 13 (2006), no. 1, c International Press 2006 ENERGY IDENTITY FOR ANTI-SELF-DUAL INSTANTONS ON C Σ.

Math. Res. Lett. 3 (2006), no., 6 66 c International Press 2006 ENERY IDENTITY FOR ANTI-SELF-DUAL INSTANTONS ON C Σ Katrin Wehrheim Abstract. We establish an energy identity for anti-self-dual connections

### 4-MANIFOLDS AS FIBER BUNDLES

4-MANIFOLDS AS FIBER BUNDLES MORGAN WEILER Structures on 4-Manifolds I care about geometric structures on 4-manifolds because they allow me to obtain results about their smooth topology. Think: de Rham

### arxiv:math/ v1 [math.dg] 4 Feb 2003 Nikolay Tyurin

1 TWO LECTURES ON LOCAL RIEMANNIAN GEOMETRY, Spin C - STRUCTURES AND SEIBERG - WITTEN EQUATION arxiv:math/0302034v1 [math.dg] 4 Feb 2003 Nikolay Tyurin Introduction The mathematical part of Seiberg - Witten

### TITLE AND ABSTRACT OF HALF-HOUR TALKS

TITLE AND ABSTRACT OF HALF-HOUR TALKS Speaker: Jim Davis Title: Mapping tori of self-homotopy equivalences of lens spaces Abstract: Jonathan Hillman, in his study of four-dimensional geometries, asked

### B 1 = {B(x, r) x = (x 1, x 2 ) H, 0 < r < x 2 }. (a) Show that B = B 1 B 2 is a basis for a topology on X.

Math 6342/7350: Topology and Geometry Sample Preliminary Exam Questions 1. For each of the following topological spaces X i, determine whether X i and X i X i are homeomorphic. (a) X 1 = [0, 1] (b) X 2

### Embedded contact homology and its applications

Embedded contact homology and its applications Michael Hutchings Department of Mathematics University of California, Berkeley International Congress of Mathematicians, Hyderabad August 26, 2010 arxiv:1003.3209

### Chern numbers and Hilbert Modular Varieties

Chern numbers and Hilbert Modular Varieties Dylan Attwell-Duval Department of Mathematics and Statistics McGill University Montreal, Quebec attwellduval@math.mcgill.ca April 9, 2011 A Topological Point

### arxiv:math/ v2 [math.gt] 18 Apr 2007

arxiv:math/0703480v2 [math.gt] 18 Apr 2007 CONSTRUCTING INFINITELY MANY SMOOTH STRUCTURES ON SMALL 4-MANIFOLDS ANAR AKHMEDOV, R. İNANÇ BAYKUR, AND B. DOUG PARK Abstract. The purpose of this article is

### LANTERN SUBSTITUTION AND NEW SYMPLECTIC 4-MANIFOLDS WITH b 2 + = 3 arxiv: v2 [math.gt] 24 May 2014

LANTERN SUBSTITUTION AND NEW SYMPLECTIC -MANIFOLDS WITH b 2 + = arxiv:207.068v2 [math.gt] 2 May 20 ANAR AKHMEDOV AND JUN-YONG PARK Abstract. Motivated by the construction of H. Endo and Y. Gurtas, changing

### K-stability and Kähler metrics, I

K-stability and Kähler metrics, I Gang Tian Beijing University and Princeton University Let M be a Kähler manifold. This means that M be a complex manifold together with a Kähler metric ω. In local coordinates

### Nodal symplectic spheres in CP 2 with positive self intersection

Nodal symplectic spheres in CP 2 with positive self intersection Jean-François BARRAUD barraud@picard.ups-tlse.fr Abstract 1 : Let ω be the canonical Kähler structure on CP 2 We prove that any ω-symplectic

### ON STABILITY OF NON-DOMINATION UNDER TAKING PRODUCTS

ON STABILITY OF NON-DOMINATION UNDER TAKING PRODUCTS D. KOTSCHICK, C. LÖH, AND C. NEOFYTIDIS ABSTRACT. We show that non-domination results for targets that are not dominated by products are stable under

### DEFINITE MANIFOLDS BOUNDED BY RATIONAL HOMOLOGY THREE SPHERES

DEFINITE MANIFOLDS BOUNDED BY RATIONAL HOMOLOGY THREE SPHERES BRENDAN OWENS AND SAŠO STRLE Abstract. This paper is based on a talk given at the McMaster University Geometry and Topology conference, May

### 4-MANIFOLDS: CLASSIFICATION AND EXAMPLES. 1. Outline

4-MANIFOLDS: CLASSIFICATION AND EXAMPLES 1. Outline Throughout, 4-manifold will be used to mean closed, oriented, simply-connected 4-manifold. Hopefully I will remember to append smooth wherever necessary.

### APPROXIMATE YANG MILLS HIGGS METRICS ON FLAT HIGGS BUNDLES OVER AN AFFINE MANIFOLD. 1. Introduction

APPROXIMATE YANG MILLS HIGGS METRICS ON FLAT HIGGS BUNDLES OVER AN AFFINE MANIFOLD INDRANIL BISWAS, JOHN LOFTIN, AND MATTHIAS STEMMLER Abstract. Given a flat Higgs vector bundle (E,, ϕ) over a compact

### Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem

PETER B. GILKEY Department of Mathematics, University of Oregon Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem Second Edition CRC PRESS Boca Raton Ann Arbor London Tokyo Contents

### N = 2 supersymmetric gauge theory and Mock theta functions

N = 2 supersymmetric gauge theory and Mock theta functions Andreas Malmendier GTP Seminar (joint work with Ken Ono) November 7, 2008 q-series in differential topology Theorem (M-Ono) The following q-series

### ON NEARLY SEMIFREE CIRCLE ACTIONS

ON NEARLY SEMIFREE CIRCLE ACTIONS DUSA MCDUFF AND SUSAN TOLMAN Abstract. Recall that an effective circle action is semifree if the stabilizer subgroup of each point is connected. We show that if (M, ω)

### η = (e 1 (e 2 φ)) # = e 3

Research Statement My research interests lie in differential geometry and geometric analysis. My work has concentrated according to two themes. The first is the study of submanifolds of spaces with riemannian

### J-curves and the classification of rational and ruled symplectic 4-manifolds

J-curves and the classification of rational and ruled symplectic 4-manifolds François Lalonde Université du Québec à Montréal (flalonde@math.uqam.ca) Dusa McDuff State University of New York at Stony Brook

### arxiv:math/ v1 [math.dg] 6 Feb 1998

Cartan Spinor Bundles on Manifolds. Thomas Friedrich, Berlin November 5, 013 arxiv:math/980033v1 [math.dg] 6 Feb 1998 1 Introduction. Spinor fields and Dirac operators on Riemannian manifolds M n can be

### Rational Curves On K3 Surfaces

Rational Curves On K3 Surfaces Jun Li Department of Mathematics Stanford University Conference in honor of Peter Li Overview of the talk The problem: existence of rational curves on a K3 surface The conjecture:

### Porteous s Formula for Maps between Coherent Sheaves

Michigan Math. J. 52 (2004) Porteous s Formula for Maps between Coherent Sheaves Steven P. Diaz 1. Introduction Recall what the Thom Porteous formula for vector bundles tells us (see [2, Sec. 14.4] for

### Riemann surfaces. Paul Hacking and Giancarlo Urzua 1/28/10

Riemann surfaces Paul Hacking and Giancarlo Urzua 1/28/10 A Riemann surface (or smooth complex curve) is a complex manifold of dimension one. We will restrict to compact Riemann surfaces. It is a theorem

### Cohomology of the Mumford Quotient

Cohomology of the Mumford Quotient Maxim Braverman Abstract. Let X be a smooth projective variety acted on by a reductive group G. Let L be a positive G-equivariant line bundle over X. We use a Witten

### arxiv:math/ v1 [math.ag] 18 Oct 2003

Proc. Indian Acad. Sci. (Math. Sci.) Vol. 113, No. 2, May 2003, pp. 139 152. Printed in India The Jacobian of a nonorientable Klein surface arxiv:math/0310288v1 [math.ag] 18 Oct 2003 PABLO ARÉS-GASTESI

### Introduction To K3 Surfaces (Part 2)

Introduction To K3 Surfaces (Part 2) James Smith Calf 26th May 2005 Abstract In this second introductory talk, we shall take a look at moduli spaces for certain families of K3 surfaces. We introduce the

### Tesi di Laurea Magistrale in Matematica presentata da. Claudia Dennetta. Symplectic Geometry. Il Relatore. Prof. Massimiliano Pontecorvo

UNIVERSITÀ DEGLI STUDI ROMA TRE FACOLTÀ DI SCIENZE M.F.N. Tesi di Laurea Magistrale in Matematica presentata da Claudia Dennetta Symplectic Geometry Relatore Prof. Massimiliano Pontecorvo Il Candidato

### Combinatorial Heegaard Floer Theory

Combinatorial Heegaard Floer Theory Ciprian Manolescu UCLA February 29, 2012 Ciprian Manolescu (UCLA) Combinatorial HF Theory February 29, 2012 1 / 31 Motivation (4D) Much insight into smooth 4-manifolds

### Vortex Equations on Riemannian Surfaces

Vortex Equations on Riemannian Surfaces Amanda Hood, Khoa Nguyen, Joseph Shao Advisor: Chris Woodward Vortex Equations on Riemannian Surfaces p.1/36 Introduction Yang Mills equations: originated from electromagnetism

### Conformal field theory in the sense of Segal, modified for a supersymmetric context

Conformal field theory in the sense of Segal, modified for a supersymmetric context Paul S Green January 27, 2014 1 Introduction In these notes, we will review and propose some revisions to the definition

### CHARACTERISTIC CLASSES

1 CHARACTERISTIC CLASSES Andrew Ranicki Index theory seminar 14th February, 2011 2 The Index Theorem identifies Introduction analytic index = topological index for a differential operator on a compact

### SYMPLECTIC GEOMETRY: LECTURE 5

SYMPLECTIC GEOMETRY: LECTURE 5 LIAT KESSLER Let (M, ω) be a connected compact symplectic manifold, T a torus, T M M a Hamiltonian action of T on M, and Φ: M t the assoaciated moment map. Theorem 0.1 (The

### An observation concerning uniquely ergodic vector fields on 3-manifolds. Clifford Henry Taubes

/24/08 An observation concerning uniquely ergodic vector fields on 3-manifolds Clifford Henry Taubes Department of athematics Harvard University Cambridge, A 0238 Supported in part by the National Science

### Projective Images of Kummer Surfaces

Appeared in: Math. Ann. 299, 155-170 (1994) Projective Images of Kummer Surfaces Th. Bauer April 29, 1993 0. Introduction The aim of this note is to study the linear systems defined by the even resp. odd

### Mathematische Annalen

Math. Ann. 319, 707 714 (2001) Digital Object Identifier (DOI) 10.1007/s002080100175 Mathematische Annalen A Moebius characterization of Veronese surfaces in S n Haizhong Li Changping Wang Faen Wu Received